# Morse Index Bound for Minimal Torus

Yuchin Sun

###### Abstract.

The min-max construction of minimal spheres using harmonic replacement is
introduced by Colding and Minicozzi [CM08] and generalized by Zhou [Zho10] to
conformal harmonic torus. We prove that the Morse index of the min-max
conformal harmonic torus is bounded by one.

##  1\. Introduction

Morse theory studies the topology of a manifold by analysing proper functions
defined on it. Finite-dimensional Morse theory was developed by Morse [Mil63]
to study geodesics. A natural extension of it would be a Morse theory for
harmonic surfaces. Colding and Minicozzi proved the existence for min-max
harmonic spheres whose sum of areas realizes the width [CM08], the Morse index
of such min-max harmonic sphere is bounded by one [Sun20]. The approach by
Colding and Minicozzi [CM08] was extended by Zhou to min-max minimal torus
[Zho10] and min-max minimal surface of genus $g\geq 2$ [Zho17]. We prove that
the Morse index of such min-max tori is bounded by one.

######  Theorem 1.1.

Let $M$ be a closed Riemannian manifold of dimension at least three. For a
generic choice of metric on $M$ and any homotopically nontrivial path
$\beta\in\Lambda$, if the corresponding width $W$ is positive. Then there are
possibly a conformal torus $v_{0}:\mathcal{T}_{\tau_{0}}\to M$ and finitely
many harmonic spheres $v_{j}:S^{2}\to M$ such that

  1. (1)

$\text{Area}(v_{0})+\sum_{j}\text{Area}(v_{j})=W,$

  2. (2)

$\text{Index}(v_{0})+\sum_{j}Index(v_{j})\leq 1.$

Morse index of min-max minimal surface is studied under many different
settings. We list a few in the following and compare those result with Theorem
1.1. Sacks and Uhlenbeck introduced $\alpha$-energy [SU81], which can be
perturbed to be a Morse function. The corresponding min-max critical points
$f_{\alpha}$ of $\alpha$-energy converge to a harmonic sphere $f_{\infty}$ as
$\alpha\to 1$. The Morse index of the harmonic sphere $f_{\infty}$ is bounded
by the limit inferior of Morse index of $f_{\alpha}$ as $\alpha\to 1$, this
result serves as a key step in the proof of the sphere theorem [MM88].
However, the harmonic sphere $f_{\infty}$ fails to realize the energy identity
in general. An alternative to $\alpha$-energy of Sacks and Uhlenbeck is done
by Rivière using viscosity method [Riv17]. The parametric approach produces
immersed (rather than embedded) minimal surfaces with possible branch points
for arbitrary codimension. The Morse index bound of such immersed minimal
surfaces is shown in [PR20]. We like to remark here that the Morse index in
[PR20] is for the area functional, while the Morse index in Theorem 1.1 is for
the energy functional. For a conformal harmonic map, its Morse index for area
is bounded by its Morse index for energy plus a constant depending on the
genus and branch points [EM08]. Zhou and Cheng [CZ21] also use the idea of
perturbed energy and generalize the existence theory of branched immersed
minimal 2-spheres by Sacks-Uhlenbeck [SU81] to the CMC setting in Riemannian
3-spheres. They obtain the Morse index bound by showing the Morse index bound
passes through the limit as the perturbed Dirichlet energy approaches the
normal energy. On the other hand, Almgren-Pitts min-max theory gives the
existence of embedded minimal hypersurfaces in a closed manifold of dimension
between three and seven [Pit81]. The Morse index bound of such min-max
hypersurfaces is given by Marques and Neves [MN16], whom later show that there
exist such min-max hypersurfaces whose Morse index is the number of parameters
(dimension of the families used in the min-max process), under the assumption
of multiplicity one [MN18]. The multiplicity one conjecture is proved by Zhou
[Zho20] and states that for a generic metric on a closed manifold $M^{n}$,
$3\leq n\leq 7$, there exists an embedded minimal hypersurface of multiplicity
one. The conjecture is also shown to be true under the Allen-Cahn setting by
Chodosh and Mantoulidis [CM18] in dimension three.

Theorem 1.1 is a generalization of the author’s previous work [Sun20] of
proving the Morse index bound for the min-max harmonic sphere. The novelty of
Theorem 1.1 lies in two points. First, while the sphere has a unique conformal
structure, the one parameter family of mappings from tori has varying
conformal structures. The technical step of proving Theorem 1.1 is
constructing the vector fields corresponding to the varying conformal
structures, such that we can perturb the one parameter family of mappings away
from those minimal tori with Morse index larger than one. Second, we show that
the space of conformal harmonic tori in a closed manifold of dimension at
least three is countable. Unlike the compactness result for the space of
embedded minimal hypersurfaces [Sha17], we don’t have codimension one
condition one requirement for the space of conformal harmonic tori. Moreover,
unlike the harmonic sphere which has a unique conformal structure. The
Teichmüller space for tori is uncountable. We overcome this difficulty by
using the bumpy metric theorem for immersed minimal surfaces by [Whi17] and
for conformal harmonic maps by [Moo17].

The organization of the paper is as follows. In section 2, we give the basic
definition of harmonic map, Morse index, and bubble convergence for varying
conformal structures. In section 3, we prove a technical lemma 3.2. In section
4, we state Zhou’s existence result [Zho10] in the form of bubble convergence.
In section 5, we prove the main result Theorem 1.1.

##  2\. Preliminaries

###  2.1. Harmonic Maps

Let $(\Sigma,h)$ be a Riemann surface with a metric $h=\lambda^{2}dzd\bar{z}$
in conformal coordiantes $z=x+\sqrt{-1}y$. Let $(M,g)$ be a closed Riemannian
manifold of dimension $d$, and let its metric in local coordinated be given by
$g_{ij}$, with Christoffel symbols $\Gamma^{i}_{kl}.$

For $u\in W^{1,2}(\Sigma,M)$, the energy of $u$ on $\Sigma$ is

(1)
$E(u,\Sigma)=\frac{1}{2}\int_{\Sigma}g_{ij}(u^{i}_{x}u^{j}_{x}+u^{i}_{y}u^{j}_{y})dxdy.$

A solution of the corresponding Euler-Lagrange equations

(2) $\Delta
u^{i}+\Gamma^{i}_{kl}(u^{k}_{x}u^{l}_{x}+u^{k}_{y}u^{l}_{y})=0,\quad
i=1,...,d,$

is called a harmonic map. Note that $\eqref{energy}$ and $\eqref{EL}$ are
conformally invariant. If we isometrically embed $M$ into some Euclidean space
$\mathbb{R}^{N}$, then (2) can be written as the following:

(3) $\Delta u+A(u)(\nabla u,\nabla u)=0,$

where $A(\cdot,\cdot)$ is the second fundamental form of $M$ in
$\mathbb{R}^{N}$. Any $u\in W^{1,2}(\Sigma,M)$ that satisfies (3) weakly is
smooth. We introduce nearest point projection $\Pi:\mathbb{R}^{N}\to M$ which
maps a point $x\in\mathbb{R}^{N}$ to the nearest point of $M$. There is a
tubular neighborhood of $M$, $M_{\delta}=\big{\\{}x\in\mathbb{R}^{N}:\text{
dist}(x,M)<\delta\big{\\}},$ on which $\Pi$ is well-defined and smooth. For a
given $X\in C^{\infty}(\Sigma,\mathbb{R}^{N})$, we consider the variation of
$u$ with respect to $X$ defined as the following:

(4) $\Pi\circ(u+sX),\quad s\in\mathbb{R}.$

$\Pi\circ(u+sX)$ is well-defined for $s$ sufficiently small such that
$(u+sX)(\Sigma)\subset M_{\delta}$. The Taylor polynomial expansion of $\Pi$
is the following:

(5)
$\Pi\circ(u+sX)=u+sd\Pi_{u}(X)+\frac{s^{2}}{2}\text{Hess}\Pi_{u}(X,X)+o(s^{3}).$

From [Sim96, 2.12.3], we know that the following properties of the nearest
point projection $\Pi$ hold:

(6) $d\Pi_{u}(X)=p_{u}(X),$

where $p_{u}$ denotes orthogonal projection of $\mathbb{R}^{N}$ onto $T_{u}M$,

(7) $v_{1}\cdot\text{Hess}\Pi_{u}(v_{2},v_{3})=\frac{1}{2}\sum
v_{\sigma_{1}}^{\perp}\cdot\text{Hess}\Pi_{u}(v^{\ThisStyle{\abovebaseline[0pt]{\scalebox{-1.0}{$\SavedStyle\perp$}}}}_{\sigma_{2}},v^{\ThisStyle{\abovebaseline[0pt]{\scalebox{-1.0}{$\SavedStyle\perp$}}}}_{\sigma_{3}}),\quad
v_{1},v_{2},v_{3}\in\mathbb{R}^{N},$

where $v^{\perp}|_{u}=p_{u}(v)$,
$v^{\ThisStyle{\abovebaseline[0pt]{\scalebox{-1.0}{$\SavedStyle\perp$}}}}=v-v^{\perp}$;
the sum on the right is over all $6$ permutations
$\sigma_{1},\sigma_{2},\sigma_{3}$ of $1,2,3$ and $\text{Hess}\Pi_{u}$ denotes
the Hessian of $\Pi$ at $u$,

(8) $\text{Hess}\Pi_{u}(v_{1},v_{2})=-A_{u}(v_{1},v_{2}),\quad v_{1},v_{2}\in
T_{u}M,$

where $A_{u}$ is the second fundamental form of $M$ at $u$. By applying
$\nabla$ to (5) we have

(9) $\begin{split}\nabla\Big{(}\Pi\circ(u+sX)\Big{)}=&\nabla
u+s(d\Pi_{u}(\nabla X)+\text{Hess}\Pi_{u}(X,\nabla u))\\\
&+\frac{s^{2}}{2}\big{(}2\text{Hess}\Pi_{u}(X,\nabla
X)+\nabla^{3}\Pi_{u}(X,X,\nabla u)\big{)}+o(s^{3}),\end{split}$

and the energy of $\Pi\circ(u+sX)$ is

(10)
$\begin{split}E(\Pi\circ(u+sX),\Sigma)=&\int_{\Sigma}\langle\nabla\Big{(}\Pi\circ(u+sX)\Big{)},\nabla\Big{(}\Pi\circ(u+sX)\Big{)}\rangle\\\
=&\int_{\Sigma}\langle\nabla u,\nabla u\rangle\\\
+2s&\int_{\Sigma}\langle\nabla u,d\Pi_{u}(\nabla X)\rangle+\langle\nabla
u,\text{Hess}\Pi_{u}(X,\nabla u)\rangle\\\
+s^{2}&\Big{\\{}\int_{\Sigma}\langle d\Pi_{u}(\nabla X),d\Pi_{u}(\nabla
X)\rangle\\\ &+\int_{\Sigma}2\langle\text{Hess}\Pi_{u}(X,\nabla
u),d\Pi_{u}(\nabla X)\rangle\\\
&+\int_{\Sigma}\langle\text{Hess}\Pi_{u}(X,\nabla
u),\text{Hess}\Pi_{u}(X,\nabla u)\rangle\\\
&+\frac{1}{2}\int_{\Sigma}\langle\nabla u,\big{(}2\text{Hess}\Pi_{u}(X,\nabla
X)+\nabla^{3}\Pi_{u}(X,X,\nabla u)\big{)}\rangle\Big{\\}}\\\
+&o(s^{3}).\end{split}$

Then we have

(11)
$\begin{split}\frac{d}{ds}E(\Pi\circ(u+sX),\Sigma)=&\frac{d}{ds}\int_{\Sigma}\langle\nabla\Big{(}\Pi\circ(u+sX)\Big{)},\nabla\Big{(}\Pi\circ(u+sX)\Big{)}\rangle\\\
=&2\int_{\Sigma}\langle\nabla u,d\Pi_{u}(\nabla X)\rangle+\langle\nabla
u,\text{Hess}\Pi_{u}(X,\nabla u)\rangle+o(s).\end{split}$

For such a variation we have that the equation
$\frac{d}{ds}\Big{|}_{s=0}E(\Pi\circ(u+sX),\Sigma)=0$, and using (6), (7), (8)
gives the following integral identity

(12) $\int_{\Sigma}\langle X,\Delta u+A_{u}(\nabla u,\nabla u)\rangle=0,$

which coincides with (3). Now we look at second variation of energy:

(13)
$\begin{split}\frac{d^{2}}{ds^{2}}E(\Pi\circ(u+sX),\Sigma)=&2\Big{\\{}\int_{\Sigma}\langle
d\Pi_{u}(\nabla X),d\Pi_{u}(\nabla X)\rangle\\\
&+\int_{\Sigma}2\langle\text{Hess}\Pi_{u}(X,\nabla u),d\Pi_{u}(\nabla
X)\rangle\\\ &+\int_{\Sigma}\langle\text{Hess}\Pi_{u}(X,\nabla
u),\text{Hess}\Pi_{u}(X,\nabla u)\rangle\\\ &+\int_{\Sigma}\langle\nabla
u,\big{(}2\text{Hess}\Pi_{u}(X,\nabla X)+\nabla^{3}\Pi_{u}(X,X,\nabla
u)\big{)}\rangle\Big{\\}}\\\ &+o(s).\end{split}$

If $u$ is harmonic and assume that $X(x)\in T_{u(x)}M$, $\forall x\in\Sigma$.
By (3), (7), (8) we have the following

(14) $\langle\text{Hess}\Pi_{u}(X,\nabla u),d\Pi_{u}(\nabla X)\rangle=0,$ (15)
$\langle\text{Hess}\Pi_{u}(X,\nabla u),\text{Hess}\Pi_{u}(X,\nabla
u)\rangle=\langle A(X,\nabla u),A(X,\nabla u)\rangle,$ (16)
$\begin{split}\int_{\Sigma}\langle\nabla u,\big{(}2\text{Hess}\Pi_{u}(X,\nabla
X)+\nabla^{3}\Pi_{u}(X,X,\nabla u)\big{)}\rangle&=\int_{\Sigma}\langle\nabla
u,\nabla(\text{Hess}\Pi_{u}(X,X))\rangle\\\ &=-\int_{\Sigma}\langle\Delta
u,\text{Hess}\Pi_{u}(X,X)\rangle\\\ &=\int_{\Sigma}\langle A(\nabla u,\nabla
u),\text{Hess}\Pi_{u}(X,X)\rangle\\\ &=-\int_{\Sigma}\langle A(\nabla u,\nabla
u),A(X,X)\rangle.\end{split}$

Then the second variation of a harmonic map $u$ is

(17)
$\begin{split}\frac{d^{2}}{ds^{2}}\Big{|}_{s=0}E(\Pi\circ(u+sX),\Sigma)=&\int_{\Sigma}\langle
d\Pi_{u}(\nabla X),d\Pi_{u}(\nabla X)\rangle\\\ -&\int_{\Sigma}\langle
A(\nabla u,\nabla u),A(X,X)\rangle-\langle A(X,\nabla u),A(X,\nabla
u)\rangle\\\ =&\int_{\Sigma}\langle\nabla X,\nabla X\rangle-\langle
R^{M}(\nabla u,X)X,\nabla u\rangle,\end{split}$

where $R^{M}(\cdot,\cdot)$ is the Riemann curvature tensor of $M$.

######  Definition 2.1 (Index Form).

The index form of a harmonic map $u:\Sigma\to M$ is defined as the following

(18) $I(X,Y)=\int_{\Sigma}\langle\nabla X,\nabla Y\rangle-\int_{\Sigma}\langle
R^{M}(\nabla u,X)Y,\nabla u\rangle,\quad X,Y\in\Gamma(u^{-1}TM),$

where $u^{-1}TM$ is a bundle over $\Sigma$ with metric $g_{ij}(u(x))$.

######  Definition 2.2 (Index).

The index of a harmonic map $u:\Sigma\to M$ is the maximal dimension of the
subspace $X$ of $\Gamma(u^{-1}TM)$ on which the index form is negative
definite.

###  2.2. Convergence Result for Harmonic Torus

Any flat torus $T^{2}$ can be viewed as the quotient space of $\mathbb{C}$
moduled by a lattice generated by bases $\\{\omega_{1},\omega_{2}\\}$. After
some conformal linear transformation, we can assume $\omega_{1}=1$ and
$\omega_{2}=\tau\in\mathbb{H}$. Let $T_{\tau}$ be the torus by gluing edges of
the lattice $\\{1,\tau\\}$ with the plan metric $dzd\bar{z}$. Denote
$\iota_{\tau_{1},\tau_{2}}$ to be the diffeomorphism from
$\mathcal{T}_{\tau_{1}}$ to $\mathcal{T}_{\tau_{2}}$, which is the quotient
map of the linear map of $\mathbb{C}$ keeping $1$ and sending $\tau_{1}$ to
$\tau_{2}$. We denote $T_{0}$ by $T_{\sqrt{-1}}$ and $\iota_{\tau}$ to be the
linear map of $\mathbb{C}$ keeping $1$ and sending $\sqrt{-1}$ to $\tau$. For
a given manifold $M$, we denote $(u,T_{\tau})$ by the map $u:T_{\tau}\to M$
with the flat torus $T_{\tau}$ as its domain.

######  Proposition 2.3.

[Zho10, Proposition 3.1] Let $g$ be a $C^{1}$ metric on $T_{0}$. There exists
a unique mark $\tau\in\mathbb{H}$, and a unique orientation preserving
$C^{1,\frac{1}{2}}$ conformal diffeomorphism $h:T_{\tau}\to(T_{0},g)$ , such
that $h$ is isotopic to $(\iota_{\tau})^{-1}$.

To discuss bubble convergence for a sequence of maps
$\\{(u_{i},\mathcal{T}_{\tau_{i}})\\}_{i\in\mathbb{N}}$, we need to consider
the convergence of the metrics given by $\mathcal{T}_{\tau_{i}}$. In fact,
metrics of $T_{\tau}$ and $\mathcal{T}_{\tau^{\prime}}$ separately are
conformally equivalent if they lie in the same orbit of PSL($2,\mathbb{Z}$).
Denote

$\mathcal{M}=\\{z\in\mathbb{C},|z|\geq
1,\text{Im}z>0,-\frac{1}{2}<\text{Re}z<\frac{1}{2},\text{ and if
}|z|=1,\text{Re}z\geq 0\\}$

to be fundamental domain for $\text{PSL}(2,\mathbb{Z})$ (which is also
equivalent to the quotient space $\mathbb{H}/\text{PSL}(2,\mathbb{Z})$). Since
every metric of some $T_{\tau}$ is conformally equivalent to an element in
$\mathcal{M}$ after a PSL$(2,\mathbb{Z})$ action. We say a sequence
$\\{\tau_{i}\\}$ converges to $\tau_{\infty}\in\mathcal{M}$ if after being
conformally translated to $\\{\tau^{\prime}_{i}\\}\subset\mathcal{M}$ by
actions in PSL$(2,\mathbb{Z})$, $\tau_{i}^{\prime}\to\tau_{\infty}$.

######  Theorem 2.4.

[Zho10, Theorem 5.1] Let
$\\{(u_{i},\mathcal{T}_{\tau_{i}})\\}_{i\in\mathbb{N}}$,
$u_{i}:\mathcal{T}_{\tau_{i}}\to M$, be a sequence of maps such that

  1. (1)

$u_{i}\in W^{1,2}(\mathcal{T}_{\tau_{i}},M)$, $\forall i$,

  2. (2)

$E(u_{i},\mathcal{T}_{\tau_{i}})\leq\bar{E}$, $\forall i$, for some
$\bar{E}>0$,

  3. (3)

$E(u_{i},\mathcal{T}_{\tau_{i}}))-\text{Area}(u_{i})\to 0$, as $i\to\infty,$

  4. (4)

For any finite collection of disjoint balls $\bigcup_{i}B_{i}$ on
$\mathcal{T}_{\tau_{i}}$ such that $E(u_{i},\bigcup_{i}B_{i})\leq\epsilon_{0}$
(for the choice of $\epsilon_{0}$, see [Zho10]), let $v$ be the harmonic
replacement of $u_{i}$ on $\frac{1}{8}\bigcup_{i}B_{i}$. We have

$\int_{\frac{1}{8}\bigcup_{i}B_{i}}|\nabla u_{i}-\nabla v|^{2}\to 0,\quad
i\to\infty.$

Then either of the following holds

  1. (1)

If $\tau_{i}\to\tau_{\infty}$ in the above sense, then there exists a
conformal harmonic torus $v_{0}:\mathcal{T}_{\tau_{\infty}}\to M$ and finitely
many harmonic spheres $v_{j}:S^{2}\to M$ such that $u_{i}$ bubble converge to
up to subsequence, with

(19)
$\lim_{i\to\infty}E(u_{i},\mathcal{T}_{\tau_{i}})=E(v_{0},\mathcal{T}_{\tau_{\infty}})+\sum_{j}E(v_{j}).$

  2. (2)

If $\tau_{i}$ diverge, then there exists finitely many harmonic spheres
$\\{v_{j}\\}$, such that $u_{i}$ bubble converge to up to subsequence, with
body map degenerated, and

(20) $\lim_{i\to\infty}E(u_{i},\mathcal{T}_{\tau_{i}})=\sum_{j}E(v_{j}).$

###### Sketch of proof.

Let $\\{\tau^{\prime}_{i}\\}\subset\mathcal{M}$ be the corresponding conformal
structure of $\mathcal{T}_{\tau_{i}}$ after PSL$(2,\mathbb{Z})$ actions.

Case 1: $\tau^{\prime}_{i}\to\tau_{\infty}$:

Since $\tau^{\prime}_{i}$ converge, we can identify a point
$x\in\mathcal{T}_{\tau_{\infty}}$ as on $\mathcal{T}_{\tau^{\prime}_{i}}$ by
viewing it as on the fundamental regions of lattices $\\{1,\tau_{\infty}\\}$
and $\\{1,\tau^{\prime}_{i}\\}$ of corresponding conformal structures. For any
$x\in\mathcal{T}_{\tau_{\infty}}$, we consider a sequence of energy
concentration radii $r_{i}(x)$ defined as following

$r_{i}(x)=\sup\Big{\\{}r>0,E(u_{i},B_{r}(x))\leq\epsilon_{1}\Big{\\}},$

where $\epsilon_{1}<\epsilon_{0}$ (see [Zho10] for the choice of
$\epsilon_{0}$). Such $r_{i}(x)$ exist and are positive. We say $x$ is an
energy concentration point if $\lim_{i\to\infty}r_{i}(x)=0$. If $x$ is an
energy concentration point. We have that

$\inf_{r>0}\Big{\\{}\lim_{n\to\infty}E(u_{i},B_{r}(x))\Big{\\}}\geq\epsilon_{1}.$

Since the sequence $\\{(u_{i},\tau_{i})\\}$ has uniform bounded energy. We
know that the number of the energy concentration points are bounded by
$W/\epsilon_{1}$. Denote the energy concentration points by
$\\{x_{1},...,x_{m}\\}.$ If
$x\in\mathcal{T}_{\tau_{\infty}}\setminus\\{x_{1},...,x_{m}\\}$. We can find
$r(x)>0$ such that $E(u_{i},B_{r(x)}(x))\leq\epsilon_{1}$, $\forall i$. By
assumption (4), there exists $v_{i}$, which are the energy minimizing harmonic
maps defined on $\frac{1}{8}B_{r(x)}(x)$ with the same boundary value as
$u_{i}$, we know from [SU81] that $v_{i}$ have uniform interior
$C^{2,\alpha}$-estimates on $\frac{1}{8}B_{r(x)}(x)$ hence converges to a
harmonic map $v_{0}$ on $\frac{1}{9}B_{r(x)}(x)$ up to subsequence. For any
compact subset
$K\subset\mathcal{T}_{\tau_{\infty}}\setminus\\{x_{1},...,x_{m}\\}$, we can
cover them by finitely many balls $\frac{1}{9}B_{r(x)}(x)$ and hence $u_{i}\to
v_{0}$ in $W^{1,2}(K,M)$ up to subsequence. After exhausting
$\mathcal{T}_{\tau_{\infty}}\setminus\\{x_{1},...,x_{m}\\}$ by a sequence of
compact sets and a diagonal argument, we know $v_{0}$ is a harmonic map on
$\mathcal{T}_{\tau_{\infty}}\setminus\\{x_{1},...,x_{m}\\}$ and by removable
singularity property of harmonic map [SU81] we know $v_{0}$ extends to a
harmonic map on $\mathcal{T}_{\tau_{\infty}}$.

We now see what happens near the energy concentration points. Fix an energy
concentration point $x\in\\{x_{1},...,x_{m}\\}$ and denote $r_{i}=r_{i}(x)$.
Find $r>0$ such that $E(v_{0},B_{r}(x))\leq\frac{1}{3}\epsilon_{1}$. We recale
$u_{i}$ on $r_{i}$. Define

$u_{i}^{\prime}(y):=u_{i}(x+r_{i}(y-x)).$

So $B_{r_{i}}(x)$ are rescaled to $B_{1}$. $u_{i}^{\prime}$ can be viewed as
defined on $B_{r/r_{i}}$ with $r/r_{i}\to\infty$. Since
$B_{r/r_{i}}\to\mathbb{C}$, and $\mathbb{C}$ is conformally equivalent to
$S^{2}$ without the south pole, we can consider $u_{i}^{\prime}$ as defined on
any compact subset of $S^{2}$ away from south pole for $i$ sufficiently large
enough. Since the assumption (4) is conformally invariant, we can repeat the
same argument, find finitely many energy concentration points
$\\{x_{1}^{\prime},...x_{n}^{\prime}\\}\subset S^{2}\setminus\\{\text{south
pole}\\}$, such that $u_{i}^{\prime}$ converges to a harmonic map $v_{1}$
defined on $S^{2}$ in the above sense.

We can repeat the bubbling convergence given above. There are only finitely
many such steps, and then the bubbling convergence stops. For the energy
identity (19), we need to show that the following is true:

(21) $\lim_{r\to 0,R\to\infty}\lim_{i\to\infty}E(u_{i},B_{r}(x)\setminus
B_{r_{i}R}(x))=0.$

(21) follows from [Zho10, Proposition 5.1].

Case 2: $\tau^{\prime}_{i}$ diverge:

We use $(t,\theta)$ as parameters on $\mathcal{T}_{\tau_{i}}$. Let
$\theta_{n}=\text{arg}(\tau_{n})$, and let
$z^{\prime}=t+\sqrt{-1}\theta=e^{-\sqrt{-1}(\frac{\pi}{2}-\theta_{n})}z$ be
another conformal parameter system on $\mathcal{T}_{\tau_{i}}$. We can argue
as in case 1 to show that $u_{i}$ converges to a harmonic map $v_{1}$ defined
on $S\times\mathbb{R}$. We know that $v_{1}$ must be nontrivial or else it
contradicts with [Zho10, Proposition 5.1]. As $S\times\mathbb{R}$ is
conformally equivalent to $S^{2}\setminus${north pole, south pole}, we can
extend $v_{1}$ to a harmonic map on $S^{2}$. For the energy concentration
points, we can rescale $u_{i}$ and the rescaled map will converge as we
discussed in case 1 to finitely many bubble maps $\\{v_{j}\\}$.

∎

Inspired by Theorem 2.4, we come up with the following definition of bubble
norm which is used to describe how close a given map is to a collection of a
conformal harmonic torus and finitely many harmonic spheres in the bubble tree
sense.

######  Definition 2.5 (Bubble Norm).

From Theorem 2.4, we know that if a sequence of maps $\\{(u_{i},\tau_{i})\\}$
with varying conformal structures of the domains $\mathcal{T}_{\tau_{i}}$
converges in the bubble tree sense, the domain of the body map could be either
torus or sphere depends on whether the conformal structures converge. We
discuss the torus case in case 1 and the sphere case in case 2.

Case 1:

Given a map $u:\mathcal{T}_{\tau}\to M$, with $\tau\in\mathcal{M}$, and a
collection of maps
$\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\big{\\}}$, where
$v_{0}$ is a harmonic torus, $v_{0}:\mathcal{T}_{\tau_{0}}\to M$,
$\tau_{0}\in\mathcal{M}$, and $\\{v_{j}\\}_{j=1}^{n}$ are harmonic spheres,
$v_{j}:S^{2}\to M$, we say that

$d_{B}((u,T_{\tau}),\big{\\{}(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}\big{\\}})<\epsilon$

if there exist $B_{r_{j}}(x_{j})\subset\mathcal{T}_{\tau}$, $r_{j}<1$, $1\leq
j\leq n$, $D_{0}\in S^{1}\times S^{1}$, $D_{j}\in\text{PSL}(2,\mathbb{C})$,
such that the following holds:

  1. (1):

for $i\neq j$, if $B_{r_{i}}(x_{i})\cap B_{r_{j}}(x_{j})\neq\emptyset$, we
have either $B_{r_{i}}(x_{i})\subset B_{r_{j}^{2}}(x_{j})$ or
$B_{r_{j}}(x_{j})\subset B_{r_{i}^{2}}(x_{i})$. Let

$\Omega_{j}=B_{r_{j}^{2}}(x_{j})\setminus\Big{\\{}\bigcup
B_{r_{i}}(x_{i}),\>B_{r_{i}}(x_{i})\subset B_{r_{j}^{2}}(x_{j})\Big{\\}}.$

$\\{\Omega_{j}\\}_{j=1}^{n}$ is a collection of pairwise disjoint domains in
$T_{\tau}$.

  2. (2):

Since $\mathbb{C}$ is confromally equivalent to $S^{2}\setminus\\{\text{north
pole}\\}$, we can regard $v_{j}$ as harmonic maps defined on $\mathbb{C}$ via
stereographic projection.

(22) $|\tau-\tau_{0}|<\epsilon,$ (23)
$\begin{split}\Big{(}\int_{T_{\tau}\setminus\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla
u-&\nabla(v_{0}\circ D_{0}\circ\iota_{\tau,\tau_{0}})|^{2}\Big{)}^{1/2}\\\
&+\Big{(}\int_{\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla(v_{0}\circ
D_{0}\circ\iota_{\tau,\tau_{0}})|^{2}\Big{)}^{1/2}<\epsilon/3,\end{split}$
(24) $\sum_{j=1}^{n}\Big{(}\int_{\Omega_{j}}|\nabla u-\nabla(v_{j}\circ
D_{j})|^{2}\Big{)}^{1/2}+\Big{(}\int_{\mathbb{C}\setminus\Omega_{j}}|\nabla(v_{j}\circ
D_{j})|^{2}\Big{)}^{1/2}<\epsilon/3,$ (25)
$\Big{(}\int_{\bigcup_{j}\\{B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})\\}}|\nabla u|^{2}\Big{)}^{1/2}<\epsilon/3.$

(22) shows that the confromal structure of $T_{\tau}$ is ”close” to
$\mathcal{T}_{\tau_{0}}$, which is the domain of $v_{0}$. (23) and (24) are
for describing the convergence of the body map and bubble map in $W^{1,2}$
sense on the restricted domain $T_{\tau}\setminus\bigcup_{j}B_{r_{j}}(x_{j})$
and $\Omega_{j}$, while (25) represents the energy loss on the neck domain
$B_{r_{j}}(x_{j})\setminus B_{r_{j}^{2}}(x_{j})$. From Theorem 2.4 we know
that if a sequence of maps $\\{(u_{i},\mathcal{T}_{\tau_{i}})\\}$ converges to
$\big{\\{}(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\big{\\}}$ in a
bubble tree sense, then

$d_{B}((u_{i},\mathcal{T}_{\tau_{i}}),\big{\\{}(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\big{\\}})\to
0,\quad i\to\infty.$

Case 2:

Given a map $u:T_{\tau}\to M$, with $\tau\in\mathcal{M}$, and a collection of
harmonic spheres $\\{v_{j}\\}_{j=1}^{n}$, $v_{j}:S^{2}\to M$, we say that

$d_{B}((u,T_{\tau}),\\{v_{j}\\}_{j=1}^{n})<\epsilon$

if there exist $B_{r_{j}}(x_{j})\subset\mathcal{T}_{\tau}$, $r_{j}<1$,
$D_{j}\in\text{PSL}(2,\mathbb{C})$, for $1\leq j\leq n$, such that the
following holds:

  1. (1):

For $i\neq j$, if $B_{r_{i}}(x_{i})\cap B_{r_{j}}(x_{j})\neq\emptyset$, we
have either $B_{r_{i}}(x_{i})\subset B_{r_{j}^{2}}(x_{j})$ or
$B_{r_{j}}(x_{j})\subset B_{r_{i}^{2}}(x_{i})$. Let

$\Omega_{j}=B_{r_{j}^{2}}(x_{j})\setminus\Big{\\{}\bigcup
B_{r_{i}}(x_{i}),\>B_{r_{i}}(x_{i})\subset B_{r_{j}^{2}}(x_{j})\Big{\\}}.$

$\\{\Omega_{j}\\}_{j=1}^{n}$ is a collection of pairwise disjoint domains in
$T_{\tau}$.

  2. (2):

We argue as above and regard $v_{j}$ as harmonic maps defined on $\mathbb{C}$.
Since we are considering the case that body map is degenerated, we use
exponential map to conformally transform $\mathbb{C}\setminus\\{0\\}\simeq
S^{1}\times\mathbb{R}$, and regard the body map $v_{1}$ as harmonic map
defined on $S^{1}\times\mathbb{R}$. We don’t strictly distinguish $v_{1}$ and
$v_{1}\circ e^{z}$ here in abuse of notation. Also by changing parameters (see
the proof of Theorem 2.4) we can consider $T_{\tau}\simeq
S^{1}\times[-L,L]\subset S^{1}\times\mathbb{R}.$

(26)
$\begin{split}\Big{(}\int_{T_{\tau}\setminus\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla
u-&\nabla(v_{1}\circ D_{1})|^{2}\Big{)}^{1/2}\\\
&+\Big{(}\int_{S^{1}\times\mathbb{R}\setminus(T_{\tau}\setminus\bigcup_{j}B_{r_{j}}(x_{j})))}|\nabla(v_{1}\circ
D_{1})|^{2}\Big{)}^{1/2}<\epsilon/3,\end{split}$ (27)
$\sum_{j=1}^{n}\Big{(}\int_{\Omega_{j}}|\nabla u-\nabla(v_{j}\circ
D_{j})|^{2}\Big{)}^{1/2}+\Big{(}\int_{\mathbb{C}\setminus\Omega_{j}}|\nabla(v_{j}\circ
D_{j})|^{2}\Big{)}^{1/2}<\epsilon/3,$ (28)
$\Big{(}\int_{\bigcup_{j}\\{B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})\\}}|\nabla u|^{2}\Big{)}^{1/2}<\epsilon/3.$

Similar to Case 1, we see that (26) and (27) are for describing the
convergence of the degenerated body map and bubble map in $W^{1,2}$ sense on
the restricted domain $T_{\tau}\setminus\bigcup_{j}B_{r_{j}}(x_{j})$ and
$\Omega_{j}$, while (28) represents the energy loss on the neck domain
$B_{r_{j}}(x_{j})\setminus B_{r_{j}^{2}}(x_{j})$. From Theorem 2.4 we know
that if a sequence of maps $\\{(u_{i},\mathcal{T}_{\tau_{i}})\\}$ converges to
$\\{v_{j}\\}_{j=1}^{n}$ in a bubble tree sense with degenerated body map, then

$d_{B}((u_{i},\mathcal{T}_{\tau_{i}}),\\{v_{j}\\}_{j=1}^{n})\to 0,\quad
i\to\infty.$

######  Definition 2.6 (Equivalent Class).

Given harmonic maps $u:\Sigma\to M$ and $v:\Sigma\to M$. We have $v\sim u$ if
there exists conformal automorphism $g$ of $\Sigma$ such that $v\circ g=u$. We
denote $[u]$ by the equvalent class of $u$.

Although the bubble norm $d_{B}(\cdot,\cdot)$ is not a norm, but if a map
$u:T_{\tau}\to M$ is close to a collection of harmonic maps in the bubble tree
sense, then for any map $u^{\prime}:\mathcal{T}_{\tau^{\prime}}\to M$ that’s
sufficiently close to $u$ in $W^{1,2}$ sense, $u^{\prime}$ is also close to
the same collection of harmonic maps in the bubble tree sense.

######  Claim 2.7.

Given $u:T_{\tau}\to M$, and a collection of harmonic maps
$\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\big{\\}}$. If
$d_{B}((u_{i},\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\big{\\}})<\epsilon$
Then there exists $\delta(u)>0$, which depends on $(u,T_{\tau})$ and
$\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\big{\\}}$, such that
for any map $u^{\prime}:\mathcal{T}_{\tau^{\prime}}\to M$, if
$|\tau-\tau^{\prime}|<\delta(u)$ and

$\int_{\mathcal{T}_{\tau^{\prime}}}|\nabla
u^{\prime}-\nabla(u\circ\iota_{\tau,\tau^{\prime}})|^{2}<\delta(u),$

then

$d_{B}((u^{\prime},\mathcal{T}_{\tau^{\prime}}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\big{\\}})<\epsilon.$

###### Proof.

$d_{B}((u_{i},\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\big{\\}})<\epsilon$
implies that there exist $B_{r_{j}}(x_{j})\subset T_{\tau}$, pairwise disjoint
domains $\\{\Omega_{j}\\}$, and conformal automorphisms $\\{D_{j}\\}$ such
that the inequalities (23), (24), and (25) hold. We can assume $\tau^{\prime}$
is sufficiently close to $\tau$ such that
$B_{r_{j}}(x_{j})\subset\mathcal{T}_{\tau^{\prime}}$ and
$E(u\circ\iota_{\tau^{\prime},\tau},\mathcal{T}_{\tau^{\prime}})-E(u,T_{\tau})$
is sufficiently small. Then we have

(29)
$\begin{split}&\Big{(}\int_{\mathcal{T}_{\tau^{\prime}}\setminus\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla
u^{\prime}-\nabla(v_{0}\circ
D_{0}\circ\iota_{\tau^{\prime},\tau_{0}})|^{2}\Big{)}^{1/2}\leq\\\
&\Big{(}\int_{\mathcal{T}_{\tau^{\prime}}}|\nabla
u^{\prime}-\nabla(u\circ\iota_{\tau,\tau^{\prime}})|^{2}\Big{)}^{1/2}+\Big{(}\int_{T_{\tau}\setminus\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla
u-\nabla(v_{0}\circ
D_{0}\circ\iota_{\tau,\tau_{0}})|^{2}\Big{)}^{1/2}.\end{split}$

Thus for $\delta(u)>0$ sufficiently small we have

(30)
$\begin{split}\Big{(}\int_{\mathcal{T}_{\tau^{\prime}}\setminus\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla
u^{\prime}-&\nabla(v_{0}\circ
D_{0}\circ\iota_{\tau^{\prime},\tau_{0}})|^{2}\Big{)}^{1/2}\\\
&+\Big{(}\int_{\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla(v_{0}\circ
D_{0}\circ\iota_{\tau^{\prime},\tau_{0}})|^{2}\Big{)}^{1/2}<\epsilon/3.\end{split}$

Similarly, (24) and (25) hold for the same domain and conformal automorphisms
with $u$ replaced by $u^{\prime}$. Thus we have
$d_{B}((u^{\prime},\mathcal{T}_{\tau^{\prime}}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\big{\\}})<\epsilon.$
∎

##  3\. Unstable Lemma

The main focus of the section is Lemma 3.2 and 3.5: proving the energy is
concave for any map with some flat torus as its domain, that are sufficiently
close to either a finite collection of one harmonic torus and possibly several
harmonic spheres or a finite collection of harmonic spheres in bubble tree
sense. This corresponds to the two possible convergence result given by
Theorem 2.4. Now we focus on the case where the conformal structure converges
and the limit maps are one harmonic torus and possibly several harmonic
spheres. Consider a closed manifold $M$ of dimension at least three which is
isometrically embedded in $\mathbb{R}^{N}$, given a collection of harmonic
maps $\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\}$, where
$v_{0}$ is a harmonic torus, $v_{0}:\mathcal{T}_{\tau_{0}}\to M$,
$\tau_{0}\in\mathcal{M}$, and $\\{v_{j}\\}_{j=1}^{n}$ are harmonic spheres.
Assume that $\text{Index}(v_{j})=k_{j}$ for each $j\in\\{0,1,...,n\\}$, by
Definition 2.2, there exists subspace
$\\{X_{j,i}\\}_{i=1}^{k_{j}}\subset\Gamma(v_{j}^{-1}TM)$ on which the index
form is negative definite, by rescaling $\\{X_{j,i}\\}_{i=1}^{k_{j}}$ (which
we still denote by $\\{X_{j,i}\\}_{i=1}^{k_{j}}$), we have that for

$E_{v_{0}}(s_{0},\mathcal{T}_{\tau_{0}})=E(\Pi\circ\big{(}v_{0}+\sum_{i=1}^{k_{0}}s_{0,i}X_{0,i}\big{)},\mathcal{T}_{\tau_{0}}),\quad
s_{0}=(s_{0,1},...,s_{0,k_{0}})\in\bar{B}^{k_{0}},$

and

$E_{v_{j}}(s_{j},\mathbb{C})=E(\Pi\circ\big{(}v_{j}+\sum_{i=1}^{k_{j}}s_{j,i}X_{j,i}\big{)},\mathbb{C}),\quad
s_{j}=(s_{j,1},...,s_{j,k_{j}})\in\bar{B}^{k_{j}},\>j=1,...,n,$

there exist $c_{0}<1$ such that

(31)
$-\frac{1}{2c_{0}}\text{Id}<D^{2}_{s_{0}}E_{v_{0}}(s_{0},\mathcal{T}_{\tau_{0}})<-2c_{0}\text{Id},\quad\forall
s_{0}\in\bar{B}^{k_{0}},$

and

(32)
$-\frac{1}{2c_{0}}\text{Id}<D^{2}_{s_{j}}E_{v_{j}}(s_{j},\mathbb{C})<-2c_{0}\text{Id},\quad\forall
s_{j}\in\bar{B}^{k_{j}},$

for each $j=1,...,n$.

######  Claim 3.1.

Let $\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\}$ and
$\\{X_{j,i}\\}$ be given as above. Given $\epsilon>0$, there exists
$\delta(\epsilon)>0$, such that for any $B_{\rho}(x)\subset\mathbb{C}$, if

(33) $E(v_{j},B_{\rho}(x))<\delta(\epsilon),$

then we have

(34)
$-\epsilon\text{Id}<D^{2}_{s_{j}}E_{v_{j}}(s_{j},B_{\rho}(x))<\epsilon\text{Id},\text{
and }\int_{B_{\rho}(x)}|\nabla X_{j,i}|^{2}<\epsilon,$

for $i=1,...,k_{j}$, and $j=0,...,n.$

###### Proof.

Let

$\tilde{\rho}:=\inf\big{\\{}\rho>0,\>\int_{B_{\rho}(x)}|\nabla
X_{j,i}|^{2}>\frac{\epsilon}{2},B_{\rho}(x)\subset\mathbb{C}\big{\\}}.$

We know that $\tilde{\rho}>0$. Let

$\delta_{1}(\epsilon):=\inf\big{\\{}E(v_{j},B_{\tilde{\rho}}(x)),\>x\in\mathbb{C}\big{\\}}.$

It’s easy to check that if $E(v_{j},B_{\rho}(x))<\delta_{1}(\epsilon)$ for
some $B_{\rho}(x)\subset\mathbb{C}$, then it implies that
$\rho\leq\tilde{\rho}$, thus we have $\int_{B_{\rho}(x)}|\nabla
X_{j,i}|^{2}<\epsilon$. Similarly, we can choose $\delta_{2}(\epsilon)>0$ such
that if $E(v_{j},B_{\rho}(x))<\delta_{2}(\epsilon)$ for some
$B_{\rho}(x)\subset\mathbb{C}$, then it implies

$-\epsilon\text{Id}<D^{2}_{s_{j}}E_{v_{j}}(s_{j},B_{\rho}(x))<\epsilon\text{Id}.$

We finish the proof by choosing
$\delta(\epsilon)=\min\\{\delta_{1}(\epsilon),\delta_{2}(\epsilon)\\}.$ ∎

######  Lemma 3.2 (Unstable Lemma for torus).

Let $\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\}$,
$\\{X_{j,i}\\}$ be given as above. There exists $\epsilon>0$, such that if a
map $u:\mathcal{T}_{\tau}\to M$, $\tau\in\mathcal{M}$, $u\in
W^{1,2}(T_{\tau},M)$ has
$d_{B}((u,\mathcal{T}_{\tau}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon.$
Then we can construct $\\{\tilde{X}_{j,i}\\}\subset
C^{\infty}(T_{\tau},\mathbb{R}^{N})$, so that for the smooth function

$E_{u}(\cdot,T_{\tau}):\bar{B}^{k}\to[0,\infty),$

which is defined by

$E_{u}(s,T_{\tau})=E\big{(}\Pi\circ\big{(}u+\sum_{i=1,j=0}^{k_{j},n}s_{j,i}\tilde{X}_{j,i}\big{)},T_{\tau}\big{)},\quad
s=(s_{0,1},...,s_{j,k_{j}},s_{j+1,1},...,s_{n,k_{n}})\in\bar{B}^{k},$

where $k=\sum_{j=0}^{n}k_{j}$, the following holds:

(35) $-\frac{1}{c_{0}}\text{Id}\leq D^{2}_{s}E_{u}(s,T_{\tau})\leq-
c_{0}\text{Id},\quad\forall s\in\bar{B}^{k}.$

Moreover, $E_{u}(\cdot,T_{\tau})$ has a unique maximum at $m(u)\in
B^{k}_{c_{0}/\sqrt{10}}$ and

(36) $E_{u}(m(u),T_{\tau})-\frac{1}{2c_{0}}|s-m(u)|^{2}\leq
E_{u}(s,T_{\tau})\leq E_{u}(m(u),T_{\tau})-\frac{c_{0}}{2}|s-m(u)|^{2},$

for all $s\in\bar{B}^{k}.$

###### Proof.

We assume that
$d_{B}((u,T_{\tau}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon$
for some constant $\epsilon>0$ which will be chosen later, by Definition 2.5
we have the following

(37) $|\tau-\tau_{0}|<\epsilon,$ (38)
$\begin{split}\Big{(}\int_{T_{\tau}\setminus\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla
u-&\nabla(v_{0}\circ D_{0}\circ\iota_{\tau,\tau_{0}})|^{2}\Big{)}^{1/2}\\\
&+\Big{(}\int_{\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla(v_{0}\circ
D_{0}\circ\iota_{\tau,\tau_{0}})|^{2}\Big{)}^{1/2}<\epsilon/3,\end{split}$
(39) $\sum_{j=1}^{n}\Big{(}\int_{\Omega_{j}}|\nabla u-\nabla(v_{j}\circ
D_{j})|^{2}\Big{)}^{1/2}+\Big{(}\int_{\mathbb{C}\setminus\Omega_{j}}|\nabla(v_{j}\circ
D_{j})|^{2}\Big{)}^{1/2}<\epsilon/3,$ (40)
$\Big{(}\int_{\bigcup_{j}\\{B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})\\}}|\nabla u|^{2}\Big{)}^{1/2}<\epsilon/3.$

Now we define the following piecewise smooth cutoff function, which was
introduced by Choi and Schoen [CS85], $\eta_{j}:[0,\infty)\to[0,1]$:

(41) $\eta_{j}(r)=\left\\{\begin{array}[]{rcl}0,&\mbox{for}&r<r_{j}^{2},\\\
2-(\log r)/(\log r_{j}),&\mbox{for}&r_{j}^{2}\leq r\leq r_{j},\\\
1,&\mbox{for}&r>r_{j},\end{array}\right.$

so that

$\frac{d\eta_{j}}{dr}(r)=\left\\{\begin{array}[]{rcl}0,&\mbox{for}&r<r_{j}^{2},\\\
-1/r(\log r_{j}),&\mbox{for}&r_{j}^{2}\leq r\leq r_{j},\\\
0,&\mbox{for}&r>r_{j},\end{array}\right.$

and

(42)
$\int_{0}^{2\pi}\int_{r_{j}^{2}}^{r_{j}}\Big{(}\frac{d\eta_{j}}{dr}(r)\Big{)}^{2}rdrd\theta=-\frac{2\pi}{\log
r_{j}}.$

We can choose a local conformal coordinate $z=x+\sqrt{-1}y$ centered at
$x_{j}\in T_{\tau}$ and let $f(z)=\sqrt{x^{2}+y^{2}}$ so that we can define
the $r_{j}$-ball $B_{r_{j}}(x_{j})$ about $x_{j}$ as the set of points
satisfying $r\leq r_{j}$. The radial cutoff function we will use in our
constructions is then the map

(43) $\zeta_{j}:B_{r_{j}}(x_{j})\to[0,1],\quad\text{defined by
}\zeta_{j}=\eta_{j}\circ f.$

For distinct $j$ we have the local coordinate centered at distinct $x_{j}$,
thus the function $r$ in (43) is different for each $j$, but we don’t strictly
distinguish them and denote them by $r$ for all $j$ in abuse of notation and
terminology.

######  Claim 3.3.

There exists $\epsilon>0$, such that for
$d_{B}((u,T_{\tau}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon$,
and

(44) $\tilde{X}_{0,i}:=\prod_{j=1,...,n}\zeta_{j}X_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}},$

where $\zeta_{j}$ is given by (43), we have

(45)
$-c_{0}\text{Id}<D^{2}_{s_{0}}E(\Pi\circ(u+\sum_{i=1}^{k_{0}}s_{0,i}\tilde{X}_{0,i}),T_{\tau})<-\frac{1}{c_{0}}\text{Id},\quad\forall
s_{0}=(s_{0,1},...,s_{0,k_{0}})\in\bar{B}^{k_{0}}.$

###### proof of the claim.

To simplify the notation, let

(46)
$\displaystyle\delta^{2}_{s,t}E_{u}(s\tilde{X}_{0,l_{1}}+t\tilde{X}_{0,l_{2}},T_{\tau}):=\frac{d^{2}}{dsdt}E(\Pi\circ(u+s\tilde{X}_{0,l_{1}}+t\tilde{X}_{0,l_{2}}),T_{\tau}),$
(47)
$\displaystyle\delta^{2}_{s}E_{u}(s\tilde{X}_{0,l_{1}},T_{\tau}):=\frac{d^{2}}{ds^{2}}E(\Pi\circ(u+s\tilde{X}_{0,l_{1}}),T_{\tau}),$

for $\\{l_{1},l_{2}\\}\subset\\{1,...,k_{0}\\}$. We observe that to prove (45)
is to show that we can choose $\epsilon>0$ to make the following term
$\delta^{2}_{s,t}E_{u}(s\tilde{X}_{0,l_{1}}+t\tilde{X}_{0,l_{2}},T_{\tau})-\delta^{2}_{s,t}E_{v_{0}}(sX_{0,l_{1}}+tX_{0,l_{2}},\mathcal{T}_{\tau_{0}})$
sufficiently small for all $s,t\in[0,1]$,
$\\{l_{1},l_{2}\\}\subset\\{1,...,k_{0}\\}$. It’s enough to show the following
term

$\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},T_{\tau})-\delta^{2}_{s}E_{v_{0}}(sX_{0,i},\mathcal{T}_{\tau_{0}}),\quad
i=1,...,k_{0},$

could be made sufficiently small by choosing $\epsilon>0$. We observe that

(48)
$\begin{split}\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},T_{\tau})&=\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j}))\\\
&+\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})).\end{split}$

Which implies

(49)
$\begin{split}|\delta^{2}_{s}&E_{u}(s\tilde{X}_{0,i},T_{\tau})-\delta^{2}_{s}E_{v_{0}}(sX_{0,i},\mathcal{T}_{\tau_{0}})|\\\
&<|\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j}))-\delta^{2}_{s}E_{\tilde{v}_{0}}(sX_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}},T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j}))|\\\
&+|\delta^{2}_{s}E_{\tilde{v}_{0}}(sX_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}},\cup_{i=1}^{n}B_{r_{j}}(x_{j}))|\\\
&+|\delta^{2}_{s}E_{\tilde{v}_{0}}(sX_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}},\mathcal{T}_{\tau})-\delta^{2}_{s}E_{v_{0}}(sX_{0,i},\mathcal{T}_{\tau_{0}})|\\\
&+|\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j}))|,\end{split}$

where $\tilde{v}_{0}:=v_{0}\circ D_{0}\circ\iota_{\tau,\tau_{0}}$. We can make
$|\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},T_{\tau})-\delta^{2}_{s}E_{v_{0}}(sX_{0,i},\mathcal{T}_{\tau_{0}})|$
sufficiently small by making each term in (49) sufficiently small. We will
discuss how to bound each term in (49) in the following.

1:

$|\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j}))-\delta^{2}_{s}E_{\tilde{v}_{0}}(sX_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}},T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j}))|$

We recall from (13) for any $X\in C^{\infty}(T_{\tau},\mathbb{R}^{N})$ we have
the following,

(50)
$\begin{split}\frac{d^{2}}{ds^{2}}E(\Pi\circ(u+sX),T_{\tau})=&2\Big{\\{}\int_{T_{\tau}}\langle
d\Pi_{u}(\nabla X),d\Pi_{u}(\nabla X)\rangle\\\
&+\int_{T_{\tau}}2\langle\text{Hess}\Pi_{u}(X,\nabla u),d\Pi_{u}(\nabla
X)\rangle\\\ &+\int_{T_{\tau}}\langle\text{Hess}\Pi_{u}(X,\nabla
u),\text{Hess}\Pi_{u}(X,\nabla u)\rangle\\\ &+\int_{T_{\tau}}\langle\nabla
u,\big{(}2\text{Hess}\Pi_{u}(X,\nabla X)+\nabla^{3}\Pi_{u}(X,X,\nabla
u)\big{)}\rangle\Big{\\}}\\\ &+o(s).\end{split}$

Since $X_{0,1}\in\Gamma(v_{0}^{-1}TM)$, so $d\Pi_{v_{0}}(\nabla
X_{0,1})=\nabla X_{0,1}$, which implies

(51) $d\Pi_{\tilde{v}_{0}}\big{(}\nabla(X_{0,1}\circ
D_{0}\circ\iota_{\tau,\tau_{0}})\big{)}=\nabla(X_{0,1}\circ
D_{0}\circ\iota_{\tau,\tau_{0}}).$

Moreover, $\zeta_{j}$ has value $1$ on the domain
$T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j})$, so
$\tilde{X}_{0,1}=X_{0,1}\circ D_{0}\circ\iota_{\tau,\tau_{0}}$ on
$T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j})$, thus we have

(52)
$\int_{T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j})}|\nabla\tilde{X}_{0,1}|^{2}=\int_{T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j})}|\nabla\big{(}X_{0,1}\circ
D_{0}\circ\iota_{\tau,\tau_{0}}\big{)}|^{2}.$

We observe that the only term in (50) not bounded by the energy difference of
$u$ and $\tilde{v}_{0}$ on the domain
$T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j})$ is the term
$\int|d\Pi_{u}(\nabla X)|^{2}$. Since $M$ is a smooth manifold and
$d\Pi_{u}(\nabla X)$ is the orthogonal projection of $\nabla X$ on $T_{u}M$,
by (38) we know that $u$ is close to $\tilde{v}_{0}$ on
$T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j})$ in $W^{1,2}$ sense, thus
implies that the term $\int|d\Pi_{u}(\nabla X)|^{2}$ can be bounded by
$\int|d\Pi_{\tilde{v}_{0}}(\nabla X)|^{2}$ and (38) for any $X\in
C^{\infty}(T_{\tau},\mathbb{R}^{N})$. Then $\eqref{47}$ and $\eqref{48}$
implies that

(53)
$\begin{split}|\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j}))-\delta^{2}_{s}E_{\tilde{v}_{0}}(sX_{0,i}&\circ
D_{0}\circ\iota_{\tau,\tau_{0}},T_{\tau}\setminus\cup_{i=1}^{n}B_{r_{j}}(x_{j}))|\\\
&<C(d_{B}(u,\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\}))\end{split}$

for a continuous nonnegative function $C$ depending on the manifold $M$ and
$\\{X_{0,i}\\}_{i=1}^{n}$, such that $C(0)=0$.

2:

$|\delta^{2}_{s}E_{\tilde{v}_{0}}(sX_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}},\mathcal{T}_{\tau})-\delta^{2}_{s}E_{v_{0}}(sX_{0,i},\mathcal{T}_{\tau_{0}})|$

Since $\iota_{\tau,\tau_{0}}$ is the linear map of $\mathbb{C}$ keeping $1$
and sending $\tau$ to $\tau_{0}$, Conformal invariance of energy imply that

(54) $|\delta^{2}_{s}E_{\tilde{v}_{0}}(sX_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}},\mathcal{T}_{\tau})-\delta^{2}_{s}E_{v_{0}}(sX_{0,i},\mathcal{T}_{\tau_{0}})|\leq
C_{1}(|\tau-\tau_{0}|),$

where $C_{1}$ is a nonnegative continuous function, depending on $M$ and
$\\{X_{0,i}\\}_{i=1}^{n}$, such that $C_{1}(0)=0$.

3:

$|\delta^{2}_{s}E_{\tilde{v}_{0}}(sX_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}},\cup_{i=1}^{n}B_{r_{j}}(x_{j}))|$

By Claim 3.1, there exists $\delta(c_{0}/10)>0$, such that

$-\frac{c_{0}}{10}\text{Id}<D^{2}_{s_{j}}E_{v_{j}}(s_{j},B_{\rho}(x))<\frac{c_{0}}{10}\text{Id},$

for $E(v_{j},B_{\rho}(x))<\delta(\frac{c_{0}}{10})$. Since energy is invariant
under conformal diffeomorphism of the domain, the $\delta(\epsilon)$ chosen in
Claim 3.1 remains the same for $v_{j}\circ D_{j}$, where $D_{0}\in S^{1}\times
S^{1}$ and $D_{j}\in\text{PSL}(2,\mathbb{C})$ for each $j=1,...,n$. If the
bubble norm
$d_{B}(u,\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<(\delta(\epsilon))^{1/2}$,
then (38) give us $E(v_{0}\circ
D_{0}\circ\iota_{\tau,\tau_{0}},\cup_{i=1}^{n}B_{r_{j}}(x_{j})))<\delta(\epsilon)$,
so Claim 3.1 implies the following inequality:

(55) $|\delta^{2}_{s}E_{\tilde{v}_{0}}(sX_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}},\cup_{i=1}^{n}B_{r_{j}}(x_{j}))|<\frac{c_{0}}{10}.$

4:

$|\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j}))|$

By (50) we know that the only term not bounded by the energy of $u$ on the
domain $\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus B_{r_{j}^{2}}(x_{j}))$ is the
term $\int|d\Pi_{u}(\nabla X)|^{2}$. Since $d\Pi_{u}(\nabla X)$ is the
orthogonal projection of $\nabla X$ on $T_{u}M$, thus it’s bounded by
$\int|\nabla X|^{2}$ on $\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j}).$

(56)
$\begin{split}|\delta^{2}_{s}E_{u}&(s\tilde{X}_{0,i},\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j}))|\\\
<&C_{2}\big{(}E(u,\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j}))\big{)}+\int_{\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})}|\nabla\tilde{X}_{0,i}|^{2}\\\
=&C_{2}\big{(}E(u,\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j}))\big{)}\\\
+&\int_{\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})}|\nabla\Big{(}\prod_{j=1,...,n}\zeta_{j}X_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}}\Big{)}|^{2},\end{split}$

where $C_{2}$ is a nonnegative continuous function depending on $M$ and
$\\{X_{0,i}\\}_{i=1}^{n}$ such that $C_{2}(0)=0$. By how we construct
$\zeta_{j}$ (43) and (42), we have the following inequality

(57) $\begin{split}\int_{\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})}&|\nabla\Big{(}\prod_{j=1,...,n}\zeta_{j}X_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}}\Big{)}|^{2}\\\
&=\bigcup_{j=1}^{n}\int_{B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})}|\nabla(\zeta_{j}X_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}})|^{2}\\\
&<\bigcup_{j=1}^{n}\Big{\\{}C_{3}\int_{B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})}|\nabla\zeta_{j}|^{2}\\\
&+2\int_{B_{r_{j}}(x_{j})\setminus B_{r_{j}^{2}}(x_{j})}|\nabla(X_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}})|^{2}\Big{\\}},\end{split}$

where $C_{3}$ is a constant depending on $X_{0,i}$. Choosing $r$ such that
$-\frac{2\pi}{\log r}<\frac{c_{0}}{10nC_{3}}$.

(58) $\epsilon_{2}:=\min\\{E(v_{0}\circ
D_{0},B_{\rho}(x)),B_{\rho}(x)\subset\mathcal{T}_{\tau_{0}},\rho\geq
r/2,D_{0}\in S^{1}\times S^{1}\\}.$

We know that $\epsilon_{2}>0$. Then
$d_{B}(u,\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\min\\{\epsilon_{2}^{1/2},\delta(c_{0}/20n)\\}$
implies that

$\int_{\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla(v_{0}\circ
D_{0}\circ\iota_{\tau,\tau_{0}})|^{2}<\min\\{\epsilon_{2}^{1/2},\delta(c_{0}/20)\\},$

by (58) and Claim 3.1 we have the following inequality

(59)
$\begin{split}\bigcup_{j=1}^{n}\Big{\\{}C_{3}\int_{B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})}|\nabla\zeta_{j}|^{2}+2\int_{B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})}|&\nabla(X_{0,i}\circ
D_{0}\circ\iota_{\tau,\tau_{0}})|^{2}\Big{\\}}\\\
&<n\big{(}C_{3}\times\frac{c_{0}}{10nC_{3}}+\frac{c_{0}}{20n}\big{)}\\\
&<\frac{c_{0}}{5}.\end{split}$

By combining all the above estimates we know that if the bubble norm is
bounded by $\epsilon>0$, i.e.,
$d_{B}((u,\mathcal{T}_{\tau}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon$,
then we can choose $\epsilon$ sufficiently small so
$\delta^{2}_{s,t}E_{u}(s\tilde{X}_{0,l_{1}}+t\tilde{X}_{0,l_{2}},T_{\tau})-\delta^{2}_{s,t}E_{v_{0}}(sX_{0,l_{1}}+tX_{0,l_{2}},\mathcal{T}_{\tau_{0}})$
is sufficiently small for all ${l_{1},l_{2}}\subset\\{1,...,n\\}$ thus implies

$-c_{0}\text{Id}<D^{2}_{s_{0}}E(\Pi\circ(u+\sum_{i=0}^{k_{0}}s_{0,i}\tilde{X}_{0,i}),T_{\tau})<-\frac{1}{c_{0}}\text{Id},\quad\forall
s_{0}=(s_{0,1},...,s_{0,k_{0}})\in\bar{B}^{k_{0}},$

which is the desired result. ∎

Claim 3.3 implies that for
$d_{B}((u,\mathcal{T}_{\tau}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon$,
if the body map $v_{0}:\mathcal{T}_{\tau_{0}}\to M$ has Index$(v_{0})=k_{0}$,
then we can construct $\tilde{X}_{0,i}$, $i=1,...,k_{0}$, so that
$d\Pi_{u}(\tilde{X}_{0,i})$ form a $k_{0}$ dimensional subspace of
$\Gamma(u^{-1}TM)$ on which the energy of $u$ is concave. For $v_{j}:S^{2}\to
M$ Index$(v_{j})=k_{j}$, $j=1,...,n$, we can construct $\tilde{X}_{j,i}$
similarly as we construct $\tilde{X}_{0,i}$. The only difference is the cutoff
function $\zeta_{j}$. In Claim 3.3, we argue that
$d_{B}((u,\mathcal{T}_{\tau}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon$
can imply the following term

$|\delta^{2}_{s}E_{u}(s\tilde{X}_{0,i},\cup_{i=1}^{n}B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j}))|$

to be sufficiently small. To do so, we need to bound the term in (56). If we
consider the harmonic sphere $v_{j}:S^{2}\to M$ instead of harmonic torus
$v_{0}$, with Index($v_{j}$)=$k_{j}$ and the corresponding
$\\{X_{i,j}\\}_{i=1}^{k_{j}}\subset\Gamma(v_{j}^{-1}TM)$. We know that from
Claim 3.3 that (56) can bounded by choosing the energy of $u$ on the domain
$\cup_{j=1}^{n}(B_{r_{j}}(x_{j})\setminus B_{r_{j}^{2}}(x_{j}))$ sufficiently
small, and the energy of the cutoff function which has support on
$\cup_{j=1}^{n}(B_{r_{j}}(x_{j})\setminus B_{r_{j}^{2}}(x_{j}))$. If we choose
some constant $r<1$ so that $-\frac{2\pi}{\log r}<\varepsilon$ for some given
$\varepsilon>0$ and let

$\epsilon_{2}:=\min\\{E(v_{0}\circ
D_{0},B_{\rho}(x)),B_{\rho}(x)\subset\mathcal{T}_{\tau_{0}},\rho\geq
r/2,D_{0}\in S^{1}\times S^{1}\\},$

since $\epsilon_{2}>0$, then we have the energy of the cutoff function is
bounded by $\varepsilon$ whenever the energy of the harmonic torus is bounded
by $\epsilon_{2}$ on the domain which the cutoff function is defined on. But
this estimate no longer works for harmonic sphere, since the conformal
automorphism PSL(2,$\mathbb{C}$) of $\mathbb{C}\cup\\{\infty\\}\simeq S^{2}$
is not compact. However, we can overcome this by the conformal invariance
property of energy functional. Now we fix $j\in\\{1,...,n\\}$. We choose $r$
so that $-\frac{2\pi}{\log r}<\varepsilon$ for some given $\varepsilon>0$. Let
$\Pi_{S}$ denote the stereographic projection from
$S^{2}\setminus\\{\text{north pole}\\}$ to $\mathbb{C}$, and let $\tilde{r}$
denote the radius of the geodesic ball $\Pi^{-1}_{S}(B_{r}(0))\subset S^{2}$,
now we consider the following

(60) $\epsilon_{3}:=\min\\{E(v_{j},B_{\rho}(x)),B_{\rho}(x)\subset
S^{2},\rho\geq\tilde{r}/2\\},$

where $B_{\rho}(x)$ is the geodesic ball centered at $x$ with radius $\rho$,
we don’t distinguish it from the ball in $\mathbb{C}$ in abuse of notation.
Then
$d_{B}((u,T_{\tau}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon_{3}^{1/2}$
implies that there exists $\Omega_{j}\subset\mathbb{C}$ and
$D_{j}\in\text{PSL}(2,\mathbb{C})$ (see Definition 2.5) such that

(61) $\Big{(}\int_{\Omega_{j}}|\nabla u-\nabla(v_{j}\circ
D_{j})|^{2}\Big{)}^{1/2}+\Big{(}\int_{\mathbb{C}\setminus\Omega_{j}}|\nabla(v_{j}\circ
D_{j})|^{2}\Big{)}^{1/2}<\frac{\epsilon_{3}^{1/2}}{3}.$

To make the discussion easier, we assume that there’s only one
$B_{r_{l}}(x_{l})\subset B_{r_{j}^{2}}(x_{j})$,

(62) $\begin{split}\Omega_{j}&=B_{r_{j}^{2}}(x_{j})\setminus\Big{\\{}\bigcup
B_{r_{i}}(x_{i}),\>B_{r_{i}}(x_{i})\subset B_{r_{j}^{2}}(x_{j})\Big{\\}}\\\
&=B_{r_{j}^{2}}(x_{j})\setminus B_{r_{l}}(x_{l}).\end{split}$

We can think of this case as the bubble map $v_{j}$ itself has another bubble
map $v_{l}$ (see case 1 in Theorem 2.4). In this case, we need to construct
two cutoff function, one for the bubble map $v_{j}$ and another one for
excluding the bubble map $v_{l}$ of $v_{j}$. We focus on constructing the
later first. So (61) implies that

$\int_{\mathbb{C}\setminus B_{r^{2}_{j}}(x_{j})}|\nabla(v_{j}\circ
D_{j})|^{2}+\int_{B_{r_{l}}(x_{l})}|\nabla(v_{j}\circ
D_{j})|^{2}<\epsilon_{3},$

and by the conformal invariance of energy we have

$\int_{\mathbb{C}\setminus D_{j}\circ B_{r^{2}_{j}}(x_{j})}|\nabla
v_{j}|^{2}+\int_{D_{j}\circ B_{r_{l}}(x_{l})}|\nabla v_{j}|^{2}<\epsilon_{3}.$

By how $\epsilon_{3}$ is chosen, we know that $D_{j}\circ B_{r_{l}}(x_{l})$ is
a ball with radius $r_{l}^{\prime}$ no larger than $r$, say
$B^{\prime}_{r^{\prime}_{l}}(x_{l})=D_{j}\circ B_{r_{l}}(x_{l})$. We can
construct the cutoff function $\zeta_{j,l}=\eta_{j,l}\circ f$ similarly to
(41), but with value $1$ on $\mathbb{C}\setminus D_{j}\circ
B_{r_{l}}(x_{l})=\mathbb{C}\setminus B_{r_{l}^{\prime}}(x_{l})$ instead of
$\mathbb{C}\setminus B_{r_{l}}(x_{l})$, and support on $(\mathbb{C}\setminus
B_{r_{l}^{\prime 2}}(x_{l}))\subset(\mathbb{C}\setminus D_{j}\circ
B_{r_{l}^{2}}(x_{l}))$ i.e.,

(63) $\eta_{j,l}(r)=\left\\{\begin{array}[]{rcl}0,&\mbox{for}&r<r_{l}^{\prime
2},\\\ 2-(\log r)/(\log r_{l}^{\prime}),&\mbox{for}&r_{l}^{\prime 2}\leq r\leq
r_{l}^{\prime},\\\ 1,&\mbox{for}&r>r_{l}^{\prime},\end{array}\right.$

Thus we have

(64) $\int_{D_{j}\circ
B_{r_{l}}(x_{l})}|\nabla\zeta_{j,l}|^{2}=\int_{B_{r_{l}}(x_{l})}|\nabla(\zeta_{j,l}\circ
D_{j})|^{2}<\varepsilon.$

So the inequality of (59) still holds for harmonic spheres. Now we construct
the cutoff function for $v_{j}$. Since (61) implies that

$\int_{\mathbb{C}\setminus B_{r^{2}_{j}}(x_{j})}|\nabla(v_{j}\circ
D_{j})|^{2}=\int_{\mathbb{C}\setminus D_{j}\circ B_{r^{2}_{j}}(x_{j})}|\nabla
v_{j}|^{2}<\epsilon_{3}.$

By how we choose $\epsilon_{3}$, see (60), we know that
$S^{2}\setminus\Pi_{S}^{-1}(D_{j}\circ B_{r^{2}_{j}}(0))$ is a geodesic ball
centered at north pole and has radius smaller than $\tilde{r}$. Since ${r}$ is
the radius of $\Pi^{-1}_{S}(B_{r}(0))$ for some $r$ satisfying the inequality
$-\frac{2\pi}{\log r}<\varepsilon$. Then we can construct the cutoff function
$\zeta_{j,0}$ the same way as we constructed $\zeta_{j,l}$, such that
$\zeta_{j,0}$ has value $1$ on $D_{j}\circ B_{r_{j}^{2}}(x_{j})$, with support
on $D_{j}\circ B_{r_{j}}(x_{j})$, and its energy bounded by $\varepsilon$,
i.e.,

(65) $\int_{D_{j}\circ
B_{r_{j}}(x_{l})}|\nabla\zeta_{j,0}|^{2}=\int_{B_{r_{j}}(x_{j})}|\nabla(\zeta_{j,0}\circ
D_{j})|^{2}<\varepsilon.$

Now we consider the following

(66) $\tilde{X}_{j,i}:=\Big{(}\zeta_{j,0}\zeta_{j,l}X_{j,i}\Big{)}\circ
D_{j},\quad i\in\\{1,...,k_{j}\\}.$

Since $\zeta_{j,0}\circ D_{j}$ and $\zeta_{j,l}\circ D_{j}$ both has support
on $T_{\tau}$, so $\tilde{X}_{j,i}\in C^{\infty}(T_{\tau},\mathbb{R}^{N})$. We
can argue similarly as Claim 3.3 and conclude that

(67)
$-c_{0}\text{Id}<D^{2}_{s_{j}}E(\Pi\circ(u+\sum_{i=1}^{k_{j}}s_{j,i}\tilde{X}_{j,i}),T_{\tau})<-\frac{1}{c_{0}}\text{Id},\quad\forall
s_{j}=(s_{j,1},...,s_{j,k_{j}})\in\bar{B}^{k_{j}}.$

Combining (45) and (67) implies that

(68) $-\frac{1}{c_{0}}\text{Id}\leq D^{2}_{s}E_{u}(s,T_{\tau})\leq-
c_{0}\text{Id},\quad\forall s\in\bar{B}^{k},$

where $k=\sum_{j=0}^{n}k_{j}$, and

$E_{u}(s,T_{\tau})=E\big{(}\Pi\circ\big{(}u+\sum_{i=1,j=1}^{k_{j},n}s_{j,i}\tilde{X}_{j,i}\big{)},T_{\tau}\big{)},\quad
s=(s_{0,1},...,s_{j,k_{j}},s_{j+1,1},...,s_{n,k_{n}})\in\bar{B}^{k},$

which is the desired result (35).

Since $\\{d\Pi_{u}(\tilde{X}_{j,i})\\}$ is a subspace of $\Gamma(u^{-1}TM)$ on
which the energy is concave. So $E_{u}(\cdot,T_{\tau})$ has a unique maximum
on $\bar{B}^{k}$. Since $\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}\\}$ is
a collection of harmonic maps, so for
$E_{v_{0}}(\cdot,\mathcal{T}_{\tau_{0}}):\bar{B}^{k_{0}}\to[0,\infty)$ defined
as

$E_{v_{0}}(s_{0},\mathcal{T}_{\tau_{0}})=E(\Pi\circ\big{(}v_{0}+\sum_{i=1}^{k_{0}}s_{0,i}X_{0,i}\big{)},\mathcal{T}_{\tau_{0}}),\quad
s_{0}=(s_{0,1},...,s_{0,k_{0}})\in\bar{B}^{k_{0}},$

we know that $E_{v_{0}}(\cdot,\mathcal{T}_{\tau_{0}})$ has a unique maximum at
$0\in\bar{B}^{k_{0}}$. Similarly, for
$E_{v_{j}}(\cdot,\mathbb{C}):\bar{B}^{k_{j}}\to[0,\infty)$ defined as

$E_{v_{j}}(s_{j},\mathbb{C})=E(\Pi\circ\big{(}v_{j}+\sum_{i=1}^{k_{j}}s_{j,i}X_{j,i}\big{)},\mathbb{C}),\quad
s_{j}=(s_{j,1},...,s_{j,k_{j}})\in\bar{B}^{k_{j}},\>j=1,...,n,$

$E_{v_{j}}(\cdot,\mathbb{C})$ has a unique maximum at $0\in\bar{B}^{k_{j}}$.
We can argue as in Claim 3.3 and show that
$|\frac{d}{ds}E(\Pi\circ(u+s\tilde{X}_{0,i}),T_{\tau})-\frac{d}{ds}E(\Pi\circ(v_{0}+sX_{0,i}),\mathcal{T}_{\tau_{0}})|$
and
$|\frac{d}{ds}E(\Pi\circ(u+s\tilde{X}_{j,i}),T_{\tau})-\frac{d}{ds}E(\Pi\circ(v_{j}+sX_{j,i}),\mathbb{C})|$,
for $j=1,...,n$, can be sufficiently small so that $E_{u}(\cdot,T_{\tau})$ has
a unique maximum at $m(u)\in B^{k}_{c_{0}/\sqrt{10}}$. ∎

######  Remark 3.4.

The family of vector fields $\\{\tilde{X}_{j,i}\\}$ constructed in Lemma 3.2
depends on $(u,T_{\tau})$. However, by Claim 2.7 there exists $\delta(u)>0$
such that for any $(u^{\prime},\mathcal{T}_{\tau^{\prime}})\in
W^{1,2}(\mathcal{T}_{\tau^{\prime}},M)$ with $|\tau-\tau^{\prime}|<\delta(u)$
and

$\int_{\mathcal{T}_{\tau^{\prime}}}|\nabla
u^{\prime}-\nabla(u\circ\iota_{\tau,\tau^{\prime}})|^{2}<\delta(u),$

we have
$d_{B}((u^{\prime},\mathcal{T}_{\tau^{\prime}}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon$.
Moreover, (23), (24) and (25) hold with the same domains $B_{r_{j}}(x_{j})$
and conformal automorphisms $D_{j}$ for both
$(u^{\prime},\mathcal{T}_{\tau^{\prime}})$ and $(u,\mathcal{T}_{\tau})$. Since
$\\{\tilde{X}_{j,i}\\}$ depends on $B_{r_{j}}(x_{j})$ and $D_{j}$, we know
that the variation of $(u^{\prime},\mathcal{T}_{\tau^{\prime}})$ defined as
the following

$\Pi\circ(u^{\prime}+\tilde{X}_{j,i}\circ\iota_{\tau,\tau^{\prime}})$

also satisfies (35) and (36).

Now we focus on the case where the conformal structure diverges and the limit
maps are possibly several harmonic spheres. Given a collection of harmonic
sheres $\\{v_{j}\\}_{j=1}^{n}$, $v_{j}:S^{2}\to M$. Assume that
$\text{Index}(v_{j})=k_{j}$, by Definition 2.2, there exists subspace
$\\{X_{j,i}\\}_{i=1}^{k_{j}}\subset\Gamma(v_{j}^{-1}M)$ on which the index
form is negative definite, by rescaling $\\{X_{j,i}\\}_{i=1}^{k_{j}}$ (which
we still denote by $\\{X_{j,i}\\}_{i=1}^{k_{j}}$), we have that for

$E_{v_{j}}(s_{j},\mathbb{C})=E(\Pi\circ\big{(}v_{j}+\sum_{i=1}^{k_{j}}s_{j,i}X_{j,i}\big{)},\mathbb{C}),\quad
s_{j}=(s_{j,1},...,s_{j,k_{j}})\in\bar{B}^{k_{j}},\>j=1,...,n,$

there exist $c_{0}<1$ such that

(69)
$-\frac{1}{2c_{0}}\text{Id}<D^{2}_{s_{j}}E_{v_{j}}(s_{j},\mathbb{C})<-2c_{0}\text{Id},\quad\forall
s_{j}\in\bar{B}^{k_{j}},$

for each $j=1,...,n$.

######  Lemma 3.5 (Unstable lemma for degenerated torus).

Let $\\{v_{j}\\}_{j=1}^{n}$, $v_{j}:S^{2}\to M$, be given as above. There
exists $\epsilon>0$, such that if a map $u:\mathcal{T}_{\tau}\to M$,
$\tau\in\mathcal{M}$, $u\in W^{1,2}(T_{\tau},M)$ has sufficiently small bubble
norm with $\\{v_{j}\\}_{j=1}^{n}$, i.e.,

$d_{B}((u,\mathcal{T}_{\tau}),\\{v_{j}\\}_{j=1}^{n})<\epsilon.$

Then we can construct $\\{\tilde{X}_{j,i}\\}\subset
C^{\infty}(T_{\tau},\mathbb{R}^{N})$, so that for the smooth function

$E_{u}(\cdot,T_{\tau}):\bar{B}^{k}\to[0,\infty),$

which is defined by

$E_{u}(s,T_{\tau})=E\big{(}\Pi\circ\big{(}u+\sum_{i=1,j=1}^{k_{j},n}s_{j,i}\tilde{X}_{j,i}\big{)},T_{\tau}\big{)},\quad
s=(s_{1,1},...,s_{j,k_{j}},s_{j+1,1},...,s_{n,k_{n}})\in\bar{B}^{k},$

where $k=\sum_{j=1}^{n}k_{j}$, the following holds:

(70) $-\frac{1}{c_{0}}\text{Id}\leq D^{2}_{s}E_{u}(s,T_{\tau})\leq-
c_{0}\text{Id},\quad\forall s\in\bar{B}^{k}.$

Moreover, $E_{u}(\cdot,T_{\tau})$ has a unique maximum at $m(u)\in
B^{k}_{c_{0}/\sqrt{10}}$ and

(71) $E_{u}(m(u),T_{\tau})-\frac{1}{2c_{0}}|s-m(u)|^{2}\leq
E_{u}(s,T_{\tau})\leq E_{u}(m(u),T_{\tau})-\frac{c_{0}}{2}|s-m(u)|^{2},$

for all $s\in\bar{B}^{k}.$

###### Proof.

From Definition 2.5, we know that
$d_{B}((u,\mathcal{T}_{\tau}),\\{v_{j}\\}_{j=1}^{n})<\epsilon$ implies the
following:

(72)
$\begin{split}\Big{(}\int_{T_{\tau}\setminus\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla
u-&\nabla(v_{1}\circ D_{1})|^{2}\Big{)}^{1/2}\\\
&+\Big{(}\int_{S^{1}\times\mathbb{R}\setminus(T_{\tau}\setminus\bigcup_{j}B_{r_{j}}(x_{j})))}|\nabla(v_{1}\circ
D_{1})|^{2}\Big{)}^{1/2}<\epsilon/3,\end{split}$ (73)
$\sum_{j=1}^{n}\Big{(}\int_{\Omega_{j}}|\nabla u-\nabla(v_{j}\circ
D_{j})|^{2}\Big{)}^{1/2}+\Big{(}\int_{\mathbb{C}\setminus\Omega_{j}}|\nabla(v_{j}\circ
D_{j})|^{2}\Big{)}^{1/2}<\epsilon/3,$ (74)
$\Big{(}\int_{\bigcup_{j}\\{B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})\\}}|\nabla u|^{2}\Big{)}^{1/2}<\epsilon/3.$

So we can argue as Lemma 3.2, use cutoff functions to construct
$\tilde{X}_{j,i}$. For the bubble map $\\{v_{j}\\}_{j=2}^{n}$, (73) and (74)
give us the same condition as (39) and (40), so we can define cutoff functions
as we did in Lemma 3.2 and construct $\\{\tilde{X}_{j,i}\\}$ for $j\geq 2$,
$i\in\\{1,...,k_{j}\\}$. The main difference is (72). We use $(t,\theta)$ as
parameters on $\mathcal{T}_{\tau}$. Let $\theta^{\prime}=\text{arg}(\tau)$,
and let
$z^{\prime}=t+\sqrt{-1}\theta=e^{-\sqrt{-1}(\frac{\pi}{2}-\theta^{\prime})}z$
be another conformal parameter system on $\mathcal{T}_{\tau}$. We can
conformally expand the torus such that the length of the circle of parameter
$\theta$ is one, and we denote the length of the parameter $t$ by $2L$. So we
can regard $u$ as a map defined on $S^{1}\times[-L,L]$. Recall that
$S^{2}\setminus\\{$north pole, south
pole$\\}\simeq\mathbb{C}\setminus\\{0\\}\simeq S^{1}\times\mathbb{R}$ by
conformal transformation. So we may regard the harmonic sphere $v_{1}$ as a
map defined on $S^{1}\times\mathbb{R}$. For $j\geq 2$, we can define
$\zeta_{j}$ similar to (63) with $S^{1}\times\mathbb{R}$ instead of
$\mathbb{C}$, i.e., $\zeta_{j}$ has value $1$ on
$S^{1}\times\mathbb{R}\setminus D_{1}\circ B_{r_{j}}(x_{j})$

$\int_{S^{1}\times[-L,L]}|\nabla(\zeta_{j}\circ
D_{1})|^{2}=\int_{B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})}|\nabla(\zeta_{j}\circ D_{1})|^{2}<-\frac{2\pi}{\log r},$

for some chosen $r$ which can be made sufficiently small when the energy of
$v_{1}\circ D_{1}$ on the annuli $B_{r_{j}}(x_{j})\setminus
B_{r_{j}^{2}}(x_{j})$ is sufficiently small. (See how we choose $\epsilon_{3}$
(60) in Lemma 3.2.) For $j\geq 2$, $\zeta_{j}$ is constructed to exclude the
bubble map $v_{j}$ on $v_{1}$, now we construct the cutoff function
$\zeta_{1}$ for $v_{1}$. (72) implies that

(75)
$\begin{split}\int_{S^{1}\times\mathbb{R}\setminus(T_{\tau}\setminus\bigcup_{j}B_{r_{j}}(x_{j})))}|\nabla(v_{1}\circ
D_{1})|^{2}&\\\ =\int_{S^{1}\times\mathbb{R}\setminus
T_{\tau}}|\nabla(v_{1}\circ
D_{1})|^{2}&+\int_{\bigcup_{j}B_{r_{j}}(x_{j})}|\nabla(v_{1}\circ
D_{1})|^{2}<\epsilon^{2}/9.\end{split}$

So we have

(76) $\int_{S^{1}\times\mathbb{R}\setminus T_{\tau}}|\nabla(v_{1}\circ
D_{1})|^{2}=\int_{D_{1}\circ\\{S^{1}\times\mathbb{R}\setminus
T_{\tau}\\}}|\nabla v_{1}|^{2}<\epsilon^{2}/9.$

We choose $r$ so that $-\frac{2\pi}{\log r^{1/2}}<\varepsilon$ for some given
$\varepsilon>0$. Let $\tilde{v}_{1}=v_{1}\circ e^{z}\circ\Pi_{S}$, so that
$\tilde{v}_{1}$ is defined on $S^{2}\setminus\\{$north pole, south pole$\\}$.
By the removable singularity property of harmonic map we can extend
$\tilde{v}$ to be defined on $S^{2}$. Let $\tilde{r}$ denote the radius of the
geodesic ball $\Pi^{-1}_{S}(B_{r}(0))\subset S^{2}$, now we consider the
following

(77)
$\epsilon_{4}:=\min\Big{\\{}E(\tilde{v}_{1},B_{\rho}(x)),B_{\rho}(x)\subset
S^{2},\rho\geq\tilde{r}/2\Big{\\}},$

where $B_{\rho}(x)$ is the geodesic ball centered at $x$ with radius $\rho$.
We know that since

$\Pi_{S}^{-1}\circ(e^{z})^{-1}\circ\\{S^{1}\times\mathbb{R}\setminus
T_{\tau}\\}=\Pi_{S}^{-1}\circ(e^{z})^{-1}\circ\\{S^{1}\times(-\infty,-L)\cup
S^{1}\times(L,\infty)\\}$

is the two geodesic ball centered at north pole and south pole seperately, we
denote them by $B_{\rho_{1}}(\hat{n})$ and $B_{\rho_{2}}(\hat{s})$. If we
choose $\epsilon$ so that $\epsilon^{2}/9<\epsilon_{4}$, then by (77) we know
that there wouldn’t be any geodesic balls with radius larger than $\tilde{r}$
contained in $D_{1}\circ B_{\rho_{1}}(\hat{n})$ or $D_{1}\circ
B_{\rho_{2}}(\hat{s})$. Now we can define $\zeta_{j}$ which has energy
supported in $D_{1}\circ B_{\rho_{1}}(\hat{n})$ and $D_{1}\circ
B_{\rho_{2}}(\hat{s})$, and with energy smaller than the given $\varepsilon$.
Thus we can construct $\tilde{X}_{1,i}=(\prod_{j=1}^{n}\zeta_{j}X_{1,i})\circ
D_{1}$ and argue similarly as Lemma 3.2. ∎

######  Lemma 3.6.

Let $\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\}$,
$\\{X_{j,i}\\}$, $\epsilon>0$ be given by Lemma 3.2. For any $u\in
W^{1,2}(T_{\tau},M)$, if
$\epsilon/2<d_{B}((u,\mathcal{T}_{\tau}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon$,
then there exists a constant $\nu(\epsilon)>0$ such that if

(78)
$E\big{(}\Pi\circ\big{(}u+\sum_{i=1,j=0}^{k_{j},n}s_{j,i}\tilde{X}_{j,i}\big{)},T_{\tau}\big{)}\leq
E(u,T_{\tau})+\nu(\epsilon),$

for some
$s=(s_{0,1},...,s_{j,k_{j}},s_{j+1,1},...,s_{n,k_{n}})\in\bar{B}^{k}$, where
$\\{X_{j,i}\\}$ is given by Lemma 3.2. Then we have

(79)
$d_{B}((\Pi\circ\big{(}u+\sum_{i=1,j=0}^{k_{j},n}s_{j,i}\tilde{X}_{j,i}\big{)},\mathcal{T}_{\tau}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})>\nu(\epsilon).$

###### Proof.

We argue by contradiction and assume that there’s a sequence of maps
$\\{u_{m}\\}_{m\in\mathbb{N}}$, $u_{m}:\mathcal{T}_{\tau_{m}}\to M$, with
$\epsilon/2<d_{B}((u_{m},\mathcal{T}_{\tau_{m}}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon$,

(80)
$E\big{(}\Pi\circ\big{(}u_{m}+\sum_{i=1,j=0}^{k_{j},n}s^{m}_{j,i}\tilde{X}^{m}_{j,i}\big{)},\mathcal{T}_{\tau_{m}}\big{)}\leq
E(u,\mathcal{T}_{\tau_{m}})+\frac{1}{m},$

for some
$s^{m}=(s^{m}_{0,1},...,s^{m}_{j,k_{j}},s^{m}_{j+1,1},...,s^{m}_{n,k_{n}})\in\bar{B}^{k}$,
where $\\{\tilde{X}^{m}_{j,i}\\}$ is given by Lemma 3.2 for each $u_{m}$, and

(81)
$d_{B}((\Pi\circ\big{(}u_{m}+\sum_{i=1,j=0}^{k_{j},n}s^{m}_{j,i}\tilde{X}^{m}_{j,i}\big{)},\mathcal{T}_{\tau_{m}}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})\to
0,\text{ as }m\to\infty.$

(81) implies that for each $u_{m}$ we can find
$B_{r_{j,m}}(x_{j,m})\subset\mathcal{T}_{\tau_{m}}$ and
$\\{D_{0}^{m}\\}\subset S^{1}\times S^{1}$,
$\\{D_{j}^{m}\\}\subset\text{PSL}(2,\mathbb{C})$, such that

(82)
$\int_{\mathcal{T}_{\tau_{m}}\setminus\bigcup_{j}B_{r_{j,m}}(x_{j,m})}|\nabla(\Pi\circ(u_{m}+\sum_{i=1}^{k_{0}}s^{m}_{0,i}\tilde{X}^{m}_{0,i}))-\nabla(v_{0}\circ
D^{m}_{0}\circ\iota_{\tau_{m},\tau_{0}})|^{2}\to 0,$ (83)
$\int_{\Omega^{m}_{j}}|\nabla(\Pi\circ(u_{m}+\sum_{i=1}^{k_{j}}s^{m}_{j,i}\tilde{X}^{m}_{j,i}))-\nabla(v_{j}\circ
D_{j}^{m})|^{2}\to 0,$

where $\Omega^{m}_{j}=B_{r_{j,m}^{2}}(x_{j,m})\setminus\Big{\\{}\bigcup
B_{r_{i,m}}(x_{i,m}),\>B_{r_{i,m}}(x_{i,m})\subset
B_{r_{j,m}^{2}}(x_{j,m})\Big{\\}},$ and

(84) $\int_{\bigcup_{j}B_{r_{j,m}}(x_{j,m})}|\nabla(v_{0}\circ
D^{m}_{0}\circ\iota_{\tau,\tau_{0}})|^{2}+\int_{\mathbb{C}\setminus\Omega^{m}_{j}}|\nabla(v_{j}\circ
D_{j}^{m})|^{2}\to 0,$ (85) $\int_{\bigcup_{j}\\{B_{r_{j,m}}(x_{j,m})\setminus
B_{r_{j,m}^{2}}(x_{j,m})\\}}|\nabla\Pi\circ\big{(}u_{m}+\sum_{i=1,j=0}^{k_{j},n}s^{m}_{j,i}\tilde{X}^{m}_{j,i}\big{)}|^{2}\to
0,$

as $m\to 0$. By how we construct $\tilde{X}^{m}_{j,i}\in
C^{\infty}(\mathcal{T}_{\tau_{m}},M)$ in Lemma 3.2, we know (84) implies that
the energy of $\tilde{X}^{m}_{j,i}$ tends to $0$ on
$\bigcup_{j}\\{B_{r_{j,m}}(x_{j,m})\setminus B_{r_{j,m}^{2}}(x_{j,m})\\}$ as
$m\to\infty$. (See part 4 in Claim 3.3.) Thus (84) and (85) imply that as
$m\to\infty$ we have

(86) $\int_{\bigcup_{j}\\{B_{r_{j,m}}(x_{j,m})\setminus
B_{r_{j,m}^{2}}(x_{j,m})\\}}|\nabla u_{m}|^{2}\to 0.$

Assume that $\lim_{m\to\infty}D_{0}^{m}=$Id. For almost all
$x\in\mathcal{T}_{\tau_{0}}$, (82) implies that

$\lim_{m\to\infty}\Pi\circ\big{(}u_{m}\circ\iota_{\tau_{0},\tau_{m}}(x)+\sum_{i=1}^{k_{0}}(s^{m}_{0,i}\tilde{X}^{m}_{0,i})\circ\iota_{\tau_{0},\tau_{m}}(x)\big{)}=v_{0}(x).$

(83) implies that

$\lim_{m\to\infty}\Pi\circ\big{(}u_{m}\circ(D^{m}_{j})^{-1}(x)+\sum_{i=1}^{k_{j}}(s^{m}_{j,i}\tilde{X}^{m}_{j,i}\circ(D^{m}_{j})^{-1}(x))\big{)}=v_{j}(x),\quad
j\in\\{1,...,n\\},$

for almost all $x\in\mathbb{C}$. We recall that $x-\Pi(x)\in
T_{\Pi(x)}^{\perp}M$, then there exist functions
$\sigma_{0}:\mathcal{T}_{\tau_{0}}\to T_{v_{0}}^{\perp}M$,
$\sigma_{j}:\mathbb{C}\to T_{v_{j}}^{\perp}M$, such that

$\lim_{m\to\infty}u_{m}\circ\iota_{\tau_{0},\tau_{m}}(x)+\sum_{i=1}^{k_{0}}(s^{m}_{0,i}\tilde{X}^{m}_{0,i})\circ\iota_{\tau_{0},\tau_{m}}(x)=v_{0}(x)+\sigma_{0}(x),$

and

$\lim_{m\to\infty}u_{m}\circ(D^{m}_{j})^{-1}(x)+\sum_{i=1}^{k_{0}}(s^{m}_{j,i}\tilde{X}^{m}_{j,i})\circ(D^{m}_{j})^{-1}(x)=v_{j}(x)+\sigma_{j}(x).$

Now we consider $Y^{m}_{0}=u_{m}\circ\iota_{\tau_{0},\tau_{m}}-v_{0}$ and
$Y^{m}_{j}=u_{m}\circ(D^{m}_{j})^{-1}-v_{j}$, since $d\Pi_{v_{0}}$ is the
orthogonal projection of $\mathbb{R}^{N}$ onto $T_{v_{0}}M$, so

(87)
$\begin{split}\lim_{m\to\infty}d\Pi_{v_{0}}(-Y^{m}_{0})(x)&=\lim_{m\to\infty}d\Pi_{v_{0}}\big{(}\sum_{i=1}^{k_{0}}(s^{m}_{0,i}\tilde{X}^{m}_{0,i})\circ\iota_{\tau_{0},\tau_{m}}(x)-\sigma_{0}(x)\big{)}\\\
&=\lim_{m\to\infty}d\Pi_{v_{0}}\big{(}\sum_{i=1}^{k_{0}}(s^{m}_{0,i}\tilde{X}^{m}_{0,i})\circ\iota_{\tau_{0},\tau_{m}}(x)\big{)}\\\
&=d\Pi_{v_{0}}\big{(}\sum_{i=1}^{k_{0}}(s_{0,i}X_{0,i})(x)\big{)},\end{split}$

where $s_{0,i}:=\lim_{m\to\infty}s^{m}_{0,i}$, and similarly we have

(88)
$\lim_{m\to\infty}d\Pi_{v_{j}}(-Y^{m}_{j})(x)=d\Pi_{v_{j}}\big{(}\sum_{i=1}^{k_{j}}(s_{j,i}X_{j,i})(x)\big{)},$

where $s_{j,i}:=\lim_{m\to\infty}s^{m}_{j,i}.$

(89)
$\begin{split}\lim_{m\to\infty}E(u_{m})&=\lim_{m\to\infty}\Big{\\{}\sum_{j=1}^{n}\int_{\Omega_{j}^{m}}|\nabla(\Pi\circ(v_{j}\circ(D^{m}_{j})^{-1}+(u_{m}-v_{j}\circ
D^{m}_{j}))|^{2}\\\
&+\int_{{\mathcal{T}_{\tau_{m}}\setminus\bigcup_{j}B_{r_{j,m}}(x_{j,m})}}|\nabla(\Pi\circ(v_{0}\circ\iota_{\tau_{m},\tau_{0}}+(u_{m}-v_{0}\circ\iota_{\tau_{m},\tau_{0}}))|^{2}\\\
&+\int_{\bigcup_{j}\\{B_{r_{j,m}}(x_{j,m})\setminus
B_{r_{j,m}^{2}}(x_{j,m})\\}}|\nabla u_{m}|^{2}\Big{\\}}\\\
&=\lim_{m\to\infty}\Big{\\{}\sum_{j=1}^{n}\int_{D^{m}_{j}\circ\Omega_{j}^{m}}|\nabla(\Pi\circ(v_{j}+Y^{m}_{j})|^{2}\\\
&+\int_{{\mathcal{T}_{\tau_{m}}\setminus\bigcup_{j}B_{r_{j,m}}(x_{j,m})}}|\nabla(\Pi\circ(v_{0}+Y^{m}_{0})\circ\iota_{\tau_{m},\tau_{0}}|^{2}\Big{\\}}\\\
&\leq
E(\Pi\circ(v_{0}+Y^{m}_{0}),\mathcal{T}_{\tau_{0}})+\sum_{j=1}^{n}E(\Pi\circ(v_{j}+Y^{m}_{j}),\mathbb{C})\\\
&\leq
E(v_{0},\mathcal{T}_{\tau_{0}})+\sum_{j=1}^{n}E(v_{j},\mathbb{C}).\end{split}$

Since $\\{X_{j,i}\\}_{i=1}^{k_{j}}$ is the subspace on $\Gamma(v_{j}^{-1}TM)$
on which the index form is negative definite, so the last inequality of (89)
follows from (87) and (88), and the equality holds if and only if $s_{0,i}=0$
and $s_{j,i}=0$. On the other hand, by (80) and (81) we have the following

(90)
$\begin{split}E(v_{0},\mathcal{T}_{\tau_{0}})+\sum_{i=1}^{n}E(v_{i},\mathbb{C})=\lim_{m\to\infty}E(\Pi\circ\big{(}u_{m}+&\sum_{i=1,j=0}^{k_{j},n}s^{m}_{j,i}\tilde{X}^{m}_{j,i}\big{)},\mathcal{T}_{\tau_{m}})\\\
&\leq\lim_{m\to\infty}E(u_{m},\mathcal{T}_{\tau_{m}}),\end{split}$

which forces $s_{0,i}=0$ and $s_{j,i}=0$. Thus we know that

$\lim_{m\to\infty}d_{B}((u_{m},\mathcal{T}_{\tau_{m}}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})=0,$

which is the desired contradiction. ∎

##  4\. Min-Max Minimal Torus

Let $\tilde{\Lambda}=\\{(\gamma(t),\mathcal{T}_{\tau(t)});\gamma(t)\in
C^{0}([0,1],C^{0}\cap W^{1,2}(\mathcal{T}_{\tau(t)},M))\\}$, $\tau(t)\in
C^{0}([0,1],\mathcal{M})$, and $\Lambda=\\{\gamma(t)\in C^{0}([0,1],C^{0}\cap
W^{1,2}(T_{0},M))\\}$. We assume $\gamma(0)$ and $\gamma(1)$ are constant
mappings or map the torus to some circles in $M$. And $\tau(0)$,
$\tau(1)=\sqrt{-1}$, if mappings on the endpoints are constant mappings and
not restrained if not.

For the area functional, we only need to consider the variational problem in
the space $\Lambda$, since changing domain metrics will not change the area.
But for the energy functional, different conformal structures may lead to
different energy, so we have to consider the variational problem in the space
$\tilde{\Lambda}$. Fix a homotopically nontrivial path $\beta\in\Lambda$. Let
$[\beta]$ be the homotopy class of $\beta$ in $\Lambda$. Since the path
$(\beta(t),\mathcal{T}_{\tau_{\beta}(t)})\in\tilde{\Lambda}$ may have
different domains $\mathcal{T}_{\tau_{\beta}(t)}$ the homotopy equivalence
$\alpha\sim\beta$ of $\alpha(t):\mathcal{T}_{\tau_{\alpha}(t)}\to M$ and
$\beta(t):\mathcal{T}_{\tau_{\beta}(t)}\to M$ is defined as follows. We can
identify $\mathcal{T}_{\tau_{\alpha}(t)}$ and $\mathcal{T}_{\tau_{\beta}(t)}$
to $T_{0}$ by $\iota_{\tau_{\alpha}(t)}$ and $\iota_{\tau_{\beta}(t)}$, then
we can view $\alpha(t)$ and $\beta(t)$ as mappings defined on the same domain
$T_{0}$ and hence define their homotopy equivalence.

######  Definition 4.1 (Width).

Let

$W=\inf_{\gamma\in[\beta]}\max_{t\in[0,1]}\text{Area}(\gamma(t)),$

and

$W_{E}=\inf\Big{\\{}\max_{t\in[0,1]}E(\gamma,\mathcal{T}_{\tau(t)});(\gamma(t),\mathcal{T}_{\tau(t)})\in[\beta(t),\mathcal{T}_{\tau_{\beta}(t)}]\Big{\\}}.$

######  Claim 4.2.

[Zho10, Remark 3.2] $W=W_{E}$.

######  Theorem 4.3.

[Zho10, Theorem 1.1] For any homotopically nontrivial path $\beta\in\Lambda$,
if $W>0$, there exists a sequence
$(\rho_{l}(t),\mathcal{T}_{\tau_{l}(t)})\in[\beta]$, with
$\max_{t\in[0,1]}E(\rho_{l}(t),\mathcal{T}_{\tau_{l}(t)})\to W$, and
$\forall\epsilon>0$, there exist $N>0$ and $\delta>0$ such that if $l>N$, then
for any $t\in(0,1)$ satisfying

$E(\rho_{l}(t),\mathcal{T}_{\tau_{l}(t)})>W-\delta,$

there are possibly a conformal harmonic torus $v_{0}:\mathcal{T}_{\tau_{0}}\to
M$ and finitely many harmonic spheres $v_{j}:S^{2}\to M$ such that

$d_{V}(\rho_{l}(t),\cup_{j}v_{j})\leq\epsilon,$

where $d_{V}$ means varifold distance. (See [Zho10].)

######  Definition 4.4 (Minimizing sequence).

We call
$\\{(\gamma_{l}(t),\mathcal{T}_{\tau(t)})\\}_{l\in\mathbb{N}}\in\tilde{\Lambda}$
a minimizing sequence if

$\lim_{l\to\infty}\max_{t\in[0,1]}E(\gamma_{l}(t),\mathcal{T}_{\tau(t)})=W.$

######  Theorem 4.5.

For any minimizing sequence
$\\{(\gamma_{l}(t),\mathcal{T}_{\tau(t)})\\}_{l\in\mathbb{N}}$. There exist
$\\{i_{j}\\}\to\infty$ and $t_{i_{j}}\in[0,1]$ such that one of the following
case holds:

  1. (1)

there exist a conformal harmonic torus $v_{0}:\mathcal{T}_{\tau_{0}}\to M$ and
finitely many harmonic spheres $v_{j}:S^{2}\to M$ such that

$d_{B}((\gamma_{i_{j}}(t_{i_{j}}),\mathcal{T}_{\tau_{i_{j}}(t)}),\\{(v_{0},\mathcal{T}_{\tau_{0}},\\{v_{j}\\})\\})\to
0,$

  2. (2)

there exist finitely many harmonic spheres $v_{j}:S^{2}\to M$ such that

$d_{B}((\gamma_{i_{j}}(t_{i_{j}}),\mathcal{T}_{\tau_{i_{j}}(t)}),\\{v_{j}\\})\to
0.$

##  5\. Deformation Theorem

The metric $g$ is called bumpy if there’s no prime compact parametrized
minimal surface $f:\Sigma\to M$ with branch points, nontransversal crossings
or intersections, and non-trivial Jacobi field other than the Jacobi field
generated by the group of conformal automorphisms of $\Sigma.$ (See Definition
A.16.) It’s proved by Moore [Moo17] that bumpy metrics are generic in the
Baire sense. Let $\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\}$,
$\\{X_{j,i}\\}$ be given as in section 3. Moreover, we assume that the sum of
energy is equivalent to width (see Definition 4.1), i.e.,
$E(v_{0},\mathcal{T}_{\tau_{0}})+\sum_{j=1}^{n}E(v_{j},\mathbb{C})=W$. Suppose
$(M,g)$ is a bumpy metric and the given harmonic torus
$(v_{0},\mathcal{T}_{\tau_{0}})$ is conformal so its image is minimal. Since
$g$ is bumpy, $v_{j}$ has only transversal intersections, so there exists a
positive constant $C\leq 1$ such that $\int_{\Omega}|\nabla v_{j}-\nabla
v_{i}|^{2}>C^{2}\int_{\Omega}|\nabla v_{j}|^{2}$. Now we consider
$\gamma(t)\in C^{0}([0,1],C^{0}\cap W^{1,2}(\mathcal{T}_{\tau(t)},M))$, and
$\tau(t)\in C^{0}([0,1],\mathcal{M})$. Let

$\mathcal{I}_{\epsilon}=\\{t\in[0,1],d_{B}((\gamma(t),\mathcal{T}_{\tau(t)}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<\epsilon\\},$

where $\epsilon>0$ is the constant given in Lemma 3.2. We assume that
$\epsilon>0$ is smaller than the energy lower bound for a nontrivial harmonic
map in $M$.

######  Claim 5.1.

For $t\in\mathcal{I}_{C\epsilon/10}$, where $C$ is a constant which depends on
$\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\}$ and $\epsilon>0$
is the constant given in Lemma 3.2. We can choose $\\{X_{j,i}(t)\\}\subset
C^{\infty}(\mathcal{T}_{\tau(t)},\mathbb{R}^{N})$ such that it varies
continuously with respect to $t$, and the variation of $\gamma(t)$ defined as
the following

$\Pi\circ\big{(}\gamma(t)+\sum_{i=1,j=0}^{k_{j},n}s_{j,i}X_{j,i}(t)\big{)},\quad
s=(s_{0,1},...,s_{j,k_{j}},s_{j+1,1},...,s_{n,k_{n}})\in\bar{B}^{k},$

satisfies (35) and (36).

###### Proof.

Lemma 3.2 implies that for each $t\in\mathcal{I}_{\epsilon}$ there exist
$\\{X_{j,i}(t)\\}\subset C^{\infty}(\mathcal{T}_{\tau(t)},\mathbb{R}^{N})$
such that (35) and (36) hold. By how we construct $\\{X_{j,i}(t)\\}$ we know
that it depends on the pairwise disjoint domains $\Omega_{j}(t)$ and conformal
automorphisms $D_{j}(t)$. If we can find domains $\Omega_{j}(t)$ and conformal
automorphisms $D_{j}(t)$ that vary continuous with respect to $t$, and satisfy
the inequalities (23), (24) and (25). Then we can construct $\\{X_{j,i}(t)\\}$
which varies continuously with respect to $t$.

For each $a\in\mathcal{I}$, there exists $\delta(a)>0$ such that
$(a-\delta(a),a+\delta(a))\subset\mathcal{I}$ by Claim 2.7. We can cover
$\mathcal{I}$ with countably many such open intervals. Moreover, after
discarding some of the intervals, we can arrange that each $t\in\mathcal{I}$
is in at least one open interval, each interval intersects at most two others,
and the two intervals intersecting the same interval do not intersect each
other.

We assume that there’s only one bubble map, i.e.,
$\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\}=\\{(v_{0},\mathcal{T}_{\tau_{0}}),v_{1}\\}$
and we choose some
$t\in(a-\delta(a),a+\delta(a))\cap(b-\delta(b),b+\delta(b))$. There exist
$\Omega_{a},\Omega_{b}\subset\mathcal{T}_{\tau(t)}$, $D_{a}$,
$D_{b}\in$PSL$(2,\mathbb{C})$, and $D_{0}^{a},D_{0}^{b}\in S^{1}\times S^{1}$
such that (23), (24) and (25) hold. It’s easy to see that
$\Omega_{a}\cap\Omega_{b}\neq\emptyset$. If not, then

$\begin{split}\Big{(}\int_{\Omega_{a}}|\nabla(v_{1}\circ D_{a})-\nabla
v_{0}|^{2}\Big{)}^{1/2}&<\Big{(}\int_{\Omega_{a}}|\nabla(v_{1}\circ
D_{a})-\nabla\gamma(t)|^{2}\Big{)}^{1/2}\\\
&+\Big{(}\int_{\Omega_{a}}|\nabla\gamma(t)-v_{0}|^{2}\Big{)}^{1/2}\\\
&<C\epsilon/5.\end{split}$

However, since we assume that $\epsilon$ is smaller than the lower bound of
energy for nontrivial harmonic map, we have

$\Big{(}\int_{\Omega_{a}}|\nabla(v_{1}\circ
D_{a})|^{2}\Big{)}^{1/2}>\Big{(}\int_{\mathbb{C}}|\nabla(v_{1}\circ
D_{a})|^{2}\Big{)}^{1/2}-\Big{(}\int_{\mathbb{C}\setminus\Omega_{a}}|\nabla(v_{1}\circ
D_{a})|^{2}\Big{)}^{1/2}>9\epsilon/10.$

Which contradicts that $\int_{\Omega_{a}}|\nabla v_{1}\circ D_{a}-\nabla
v_{0}|^{2}>C^{2}\int_{\Omega_{a}}|\nabla v_{a}\circ D_{a}|^{2}$. Next, we
observe that

$\begin{split}\Big{(}\int_{\mathbb{C}}|\nabla(v_{1}\circ
D_{a})&-\nabla(v_{1}\circ
D_{b})|^{2}\Big{)}^{1/2}\leq\Big{(}\int_{\mathbb{C}\setminus\Omega_{a}}|\nabla(v_{1}\circ
D_{a})|^{2}\Big{)}^{1/2}\\\
&+\Big{(}\int_{\mathbb{C}\setminus\Omega_{b}}|\nabla(v_{1}\circ
D_{b})|^{2}\Big{)}^{1/2}+\Big{(}\int_{\Omega_{a}\cap\Omega_{b}}|\nabla(v_{1}\circ
D_{a})-\nabla(v_{1}\circ D_{b})|^{2}\Big{)}^{1/2}\\\
&+\Big{(}\int_{\Omega_{a}\setminus\Omega_{b}}|\nabla(v_{1}\circ
D_{a})|^{2}\Big{)}^{1/2}+\Big{(}\int_{\Omega_{b}\setminus\Omega_{a}}|\nabla(v_{1}\circ
D_{b})|^{2}\Big{)}^{1/2}.\end{split}$

Since $\Big{(}\int_{\mathbb{C}\setminus\Omega_{a}}|\nabla(v_{1}\circ
D_{a})|^{2}\Big{)}^{1/2}+\Big{(}\int_{\mathbb{C}\setminus\Omega_{b}}|\nabla(v_{1}\circ
D_{b})|^{2}\Big{)}^{1/2}$ is bounded by $\epsilon/5$ by (24), and

$\begin{split}\Big{(}\int_{\Omega_{a}\cap\Omega_{b}}|\nabla(v_{1}\circ
D_{a})-\nabla(v_{1}\circ D_{b})|^{2}\Big{)}^{1/2}&\\\
<\Big{(}\int_{\Omega_{a}}|\nabla\gamma(t)-\nabla(v_{1}\circ
D_{a})|^{2}\Big{)}^{1/2}&+\Big{(}\int_{\Omega_{b}}|\nabla\gamma(t)-\nabla(v_{1}\circ
D_{b})|^{2}\Big{)}^{1/2}<\epsilon/5.\end{split}$

We’re left to deal with
$\Big{(}\int_{\Omega_{a}\setminus\Omega_{b}}|\nabla(v_{1}\circ
D_{a})|^{2}\Big{)}^{1/2}+\Big{(}\int_{\Omega_{b}\setminus\Omega_{a}}|\nabla(v_{1}\circ
D_{b})|^{2}\Big{)}^{1/2}$. Since

$\begin{split}\Big{(}\int_{\Omega_{a}\setminus\Omega_{b}}|\nabla(v_{1}\circ
D_{a})-\nabla
v_{0}|^{2}\Big{)}^{1/2}&<\Big{(}\int_{\Omega_{a}\setminus\Omega_{b}}|\nabla(v_{1}\circ
D_{a})-\nabla\gamma(t)|^{2}\Big{)}^{1/2}\\\
&+\Big{(}\int_{\mathbb{C}\setminus\Omega_{b}}|\nabla\gamma(t)-v_{0}|^{2}\Big{)}^{1/2}\\\
&<\epsilon/5,\end{split}$

which implies $\Big{(}\int_{\Omega_{a}\setminus\Omega_{b}}|\nabla(v_{1}\circ
D_{a})|^{2}\Big{)}^{1/2}$ is bounded by $\epsilon/5$. Similarly, we can bound
$\Big{(}\int_{\Omega_{b}\setminus\Omega_{a}}|\nabla(v_{1}\circ
D_{b})|^{2}\Big{)}^{1/2}<\epsilon/5$. So we can bound
$\Big{(}\int_{\mathbb{C}}|\nabla(v_{1}\circ D_{a})-\nabla(v_{1}\circ
D_{b})|^{2}\Big{)}^{1/2}<\epsilon$, which implies that $\Omega_{a}$ is close
to $\Omega_{b}$ in PSL$(2,\mathbb{C})$, so we can define a continuous function
$D:[a,b]\to$ PSL$(2,\mathbb{C})$ with $D(a)=D_{a}$ and $D(b)=D_{b}$, such that

$\int_{\mathbb{C}\setminus\Omega_{a}\cap\Omega_{b}}|\nabla(v_{1}\circ
D(t))|^{2}<\epsilon,\quad\forall t\in[a,b],$

and define a continuous function $\Omega:[a,b]\to\mathbb{C}$,
$\Omega_{a}\cap\Omega_{b}\subseteq\Omega(t)\subseteq\Omega_{a}\cup\Omega_{b}$,
with $\Omega(a)=\Omega_{a}$ and $\Omega(b)=\Omega_{b}$, such that for
$\gamma(t)$ the inequality (24) holds with $\Omega(t)$ and
$D(t)\in\text{PSL}(2,\mathbb{C})$. Similarly, we can construct a continuous
function $D_{0}(t)\in S^{1}\times S^{1}$ such that the inequality (23) holds
with $D_{0}(t)$ and $\mathcal{T}_{\tau(t)}$. Then we can construct
$\\{X_{j,i}(t)\\}$ which varies continuously with respect to $t$. ∎

For a given $\gamma(t)\in C^{0}([0,1],C^{0}\cap
W^{1,2}(\mathcal{T}_{\tau(t)},M))$, and $\tau(t)\in C^{0}([0,1],\mathcal{M})$.
Let $I\subset\mathcal{I}_{C\epsilon/10}$ be a closed interval. We can
construct $\\{X_{j,i}(t)\\}\subset
C^{\infty}(\mathcal{T}_{\tau(t)},\mathbb{R}^{N})$ such that it varies
continuously with respect to $t\in I$ By Claim 5.1. The variation of
$\gamma(t)$ defined as the following

$\Pi\circ\big{(}\gamma(t)+\sum_{i=1,j=0}^{k_{j},n}s_{j,i}X_{j,i}(t)\big{)},\quad
s=(s_{0,1},...,s_{j,k_{j}},s_{j+1,1},...,s_{n,k_{n}})\in\bar{B}^{k},$

and we define the smooth function
$E_{\gamma(t)}(\cdot,\mathcal{T}_{\tau(t)}):\bar{B}^{k}\to[0,\infty)$ by

$E_{\gamma(t)}(s,\mathcal{T}_{\tau(t)})=E\big{(}\Pi\circ\big{(}\gamma(t)+\sum_{i=1,j=0}^{k_{j},n}s_{j,i}X_{j,i}(t)\big{)},\mathcal{T}_{\tau(t)}\big{)},$

where $s=(s_{0,1},...,s_{j,k_{j}},s_{j+1,1},...,s_{n,k_{n}})\in\bar{B}^{k}$.
By Claim 5.1 we know that $E_{\gamma(t)}(\cdot,\mathcal{T}_{\tau(t)})$
satisfies (35) and (36). Moreover, the maximum $m(\gamma(t))\in
B_{c_{0}/\sqrt{10}}^{k}$ of $E_{\gamma(t)}(\cdot,\mathcal{T}_{\tau(t)})$
varies continuously with respect to $t$. We consider the one-parameter flow

$\displaystyle\\{\phi_{t}(\cdot,x)\\}_{x\geq 0}$
$\displaystyle\in\text{Diff}(\bar{B}^{k})$
$\displaystyle\phi_{t}(\cdot,\cdot):\bar{B}^{k}$
$\displaystyle\times[0,\infty)\to\bar{B}^{k},$

generated by the vector field:

(91) $s\mapsto-(1-|s|^{2})\nabla
E_{\gamma(t)}(s,\mathcal{T}_{\tau(t)}),\>s\in\bar{B}^{k}.$

######  Claim 5.2.

For all $\kappa<\frac{1}{4}$, there is $T$ depending on
$\big{\\{}\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\},\\{X_{j,i}\\},\kappa,\epsilon\big{\\}}$
so that for any $t\in I\subset\mathcal{I}_{C\epsilon/10}$, and
$\chi\in\bar{B}^{k}$ with $|\chi-m(\gamma(t))|\geq\kappa$ we have:

(92)
$E_{\gamma(t)}(\phi_{t}(\chi,T),\mathcal{T}_{\tau(t)})<E_{\gamma(t)}(0,\mathcal{T}_{\tau(t)})-\frac{c_{0}}{10}.$

###### Proof.

By $m(\gamma(t))\in B^{k}_{\frac{c_{0}}{\sqrt{10}}}(0)$ and (36), we have:

(93)
$\sup_{s\in\bar{B}^{k}}E_{\gamma(t)}(s,\mathcal{T}_{\tau(t)})=E_{\gamma(t)}(m(\gamma(t)),\mathcal{T}_{\tau(t)})\leq
E_{\gamma(t)}(0,\mathcal{T}_{\tau(t)})+\frac{c_{0}}{20},\>\forall t\in I.$

So, to prove (92), it suffices to show the existence of $T$ such that

$|\chi-m(\gamma(t))|\geq\kappa\implies
E_{\gamma(t)}(\phi_{t}(\chi,T),\mathcal{T}_{\tau(t)})<\sup_{s\in\bar{B}^{k}}E_{\gamma(t)}(s,\mathcal{T}_{\tau(t)})-\frac{c_{0}}{5}.$

We argue by contradiction and assume that there exists a constant
$\frac{1}{4}>\kappa>0$, a sequence $\\{t_{l}\\}_{l\in\mathbb{N}}\subset I$,
and $\\{s_{l}\\}_{l\in\mathbb{N}}\subset\bar{B}^{k}$ with
$|s_{l}-m(\gamma(t_{l}))|\geq\kappa$ such that

(94) $E_{\gamma(t_{l})}(\phi_{t_{l}}(s_{l},l),\mathcal{T}_{\tau(t_{l})})\geq
E_{\gamma(t_{l})}(0,\mathcal{T}_{\tau(t_{l})})-\frac{c}{10}.$

Combining (94) with (93) we have
$E_{\gamma(t_{l})}(\phi_{t_{l}}(s_{l},l),\mathcal{T}_{\tau(t_{l})})\geq
E_{\gamma(t_{l})}(m(\gamma(t_{l})),\mathcal{T}_{\tau(t_{l})})-\frac{c_{0}}{5}$.
Since $\phi_{t}(\cdot,\cdot)$ is an energy decreasing flow, we have

$E_{\gamma(t_{l})}(\phi_{t_{l}}(s_{l},x),\mathcal{T}_{\tau(t_{l})})\geq
E_{\gamma(t_{l})}(\phi_{t_{l}}(s_{l},l),\mathcal{T}_{\tau(t_{l})})\geq
E_{\gamma(t_{l})}(m_{j}(t_{l}),\mathcal{T}_{\tau(t_{l})})-\frac{c_{0}}{5},\>\forall
0\leq x\leq l.$

Since both $I$ and $\bar{B}^{k}$ are compact, we obtain subsequential limits
$t^{\prime}\in I$ and $s\in\bar{B}^{k}$ with

(95)
$E_{\gamma(t^{\prime})}(\phi_{t^{\prime}}(s,x),\mathcal{T}_{\tau(t^{\prime})})\geq\sup_{|v|\leq
1}E_{\gamma(t^{\prime})}(v,\mathcal{T}_{\tau(t^{\prime})})-\frac{c_{0}}{5},\quad\forall
x\geq 0.$

Since $\gamma(t)\mapsto m(\gamma(t))$ is a continuous map,
$|s_{l}-m(\gamma(t_{l}))|\geq\kappa$ implies
$|s-m(\gamma(t^{\prime}))|\geq\kappa$. Thus we have
$\lim_{x\to\infty}|\phi_{t^{\prime}}(s,x)|=1$ and thus we deduce from the
equation (95) that

(96)
$\sup_{|v|=1}E_{\gamma(t^{\prime})}(v,\mathcal{T}_{\tau(t^{\prime})})\geq\sup_{|v|\leq
1}E_{\gamma(t^{\prime})}(v,\mathcal{T}_{\tau(t^{\prime})})-\frac{c_{0}}{5}.$

On the other hand, $m(\gamma(t^{\prime}))\in
B^{k}_{\frac{c_{0}}{\sqrt{10}}}(0)$ implies $|v-m(\gamma(t^{\prime}))|>2/3$
for all $v\in\bar{B}^{k}$ with $|v|=1$. Hence, we have

$\sup_{|v|=1}E_{\gamma(t^{\prime})}(v,\mathcal{T}_{\tau(t^{\prime})})\leq\sup_{|v|\leq
1}E_{\gamma(t^{\prime})}(v,\mathcal{T}_{\tau(t^{\prime})})-\frac{c_{0}}{2}\Big{(}\frac{2}{3}\Big{)}^{2}<\sup_{|v|\leq
1}E_{\gamma(t^{\prime})}(v,\mathcal{T}_{\tau(t^{\prime})})-\frac{c_{0}}{5},$

which gives us the desired contradiction. ∎

######  Theorem 5.3 (Deformation Theorem).

Let $(M,g)$, $\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}\\}$ and
$\\{X_{j,i}\\}$ be given as above. If $\sum_{j=0}^{n}\text{Index}(v_{j})=k\geq
2,$ then for a given minimizing sequence
$\\{(\gamma_{l},\mathcal{T}_{\tau_{l}(t)})\\}_{l\in\mathbb{N}}\subset
C^{0}([0,1],C^{0}\cap W^{1,2}(\mathcal{T}_{\tau_{l}(t)},M))$, and
$\tau_{l}(t)\in C^{0}([0,1],\mathcal{M})$. There exists a sequence
$\\{(\gamma^{\prime}_{l},\mathcal{T}_{\tau_{l}(t)})\\}_{l\in\mathbb{N}}\subset
C^{0}([0,1],C^{0}\cap W^{1,2}(\mathcal{T}_{\tau_{l}(t)},M))$,
$l_{0}\in\mathbb{N}$ and $\bar{\epsilon}>0$ such that

  1. (1)

$(\gamma^{\prime}_{l},\mathcal{T}_{\tau_{l}(t)})$ is homotopic to
$(\gamma^{\prime}_{l},\mathcal{T}_{\tau_{l}(t)})$,

  2. (2)

$d_{B}((\gamma^{\prime}_{l}(t),\mathcal{T}_{\tau_{l}(t)}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}\\})\geq\bar{\epsilon}$
for all $l\geq l_{0},$ $t\in[0,1]$

###### Proof.

Let

$\mathcal{I}_{l,\epsilon}=\\{t\in[0,1],d_{B}((\gamma_{l}(t),\mathcal{T}_{\tau_{l}(t)}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}_{j=1}^{n}\\})<C\epsilon/10\\},$

where the constant $C$ is given in Claim 5.1. According to Claim 5.1, for
$t\in\mathcal{I}_{l,\epsilon}$, we can choose $\\{X^{l}_{j,i}(t)\\}\subset
C^{\infty}(\mathcal{T}_{\tau_{l}(t)},\mathbb{R}^{N})$ such that it varies
continuously with respect to $t$. We define the variation of $\gamma_{l}(t)$
as the following

$\Pi\circ\big{(}\gamma_{l}(t)+\sum_{i=1,j=0}^{k_{j},n}s_{j,i}X^{l}_{j,i}(t)\big{)},\quad
s=(s_{0,1},...,s_{j,k_{j}},s_{j+1,1},...,s_{n,k_{n}})\in\bar{B}^{k},$

and we define the smooth function
$E_{\gamma_{l}(t)}(\cdot,\mathcal{T}_{\tau_{l}(t)}):\bar{B}^{k}\to[0,\infty)$
by

$E_{\gamma_{l}(t)}(s,\mathcal{T}_{\tau_{l}(t)})=E\big{(}\Pi\circ\big{(}\gamma_{l}(t)+\sum_{i=1,j=0}^{k_{j},n}s_{j,i}X^{l}_{j,i}(t)\big{)},\mathcal{T}_{\tau_{l}(t)}\big{)},$

where $s=(s_{0,1},...,s_{j,k_{j}},s_{j+1,1},...,s_{n,k_{n}})\in\bar{B}^{k}$.
By Claim 5.1 we know that $E_{\gamma_{l}(t)}(\cdot,\mathcal{T}_{\tau_{l}(t)})$
satisfies (35) and (36). Moreover, the maximum $m(\gamma_{l}(t))\in
B_{c_{0}/\sqrt{10}}^{k}$ of
$E_{\gamma_{l}(t)}(\cdot,\mathcal{T}_{\tau_{l}(t)})$ varies continuously with
respect to $t$. We define a continuous homotopy:

$H^{\prime}_{l}:\mathcal{I}_{l,\epsilon}\times[0,1]\longrightarrow
B^{k}_{1/2^{l}}(0),$

so that

$H^{\prime}_{l}(t,0)=0,\>\text{and}\>\inf_{t\in\mathcal{I}_{l,\epsilon}}|H^{\prime}_{l}(t,1)-m(\gamma_{l}(t))|\geq\kappa_{l}>0.$

We are able to define $H_{j}^{\prime}$ due to the assumption
$\sum_{j=0}^{n}\text{Index}(v_{j})=k\geq 2$. So we can choose a continuous
path in $B^{k}_{1/2^{j}}(0)$ away from the curve of $m(\gamma_{l}(t))$. By
Claim 5.2, for each $l\in\mathbb{N}$, there exists $T_{l}\in[0,\infty)$ for
$t\in\mathcal{I}_{l,\epsilon}$ such that:

$E_{\gamma_{l}(t)}\big{(}\phi_{t}^{l}(H_{l}^{\prime}(t,1),T_{l}),\mathcal{T}_{\tau_{l}(t)}\big{)}<E_{\gamma_{l}(t)}\big{(}0,\mathcal{T}_{\tau_{l}(t)}\big{)}-\frac{c_{0}}{10},$

where $\phi_{t}^{l}(\cdot,\cdot)$ is the one-parameter flow

$\displaystyle\\{\phi_{t}^{l}(\cdot,x)\\}_{x\geq 0}$
$\displaystyle\in\text{Diff}(\bar{B}^{k})$
$\displaystyle\phi_{t}^{l}(\cdot,\cdot):\bar{B}^{k}$
$\displaystyle\times[0,\infty)\to\bar{B}^{k},$

generated by the vector field:

(97) $s\mapsto-(1-|s|^{2})\nabla
E_{\gamma_{l}(t)}(s,\mathcal{T}_{\tau_{l}(t)}),\>s\in\bar{B}^{k}.$

Let $c_{l}:[0,1]\longrightarrow[0,1]$ be a cutoff function which is supported
in $\mathcal{I}_{l,\epsilon}$, and has value one in
$\mathcal{I}_{j,\epsilon/2}$, value zero in
$[0,1]\setminus\mathcal{I}_{j,\epsilon}.$ Define:

$H_{l}(t,x)=H^{\prime}_{l}(t,c_{l}(t)x),$

and

$H_{l}(t,x)=0\quad\forall t\in[0,1]\setminus\mathcal{I}_{l,\epsilon}.$

We now set $s_{l}:[0,1]\to\bar{B}^{k}$ to be

$s_{l}(t)=\phi_{t}^{l}(H_{l}(t,1),c_{l}(t)T_{l}),\quad\text{if
}t\in\mathcal{I}_{l,\epsilon},$

and

$s_{l}(t)=0,\quad\text{if }t\in[0,1]\setminus\mathcal{I}_{l,\epsilon}.$

Now we define $\gamma^{\prime}_{l}(t)$ as the following

$\gamma^{\prime}_{l}(t):=\Pi\circ\big{(}\gamma_{l}(t)+\sum_{i=1,j=0}^{k_{j},n}s^{l}_{j,i}(t)X^{l}_{j,i}(t)\big{)},$

where
$s_{l}(t)=(s^{l}_{0,1}(t),...,s^{l}_{j,k_{j}}(t),s^{l}_{j+1,1}(t),...,s^{l}_{n,k_{n}}(t))\in\bar{B}^{k}$.
Since $s_{l}$ is homotopic to the zero map in $\bar{B}^{k}$, so
$\gamma^{\prime}_{l}(t)$ is homotopic to $\gamma_{l}(t).$

######  Claim 5.4.

There exist $\bar{\epsilon}>0$ and $l_{0}\in\mathbb{N}$ such that

$d_{B}((\gamma^{\prime}_{l}(t),\mathcal{T}_{\tau_{l}(t)}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}\\})\geq\bar{\epsilon}$

for all $l\geq l_{0},$ $t\in[0,1]$.

Case 1:

$t\in[0,1]\setminus\mathcal{I}_{l,\epsilon}.$ Since we have
$\gamma_{l}(t)=\gamma_{l}^{\prime}(t)$ for
$t\in[0,1]\setminus\mathcal{I}_{l,\epsilon}$, then

$d_{B}((\gamma^{\prime}_{l}(t),\mathcal{T}_{\tau_{l}(t)}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}\\})\geq
C\epsilon/10.$

Case 2:

$t\in\mathcal{I}_{l,\epsilon}\setminus\mathcal{I}_{l,\epsilon/2}$. Since
$\phi_{t}^{l}(\cdot,\cdot)$ is the energy decreasing flow, so we have

$E_{\gamma_{l}(t)}\big{(}\phi_{t}^{l}(H_{l}(t,1),T),\mathcal{T}_{\tau_{l}(t)}\big{)}\leq
E_{\gamma_{l}(t)}\big{(}H_{l}(t,1),\mathcal{T}_{\tau_{l}(t)}\big{)},\quad\forall
T\in[0,T_{l}].$

Moreover, since $H_{l}(t,1)\in B^{k}_{1/2^{l}}$, there exists
$l_{1}\in\mathbb{N}$ such that

$E_{\gamma_{l}(t)}(s_{l}(t),\mathcal{T}_{\tau_{l}(t)})\leq
E_{\gamma_{l}(t)}(0,\mathcal{T}_{\tau_{l}(t)})+\nu(C\epsilon/10),$

for all $l\geq l_{1}$. By Lemma 3.6, we have

$d_{B}((\gamma^{\prime}_{l}(t),\mathcal{T}_{\tau_{l}(t)}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}\\})>\nu(C\epsilon/10).$

Case 3:

$t\in\mathcal{I}_{l,\epsilon/2}$. If
$d_{B}((\gamma^{\prime}_{l}(t),\mathcal{T}_{\tau_{l}(t)}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}\\})\to
0$ as $l\to\infty$, then we have
$E(\gamma^{\prime}_{l}(t),\mathcal{T}_{\tau_{l}(t)})=E(v_{0},\mathcal{T}_{\tau_{0}})+\sum_{j=1}^{n}E(v_{j},\mathbb{C})=W$.
By Claim 5.2 we have

$E_{\gamma_{l}(t)}(s_{l}(t),\mathcal{T}_{\tau_{l}(t)})\leq
E_{\gamma_{l}(t)}(0,\mathcal{T}_{\tau_{l}(t)})-c_{0}/10.$

Since $\\{(\gamma_{l}(t),\mathcal{T}_{\tau_{l}(t)})\\}$ is a minimizing
sequence, there exists $l_{2}\in\mathbb{N}$ such that

$E_{\gamma_{l}(t)}(s_{l}(t),\mathcal{T}_{\tau_{l}(t)})\leq
W-c_{0}/20,\quad\forall l\geq l_{2},$

which implies that there exists $\tilde{\epsilon}>0$ such that

$d_{B}((\gamma^{\prime}_{l}(t),\mathcal{T}_{\tau_{l}(t)}),\\{(v_{0},\mathcal{T}_{\tau_{0}}),\\{v_{j}\\}\\})\geq\tilde{\epsilon},\quad\forall
l\geq l_{2}.$

Let $l_{0}:=\max\\{l_{1},l_{2}\\}$ and
$\bar{\epsilon}:=\min\\{C\epsilon/10,\nu(C\epsilon/10),\tilde{\epsilon}\\}$,
we have proved the Theorem.

∎

###  5.1. Proof of Theorem 1.1

#### Theorem 1.1

Let $M$ be a closed Riemannian manifold of dimension at least three and
suppose $(M,g)$ is a bumpy metric. For any homotopically nontrivial path
$\beta\in\Lambda$, if $W>0$. Then there are possibly a conformal torus
$v_{0}:\mathcal{T}_{\tau_{0}}\to M$ and finitely many harmonic spheres
$\\{v_{j}\\}$ such that

  1. (1)

$E(v_{0},\mathcal{T}_{\tau_{0}})+\sum_{j}E(v_{j},\mathbb{C})=W,$

  2. (2)

$\sum_{j}Index(v_{j})\leq 1.$

###### Proof.

Denote by $\mathcal{F}^{W}$ the collection of equivalent classes of conformal
harmonic tori and harmonic spheres. By proposition A.19 and [Sun20,
Proposition B.24], $\mathcal{F}^{W}$ is countable and thus the following set
is countable:

$\mathcal{U}^{W}=\Big{\\{}\\{[(v^{1}_{0},\mathcal{T}_{\tau_{0}^{1}})],[v_{j}^{1}]\\},\\{[(v^{2}_{0},\mathcal{T}_{\tau_{0}^{2}})],[v_{j}^{2}]\\},...,\\{[(v^{n}_{0},\mathcal{T}_{\tau_{0}^{1}})],[v_{j}^{n}]\\},...\Big{\\}},$

with
$E(v_{0}^{n},\mathcal{T}_{\tau_{0}^{n}})+\sum_{j}E(v_{j}^{n},\mathbb{C})=W$
and $\sum_{j}\text{Index}(v_{j}^{n})\geq 2$ for each $n\in\mathbb{N}$.

Given a minimizing sequence
$\\{(\gamma_{j},\mathcal{T}_{\tau_{j}(t)})\\}_{j\in\mathbb{N}}$,
$\gamma_{j}(\cdot,t):\mathcal{T}_{\tau_{j}(t)}\to M$,
$(\gamma_{j},\mathcal{T}_{\tau_{j}(t)})\in\tilde{\Lambda}$ for each $j$, with
$[(\gamma_{j},\mathcal{T}_{\tau_{j}(t)})]=[\beta]$, we abuse the notation here
by using $j$ as the index for minimizing sequence and the finite collection of
harmonic spheres in $\mathcal{U}^{W}$. Considering
$\\{[(v^{1}_{0},\mathcal{T}_{\tau_{0}^{1}})],[v_{j}^{1}]\\}\in\mathcal{U}^{W}$,
and by Theorem 5.3 there exists
$\\{(\gamma^{1}_{j},\mathcal{T}_{\tau_{j}(t)})\\}_{j\in\mathbb{N}}$ so that

  1. (1)

$[(\gamma^{1}_{j},\mathcal{T}_{\tau_{j}(t)})]=[\beta]$, $\forall
j\in\mathbb{N}$,

  2. (2)

$\\{(\gamma^{1}_{j},\mathcal{T}_{\tau_{j}(t)})\\}_{j\in\mathbb{N}}$ is a
minimizing sequence,

  3. (3)

there exists $\epsilon_{1}>0$ and $j_{1}\in\mathbb{N}$ such that

$d_{B}((\gamma_{j}^{1}(t),\mathcal{T}_{\tau_{j}(t)}),\\{[(v^{1}_{0},\mathcal{T}_{\tau_{0}^{1}})],[v_{j}^{1}]\\})>\epsilon_{1},\quad\forall
t\in[0,1],\>\forall j>j_{1}.$

We can apply Theorem 5.3 again with
$\\{[(v^{2}_{0},\mathcal{T}_{\tau_{0}^{2}})],[v_{j}^{2}]\\}\in\mathcal{U}^{W}$
and obtain $\\{\gamma_{j}^{2},\mathcal{T}_{\tau_{j}(t)}\\}_{j\in\mathbb{N}}$
so that

  1. (1)

$[(\gamma^{2}_{j},\mathcal{T}_{\tau_{j}(t)})]=[\beta]$, $\forall
j\in\mathbb{N}$,

  2. (2)

$\\{(\gamma^{2}_{j},\mathcal{T}_{\tau_{j}(t)})\\}_{j\in\mathbb{N}}$ is a
minimizing sequence,

  3. (3)

there exist $\epsilon_{1},\epsilon_{2}>0$ and $j_{1},j_{2}\in\mathbb{N}$ such
that

$d_{B}((\gamma_{j}^{2}(t),\mathcal{T}_{\tau_{j}(t)}),\\{[(v^{l}_{0},\mathcal{T}_{\tau_{0}^{1}})],[v_{l}^{1}]\\})>\epsilon_{l},\quad\forall
t\in[0,1],\>\forall j>j_{l},\>l=1,2.$

Proceeding inductively we can find
$\\{(\gamma_{j}^{m},\mathcal{T}_{\tau_{j}(t)})\\}_{j\in\mathbb{N}}$ such that

  1. (1)

$[(\gamma^{m}_{j},\mathcal{T}_{\tau_{j}(t)})]=[\beta]$, $\forall
j\in\mathbb{N}$,

  2. (2)

$\\{(\gamma^{m}_{j},\mathcal{T}_{\tau_{j}(t)})\\}_{j\in\mathbb{N}}$ is a
minimizing sequence,

  3. (3)

there exist $\epsilon_{l}>0$ and $j_{l}\in\mathbb{N}$ for $l=1,...,m$ such
that

$d_{B}((\gamma_{j}^{m}(t),\mathcal{T}_{\tau_{j}(t)}),\\{[(v^{l}_{0},\mathcal{T}_{\tau_{0}^{1}})],[v_{l}^{1}]\\})>\epsilon_{l},\quad\forall
t\in[0,1],\>\forall j>j_{l},\>l=1,...,m.$

We can choose an increasing sequence $p_{m}>j_{m}$ such that

$\max_{t\in[0,1]}E(\gamma^{m}_{p_{m}}(\cdot,t),\mathcal{T}_{\tau_{p_{m}}(t)})\leq
W+\frac{1}{m}.$

The sequence
$\\{(\gamma^{m}_{p_{m}}(\cdot,t),\mathcal{T}_{\tau_{p_{m}}(t)})\\}_{m\in\mathbb{N}}$
is a minimizing sequence, thus by Theorem 4.5, there exists a sequence
$\\{t_{m}\\}_{m\in\mathbb{N}}\subset[0,1]$ such that one of the following case
holds:

  1. (1)

there exist a conformal harmonic torus $v_{0}:\mathcal{T}_{\tau_{0}}\to M$ and
finitely many harmonic spheres $v_{j}:S^{2}\to M$ such that

$d_{B}((\gamma_{p_{m}}^{m}(t_{m}),\mathcal{T}_{\tau_{p_{m}}(t)}),\\{(v_{0},\mathcal{T}_{\tau_{0}},\\{v_{j}\\})\\})\to
0,\>m\to\infty$

  2. (2)

there exist finitely many harmonic spheres $v_{j}:S^{2}\to M$ such that

$d_{B}((\gamma_{p_{m}}^{m}(t_{m}),\mathcal{T}_{\tau_{p_{m}}(t)}),\\{v_{j}\\})\to
0,\>m\to\infty.$

In either case we have

  1. (1)

$E(v_{0},\mathcal{T}_{\tau_{0}})+\sum_{j}E(v_{j},\mathbb{C})=W,$

  2. (2)

$\sum_{j}\text{Index}(v_{i})\leq 1.$

∎

##  Appendix A Countability of conformal Harmonic Torus

We are going to prove in this section that for a generic choice of metric the
space of conformal harmonic torus with bounded energy is countable.

###  A.1. Jocabi Field

######  Theorem A.1 (A Priori Estimate [SU81] Main Estimate 3.2).

Given $\Sigma$ and $M$, there exist $\epsilon_{su}>0$ and $\rho>0$ such that
if $r_{0}<\rho$, $u:\Sigma\to M$ is harmonic and

$\int_{B_{r_{0}}^{\Sigma}(y)}|\nabla u|^{2}<\delta\epsilon_{su},$

then

$|\nabla u|^{2}(y)\leq\frac{\delta}{r_{0}^{2}}.$

######  Remark A.2.

Once we know that $\nabla u\in L^{\infty}_{\text{loc}}$, it then follows from
equation (3) that $\Delta u\in L^{\infty}_{\text{loc}}$, which implies by
standard estimates on the inverse of the Laplacian that $u\in
W^{2,p}_{\text{loc}}$ for all $p<\infty$. Hence we deduce that $\Delta u\in
W^{1,p}_{\text{loc}}$ and hence $u\in W^{3,p}_{\text{loc}}$ for all $p>0$. We
can then repeat this argument to show that $u\in W^{r,p}_{\text{loc}}$,
$\forall r$, and so the smoothness of the solution follows.

######  Definition A.3 (tension field).

We call the tension field of $f$ to be the following

$\tau(f):=\text{trace}\nabla df.$

######  Remark A.4.

Intrinsically, the Euler-Lagrange equation for energy is

(98) $\tau(f)=0.$

######  Definition A.5 (Jocabi Field).

For a harmonic map $f:\Sigma\to M$, the Jacobi operator
$\mathcal{J}_{f}:\Gamma(f^{-1}TM)\to\Gamma(f^{-1}TM)$ is defined as following

(99) $\mathcal{J}_{f}(V):=-\Delta V-\text{trace}_{\Sigma}R^{M}(V,df)df,$

where $\Delta$ is the Laplacian on sections of $f^{-1}TM$ given in local
coordinates on $\Sigma$ by

$\Delta=h^{\alpha\beta}(f^{*}\nabla^{M})_{\frac{\partial}{\partial
x^{\alpha}}}(f^{*}\nabla^{M})_{\frac{\partial}{\partial x^{\beta}}},$

so we have that

$I(V,W)=\int_{\Sigma}\langle\mathcal{J}_{f}(V),W\rangle.$

We call $V\in\Gamma(f^{-1}TM)$ a Jacobi field for $f$ if
$\mathcal{J}_{f}(V)=0.$

Let

(100) $\begin{split}&f_{st}(x)=f(x,s,t),\\\
&f:\Sigma\times(-\epsilon,\epsilon)\times(-\epsilon,\epsilon)\to M\end{split}$

be a smooth family of maps between Riemannian manifolds of finite energy.
$\Sigma$ may have nonempty boundary, in which case we require
$f(x,s,t)=f(x,0,0)$ for all $x\in\partial\Sigma$ and all $s,t$.

######  Proposition A.6.

For a smooth family of maps $f_{st}:\Sigma\to M$ defined as (100), with

$\displaystyle V:=$ $\displaystyle\frac{\partial}{\partial
s}\Big{|}_{s=t=0}f_{st}$ $\displaystyle W:=$
$\displaystyle\frac{\partial}{\partial t}\Big{|}_{s=t=0}f_{st}.$

Let $f_{00}=f$ be a smooth harmonic map. We have the Jacobi operator
$\mathcal{J}_{f}$ defined as (99) to be the following:

$\mathcal{J}_{f}(V)=-\frac{\partial}{\partial s}\Big{|}_{s=0}\tau(f_{s0}).$

###### Proof.

The proposition follows from the computation below:

$\displaystyle\frac{\partial^{2}}{\partial s\partial
t}\Big{|}_{s=t=0}E(f_{st})=\Big{(}$ $\displaystyle\frac{\partial}{\partial
s}\Big{|}_{s=0}\int_{M}\langle df_{st},\nabla_{\frac{\partial}{\partial
t}}df_{st}\rangle\Big{)}\Big{|}_{t=0}$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial
s}\Big{|}_{s=0}\int_{M}\langle\tau(f_{s0}),\frac{\partial}{\partial
t}\Big{|}_{t=0}f_{st}\rangle$ $\displaystyle=$
$\displaystyle\int_{M}\langle-\frac{\partial}{\partial
s}\Big{|}_{s=0}\tau(f_{s0}),\frac{\partial}{\partial
t}\Big{|}_{s=t=0}f_{st}\rangle+\int_{M}\langle\tau(f_{00}),-\frac{\partial^{2}}{\partial
s\partial t}\Big{|}_{s=t=0}f_{st}\rangle$ $\displaystyle=$
$\displaystyle\int_{M}\langle-\frac{\partial}{\partial
s}\Big{|}_{s=0}\tau(f_{s0}),W\rangle$ $\displaystyle=$
$\displaystyle\int_{M}\langle\mathcal{J}_{f}(V),W\rangle.$

The second to last equality follows by $f_{00}$ is harmonic. Then it implies
that

$\mathcal{J}_{f}(V)=-\frac{\partial}{\partial s}\Big{|}_{s=0}\tau(f_{s0}).$

∎

Given a sequence of harmonic torus $f_{i}:\mathcal{T}_{\tau_{i}}\to M$
converges to a harmonic torus $u:T_{\tau}\to M$ in the following way:

(101)
$\tau_{i}\to\tau,\int_{\mathcal{T}_{\tau}}|\nabla(f_{i}\circ\iota_{\tau,\tau_{i}})-\nabla
u|^{2}\to 0,$

then we can use the difference of them to generate a Jacobi field of $u$
(possibly trivial or just the tangential Jacobi fields generated by the
conformal automorphisms of the domain).

$\displaystyle\tau(f_{i}\circ\iota_{\tau,\tau_{i}})-\tau(u)$
$\displaystyle=\int_{0}^{1}\frac{\partial}{\partial
s}\Big{|}_{s=t}\tau(u+s(f_{i}\circ\iota_{\tau,\tau_{i}}-u))dt$
$\displaystyle=\frac{\partial}{\partial
t}\Big{|}_{t=t_{i}}\tau(u+t(f_{i}\circ\iota_{\tau,\tau_{i}}-u)),$

so we define $w$ to be the following

(102)
$w(x):=\lim_{i\to\infty}\frac{f_{i}\circ\iota_{\tau,\tau_{i}}(x)-u(x)}{\max_{y\in\mathcal{T}_{\tau}}|f_{i}\circ\iota_{\tau,\tau_{i}}(y)-u(y)|},\quad\forall
x\in T_{\tau},$

and we observe that if $f_{i}\circ\iota_{\tau,\tau_{i}}\neq u$ then $w$ is
well defined. Thus by Proposition A.6 and (101) we get that
$\mathcal{J}_{u}(w)=0.$

###  A.2. Bumpy Metric Theorem for Minimal Tori

Now we define generic metric for a specific metric space. For
$k\in\mathbb{N}$, the space of $L^{2}_{k}$ Riemannian metrics on $M$ simply
denotes an open set of the Hilbert space of $L^{2}_{k}$-sections of the second
symmetric power of $T^{*}M$.

######  Definition A.7 (Generic Metrics, [Moo17]).

By a generic Riemannian metric on a smooth manifold $M$ we mean a Riemannian
metric that belongs to a countable intersection of open dense subsets of the
spaces of $L^{2}_{k}$ Riemannian metrics on $M$ with the $L^{2}_{k}$ topology,
for some choice of $k\in\mathbb{N}$, $k\geq 2$.

######  Remark A.8.

Notice that generic metric always implies a countable intersection of open
dense subsets of the metric space. For geodesics and harmonic maps the metric
space is $L^{2}_{k}$ [Moo17], and for minimal submanifolds it’s $C^{q}$
Riemannian metric for $q\geq 3$, [Whi17].

######  Theorem A.9 (Bumpy Metric Theorem for Minimal Submanifold, [Whi17]).

If $M$ is a compact manifold, then for a generic choice of metric of $C^{q}$
$g$ on $M$ ($q\geq 3$), there are no minimal submanifolds with nonzero normal
Jacobi fields. That is, each minimal submanifold has nullity $0$.

######  Definition A.10 (Prime Harmonic Map).

Suppose that $f,h:\Sigma\to M$ are harmonic maps. $f$ is called a branched
cover of $h$ if there exists a holomorphic map $g:\Sigma\to\Sigma$ of degree
$d\geq 2$ such that $f=h\circ g$. We call $f$ a prime harmonic map if it’s not
a branched cover of another harmonic map.

######  Definition A.11 (Branch Point).

A point $p\in\Sigma$ is a branch point for the harmonic map $f:\Sigma\to M$ if
$(\partial f/\partial z)(p)=0$, where $z$ is any complex coordinate near $p$.
If $p$ is a branch point of $f$ but there exists some neighborhood $V$
containing $p$ such that $f(V)$ is an immersed surface, then we say that $p$
is a false branch point.

######  Definition A.12 (Injective Point).

If $f:\Sigma\to M$ is a parametrized minimal surface we say that $p\in\Sigma$
is an injective point for $f$ if

$df(p)\neq 0,\text{ and}\quad f^{-1}(f(p))=0.$

If $f:\Sigma\to M$ is connected and has injective points, we say it is
somewhere injective.

######  Lemma A.13.

[Moo17, Lemma 5.2.1] If a conformal harmonic map $f:\Sigma\to M$ is prime, its
injective points form an open dense subset of $\Sigma$.

######  Theorem A.14.

[Moo17, Theorem 5.1.1] If $M$ is a compact smooth manifold of dimension at
least three, then for a generic choice of Riemannina metric on $M$, all prime
compact parametrized minimal surfaces $f:\Sigma\to M$ are free of branch
points and lie on nondegenerate critical submanifolds, each such submanifold
being an orbit of the group $G$ of conformal automorphisms of $\Sigma$ which
are homotopic to the identity.

######  Remark A.15.

In Theorem A.14, nondegeneracy of a prime confromal harmonic map means that
the Jacobi field of it are those generated by the conformal automorphisms of
$\Sigma.$

######  Definition A.16 (Bumpy Metric).

A metric $g$ on $M$ is called bumpy if

  1. (1)

there is no smooth immersed minimal submanifold with a non-trivial Jacobi
field,

  2. (2)

there’s no prime compact parametrized minimal surface with a non-trivial
Jacobi field other than the Jacobi fields generated by the conformal
automorphisms of its domain,

  3. (3)

there’s no prime compact parametrized minimal surface with transversal
crossings,

  4. (4)

the prime compact parametrized minimal surface is free without branch points.

In the case $\Sigma=\mathcal{T}_{\tau}$, the conformal automorphism of
$\mathcal{T}_{\tau}$ is $S^{1}\times S^{1}.$ Given a conformal harmonic torus
$u:\mathcal{T}_{\tau}\to M$, if $u$ is prime then it’s free of branch points
by Theorem A.14. If $u$ is not prime it can be written as a branched cover of
a prime harmonic torus, and all of its branched points are false. Theorem A.14
implies that for all conformal harmonic torus $u:T_{\tau}\to M$, $u(T_{\tau})$
is a smooth immersed minimal submanifold.

###  A.3. Geodesic coordinate and geodesic frame

For any $p\in\Sigma$, we can construct a “partial” geodesic coordinate and a
geodesic frame on a neighborhood of $p$ in $M$ as follows:

  1. (1)

Choose an oriented, orthonormal basis $\\{e_{1},e_{2}\\}$ for $T_{p}\Sigma$.
The map

$\displaystyle F_{0}:\mathbf{x}=(x^{1},x^{2})$
$\displaystyle\mapsto\exp^{\Sigma}_{p}(x^{j}e_{j})$

parametrizes an open neighborhood of $p$ in $\Sigma$, where $\exp^{\S}$ is the
exponential map of the induced metric on $\Sigma$. For any $\mathbf{x}$ of
unit length, the curve $\gamma(t)=F_{0}(t\mathbf{x})$ is called a _radial
geodesic on $\Sigma$_ (_at $p$_). By using $\nabla^{\Sigma}$ to parallel
transport $\\{e_{1},e_{2}\\}$ along these radial geodesics, we get a local
orthonormal frame for $T\Sigma$ on a neighborhood of $p$ in $\Sigma$. The
frame is still denoted by $\\{e_{1},e_{2}\\}$.

  2. (2)

Choose an orthonormal basis $\\{e_{3},\cdots,e_{n}\\}$ for $N_{p}\Sigma$. By
using $\nabla^{\bot}$ to parallel transport $\\{e_{3},\cdots,e_{n}\\}$ along
radial geodesics on $\Sigma$, we obtain a local orthonormal frame for
$N\Sigma$ on a neighborhood of $p$ in $\Sigma$. This frame is still denoted by
$\\{e_{3},\cdots,e_{n}\\}$. It is clear that
$\\{e_{1},e_{2},e_{3},\cdots,e_{n}\\}$ is a local orthonormal frame for
$TM|_{\Sigma}$.

  3. (3)

The map

$\displaystyle
F:(\mathbf{x},\mathbf{y})=\big{(}(x^{1},x^{2}),(y^{3},\cdots,y^{n})\big{)}$
$\displaystyle\mapsto\exp_{F_{0}(\mathbf{x})}(y^{\alpha}e_{\alpha})$

parametrizes an open neighborhood of $p$ in $M$. The map $\exp$ is the
exponential map of $(M,g)$. For any $\mathbf{y}$ of unit length, the curve
$\Sigma(t)=F(\mathbf{x},t\mathbf{y})=\exp_{F_{0}(\mathbf{x})}(t\mathbf{y})$ is
called a _normal geodesic for $\Sigma\subset M$_.

  4. (4)

For any $\mathbf{x}$, step (ii) gives an orthonormal basis
$\\{e_{1},\cdots,e_{n}\\}$ for $T_{F(\mathbf{x},0)}M$. By using $\nabla$ to
parallel transport it along normal geodesics, we have an orthonormal frame for
$TM$ on a neighborhood of $p$ in $M$. This frame is again denoted by
$\\{e_{1},\cdots,e_{n}\\}$.

###  A.4. The Space of Conformal Harmonic Tori with Bounded Energy is
Countable

######  Lemma A.17.

Given a closed smooth manifold $M$ of dimension $n\geq 3$. Suppose $(M,g)$ is
a bumpy metric, then for a conformal harmonic torus $u:\mathcal{T}_{\tau}\to
M$. There exists $\epsilon>0$, which depends on the equivalent class of $u$,
such that for any conformal harmonic torus $f:\mathcal{T}_{\tau^{\prime}}\to
M$, if $|\tau-\tau^{\prime}|<\epsilon$ and

$\int_{\mathcal{T}_{\tau}}|\nabla(f\circ\iota_{\tau,\tau^{\prime}})-\nabla
u|^{2}<\epsilon,$

then $[f]=[u].$

###### Proof.

Given a conformal harmonic map $u:T_{\tau}\to M$, Theorem A.14 implies that
the image $u(T_{\tau})$ is free without branch points, thus a smooth immersed
minimal surface.

We argue by contradiction. If not, then there exists a sequence of conformal
harmonic torus $\\{f_{l}\\}_{l\in\mathbb{N}}$,
$f_{l}:\mathcal{T}_{\tau_{l}}\to M$, such that $|\tau-\tau_{l}|\to 0$ and

$\int_{\mathcal{T}_{\tau}}|\nabla(f_{l}\circ\iota_{\tau,\tau_{l}})-\nabla
u|^{2}\to 0,$

and $[f_{l}]\neq[u]$ for all $l\in\mathbb{N}.$ By strong convergence in
$W^{1,2}$, Theorem A.1, and Arzelà-Ascoli theorem we know that the convergence
$f_{l}\to u$ is smooth and uniform.

First we consider the case that the image of the maps are different, i.e.,
$f_{l}(\mathcal{T}_{\tau_{l}})\neq u(T_{\tau})$ for all $l\in\mathbb{N}$.
Since the convergence is smooth, for $l$ sufficiently large,
$f_{l}(\mathcal{T}_{\tau_{l}})$ can be written as graph of $u(T_{\tau})$. In
other words, choose $p\in u(T_{\tau})$, we can use local partial geodesic
coordinate centered at $p$ (see Section A.3 for the construction) and write
$f_{l}(\mathcal{T}_{\tau_{l}})$ as the following

$\exp_{F(\mathbf{x},0)}(\eta^{l}_{\alpha}(\mathbf{x})e_{\alpha})=\phi_{l}(\mathbf{x}),$

for some smooth real functions $\eta^{l}_{\alpha}:\Sigma\to\mathbb{R}$. Thus
we have

(103)
$\begin{split}d\phi_{l}(e_{i})=&e_{i}+\big{(}\frac{\partial\eta^{l}_{\alpha}}{\partial
x_{i}}\big{)}e_{\alpha}+\eta^{l}_{\alpha}\nabla_{e_{i}}e_{\alpha}\\\
=&e_{i}+\big{(}\frac{\partial\eta^{l}_{\alpha}}{\partial
x_{i}}\big{)}e_{\alpha}\\\ &+\eta^{l}_{\alpha}\Big{(}-h_{\alpha
ij}|_{p}e_{j}-\eta^{l}_{\beta}(R_{\alpha i\beta j}+\sum_{k}h_{\alpha
ik}h_{\beta jk})|_{p}e_{j}-\eta^{l}_{\gamma}R_{\alpha\beta\gamma
i}|_{p}e_{\beta}\Big{)}\\\ &+\mathcal{O}(|\eta^{l}|^{3}),\end{split}$

where the indexes $i,j,k$ are either $1$ or $2$ and the indexes
$\alpha,\beta,\gamma$ range from $3$ to $n$. The third line of equation (103)
follows from the following (for the proof see [TW20, Proposition 2.6]).

$\displaystyle\nabla_{e_{i}}e_{\alpha}|_{\phi_{l}}=$
$\displaystyle\big{(}-h_{\alpha ij}|_{p}-\eta^{l}_{\beta}(R_{\alpha i\beta
j}+\sum_{k}h_{\alpha ik}h_{\beta jk})|_{p}\big{)}e_{j}$
$\displaystyle-\eta^{l}_{\gamma}R_{\alpha\beta\gamma
i}|_{p}e_{\beta}+\mathcal{O}(|\eta^{l}|^{2}).$

Now we compute the metric $\tilde{g}^{l}_{ij}$ for the graph
$\phi_{l}(\mathbf{x})$

$\displaystyle\tilde{g}^{l}_{ij}=$ $\displaystyle\langle
d\phi_{l}(e_{i}),d\phi_{l}(e_{j})\rangle$ $\displaystyle=$
$\displaystyle\delta_{ij}+\frac{\partial\eta^{l}_{\alpha}}{\partial
x_{i}}\frac{\partial\eta^{l}_{\alpha}}{\partial
x_{j}}-2\eta^{l}_{\alpha}h_{\alpha
ij}-2\eta^{l}_{\alpha}\eta^{l}_{\beta}R_{\alpha i\beta
j}-2\eta^{l}_{\alpha}\eta^{l}_{\beta}h_{\alpha ik}h_{\beta
jk}+\mathcal{O}(|\eta^{l}|^{3}).$

Since we will consider solutions where $|\eta^{l}|+|\nabla\eta^{l}|$ is small.
We will often write $\tilde{g}^{l}_{ij}$ as

$\tilde{g}_{ij}=\delta_{ij}-2\eta^{l}_{\alpha}h_{\alpha ij}+\mathcal{Q}_{ij},$

where $\mathcal{Q}_{ij}$ denotes a matrix that is quadratic in
$\eta^{l}_{\alpha}$ and $\nabla\eta^{l}_{\alpha}$ and will be allowed to vary
from line to line. It follows that the inverse metric is given by

$(\tilde{g}^{l}_{ij})^{-1}=\delta_{ij}+2\eta^{l}_{\alpha}h_{\alpha
ij}+\mathcal{Q}_{ij},$

then we have

$\det\tilde{g}^{l}_{ij}=(1+2\eta^{l}_{\alpha}h_{\alpha
ij})(1-2\eta^{l}_{\alpha}h_{\alpha ij})+\mathcal{Q}=1+\mathcal{Q},$

where the equality used minimality of $u(T_{\tau})$ and $\mathcal{Q}$ this
time is a scalar quadratic term. Whenever $J(s)$ is a differentiable path of
matrices, the derivative at $0$ of the determinant is given by

$\frac{d}{ds}\Big{|}_{s=0}\det J(s)=\det
J(0)\text{Trace}(J^{-1}(0)J^{\prime}(0)).$

Applying this with $J(s)=\tilde{g}^{l}_{ij}(s)$, where $\tilde{g}^{l}_{ij}(s)$
is computed with $\eta^{l}_{\alpha}+s\nu_{\alpha}$ in place of
$\eta^{l}_{\alpha}$, let $X_{l}:=\eta^{l}_{\alpha}e_{\alpha}$ we get

$\displaystyle\frac{d}{ds}\Big{|}_{s=0}\det J(s)=$
$\displaystyle(1+\mathcal{Q})(\delta_{ij}+2\eta^{l}_{\alpha}h_{\alpha
ij}+\mathcal{Q}_{ij})$
$\displaystyle\Big{(}2\frac{\partial\nu_{\alpha}}{\partial
x_{i}}\frac{\partial\eta^{l}_{\alpha}}{\partial x_{j}}-2\nu_{\alpha}h_{\alpha
ij}-2\eta^{l}_{\alpha}\nu_{\beta}R_{\alpha i\beta
j}-2\eta^{l}_{\alpha}\nu_{\beta}h_{\alpha jk}h_{\beta
ik}+\mathcal{Q}_{ij}\Big{)}$ $\displaystyle=$ $\displaystyle
2\Big{(}\frac{\partial\nu_{\alpha}}{\partial
x_{i}}\frac{\partial\eta^{l}_{\alpha}}{\partial
x_{i}}-2\eta^{l}_{\alpha}\nu_{\beta}R_{\alpha i\beta
i}-2\eta^{l}_{\alpha}\nu_{\beta}h_{\alpha ik}h_{\beta
ik}\Big{)}+\mathcal{Q}_{ij}$ $\displaystyle=$
$\displaystyle-2\langle(\Delta_{\Sigma}^{N}X_{l}+\text{Tr}[R_{M}(\cdot,X_{l})\cdot]+\tilde{A}(X_{l})),\nu_{\alpha}e_{\alpha}\rangle+\mathcal{Q}_{ij},$

where $\tilde{A}$ is the Simons’ operator and $\Delta_{\Sigma}^{N}$ is the
Laplacian on the normal bundle(see [CM11, pg.41] for the definitions). Since
$f_{l}(\mathcal{T}_{\tau_{l}})$ is minimal we get that
$X_{l}=\eta^{l}_{\alpha}e_{\alpha}$ satisfies the following equation

(104)
$\Delta_{\Sigma}^{N}X_{l}+\text{Tr}[R_{M}(\cdot,X_{l})\cdot]+\tilde{A}(X_{l}))+\mathcal{Q}_{ij}=0.$

With the assumption $f_{l}(\mathcal{T}_{\tau_{l}})\neq u(T_{\tau})$ we can
choose $q\in u(T_{\tau})$ such that $\eta^{l}_{\beta}(q)\neq 0$ for all
$l\in\mathbb{N}$ and for some $\beta\in\\{3,...,n\\}.$ We now consider

$\tilde{X}_{l}(\mathbf{x})=\frac{\eta^{l}_{\alpha}(\mathbf{x})e_{\alpha}}{\eta^{l}_{\beta}(q)},\quad\mathbf{x}\in
u(T_{\tau}),$

then $\langle\tilde{X}_{l}(q),e_{\beta}\rangle=1$ for all $l$.
$\tilde{X}_{l}(\mathbf{x})$ converges uniformly to a normal vector field $X$
on $\Sigma$. Multiplying equation (104) by $1/\eta^{l}_{\beta}(q)$ and using
the uniform convergence we can pass to limits to get that

$\Delta_{\Sigma}^{N}X+\text{Tr}[R_{M}(\cdot,X)\cdot]+\tilde{A}(X)=0.$

Thus $X$ is a Jacobi field of $\Sigma$ and it contradicts theorem A.9.

Now we consider the case that $f_{l}(\mathcal{T}_{\tau_{l}})=u(T_{\tau})$ for
all $l\in\mathbb{N}$. Since the convergence is smooth we can assume that

(105) $|f_{l}\circ\iota_{\tau_{l},\tau}(x)-u(x)|<1/l,\>\forall x\in T_{\tau},$

and

(106) $\Big{|}\frac{\partial f_{l}\circ\iota_{\tau,\tau_{l}}(x)}{\partial
x^{\alpha}}-\frac{\partial u(x)}{\partial x^{\alpha}}\Big{|}<1/l,\>\forall
x\in T_{\tau}.$

We define $w$ as the following

(107)
$w(x):=\lim_{l\to\infty}\frac{f_{l}\circ\iota_{\tau,\tau_{l}}(x)-u(x)}{\max_{y\in\mathcal{T}_{\tau}}|f_{l}\circ\iota_{\tau,\tau_{l}}(y)-u(y)|},\quad\forall
x\in T_{\tau},$

If $u$ is a prime harmonic sphere, then Proposition A.6 implies that $w$ is a
tangential Jacobi field of the map $u$. Theorem A.14 implies that $W$
generated by the conformal automorphism of $T_{\tau}$, contradicting the
assumption $[f_{l}]\neq[u]$. Now we consider the case that $u$ is not prime.

######  Claim A.18.

For $l$ sufficiently large, we have

$deg(f_{l})=deg(u).$

###### Proof of the claim.

It’s known that if two maps are homotopic then they have the same degree. The
stradegy of the proof is similar and can be found in [Hir94, Chapter 5].

For $l$ sufficiently large, by equation (105) we know that
$f_{l}\circ\iota_{\tau,\tau_{l}}$ is homotopic to $u$. That is, there exists a
continuous map $H_{l}:T_{\tau}\times[0,1]\to u(T_{\tau})$:

$H_{l}(x,t):=\gamma_{x}^{l}(t),\>x\in T_{\tau},$

such that $H_{l}(x,0)=u(x)$ and
$H_{l}(x,1)=f_{l}\circ\iota_{\tau,\tau_{l}}(x)$ for all $x\in T_{\tau}$, where
$\gamma_{x}^{j}(t)$ denotes the unique geodesic with respect to the intrinsic
metric starting from $u(x)$ with end point at
$f_{l}\circ\iota_{\tau,\tau_{l}}(x)$. Since
$f_{l}(\mathcal{T}_{\tau_{l}})=u(T_{\tau})$, and by Lemma A.13 we can choose
$y\in u(T_{\tau})$ so it satisfies the following conditions:

  1. (1)

$\forall p\in u^{-1}(y)$, $du(p)\neq 0$,

  2. (2)

$\forall p\in\big{(}f_{l}\circ\iota_{\tau,\tau_{l}}\big{)}^{-1}(y)$,
$d\big{(}f_{l}\circ\iota_{\tau,\tau_{l}}\big{)}(p)\neq 0$,

  3. (3)

$y$ is a regular value for $H_{l}$ and
$H_{l}|_{\partial(T_{\tau}\times[0,1])}$.

$H_{l}^{-1}(y)$ is a compact 1-dimensional submanifold with boundary
$(H_{j}|_{\partial(T_{\tau}\times[0,1])})^{-1}(y)$. In other words, it
contains embedded arcs which are transverse to
$\partial(T_{\tau}\times[0,1])$. By [Hir94, Chapter 5], given $p_{1}\in
H_{l}^{-1}(y)$, there is an unique $p_{2}\in H_{l}^{-1}(y)$, $p_{1}\neq
p_{2}$, and a component arc $\Gamma\in H_{l}^{-1}(y)$ with
$\partial\Gamma=\\{p_{1},p_{2}\\}$. By [Hir94, Chapter 5, Lemma 1.2], $p_{1}$
and $p_{2}$ are of opposite type for $H_{l}|_{\partial(T_{\tau}\times[0,1])}$.
This implies that for any $p_{1}\in u^{-1}(y)$, there is an unique
corresponding $p_{2}\in\big{(}f_{l}\circ\iota_{\tau,\tau_{l}}\big{)}^{-1}(y)$.
Then it follows that degree of $u$ and degree of $f_{l}$ are the same for $l$
sufficiently large. ∎

We can write $u=\tilde{u}\circ\phi$ for some conformal prime harmonic map
$\tilde{u}$ and a holomorphic map $\phi:T_{\tau}\to T_{\tau}$ with degree
$deg(u)$. By the smooth convergence of $f_{l}\circ\iota_{\tau,\tau_{l}}\to u$,
and $deg(f_{l})=deg(u)$, we can write $f_{l}$ as a branched cover of some
prime harmonic map $g_{l}:\mathcal{T}_{\tau_{l}}\to M$, i.e.,

$f_{l}=g_{l}\circ\big{(}\iota_{\tau,\tau_{l}}\circ\phi\circ\iota_{\tau_{l},\tau}\big{)}.$

So we obtain a sequence of conformal prime harmonic tori
$g_{l}:\mathcal{T}_{\tau_{l}}\to M$ that converges strongly to a conformal
prime harmonic map $\tilde{u}$ in $W^{1,2}$, and $[f_{l}]\neq[u]$ implies that
$[g_{l}]\neq[\tilde{u}]$. It is the desired contradiction. ∎

We define $\mathcal{F}^{W}$ to be the equivalent classes of all conformal
harmonic tori with energy bounded by $W$, that is:

$\mathcal{F}^{W}:=\Big{\\{}[f]\mid f:\mathcal{T}_{\tau}\rightarrow M\text{
conformal and harmonic},E(f,T_{\tau})\leq W\Big{\\}}.$

######  Proposition A.19.

The set $\mathcal{F}^{W}$ is countable.

###### Proof.

For $B_{R}\subset\mathbb{C}$, we pick a finite set
$\\{x^{1}_{n},...,x^{p^{n}}_{n}\\}$ of $\mathbb{C}$ such that
$\bar{B}_{R}\subset\bigcup_{k=1}^{p^{n}}B_{\frac{1}{n}}(x^{k}_{n}).$ We define
$\mathcal{F}_{R}^{W}(n)$ to be:

(108) $\begin{split}\mathcal{F}_{R}^{W}(n)=\Big{\\{}[f]\mid
f\in\mathcal{F}^{W}&,\>f:\mathcal{T}_{\tau}\to M,\>\\{1,\tau\\}\in B_{R}\\\
&\int_{B_{\frac{1}{n}}(x^{k}_{n})}|\nabla f|^{2}<\epsilon_{su},\>\text{for all
}k=1,...,p^{n}\Big{\\}},\end{split}$

where $\\{1,\tau\\}\in B_{R}$ means that $\bar{B}_{R}$ contains all
parallelograms generates by $\\{1,\tau\\}$, and $\epsilon_{su}>0$ is the
constant given in Theorem A.1. We can see that:

$\mathcal{F}^{W}=\bigcup_{n\in\mathbb{N},R\in\mathbb{R}}\mathcal{F}_{R}^{W}(n).$

Now we prove that $\mathcal{F}^{W}(n)$ is finite. If not, then there exists a
sequence of conformal harmonic tori
$\\{[f_{l}]\\}_{l\in\mathbb{N}}\subset\mathcal{F}_{R}^{W}(n)$,
$f_{l}:\mathcal{T}_{\tau_{l}}\to M$, with $[f_{i}]\neq[f_{j}]$. Because
$\bar{B}_{R}$ is compact we have that $\tau_{l}\to\tau$ up to subsequence and
$f_{l}$ converges to a conformal harmonic map $f:T_{\tau}\to M$, up to
subsequence. The assumption
$\\{[f_{l}]\\}_{l\in\mathbb{N}}\subset\mathcal{F}_{R}^{W}(n)$ implies that

$\int_{B_{1/n}(x^{k}_{n})}|\nabla f_{l}|^{2}<\epsilon_{su}\quad\forall
l\in\mathbb{N},k=1,...,p^{n}.$

By Theorem A.1, this implies that the convergence is strong in $W^{1,2}$ and
$f\in\mathcal{F}^{W}(n)$. Then it contradicts Lemma A.17. ∎

## References

  * [CM08] Tobias H. Colding and William P. Minicozzi, II.  Width and finite extinction time of Ricci flow.  Geom. Topol., 12(5):2537–2586, 2008.
  * [CM11] Tobias Holck Colding and William P. Minicozzi, II.  A course in minimal surfaces, volume 121 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2011.
  * [CM18] Otis Chodosh and Christos Mantoulidis.  Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates.  arXiv e-prints, page arXiv:1803.02716, Mar 2018.
  * [CS85] Hyeong In Choi and Richard Schoen.  The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature.  Invent. Math., 81(3):387–394, 1985.
  * [CZ21] Da Rong Cheng and Xin Zhou.  Existence of constant mean curvature 2-spheres in riemannian 3-spheres, 2021.
  * [EM08] Norio Ejiri and Mario Micallef.  Comparison between second variation of area and second variation of energy of a minimal surface.  Adv. Calc. Var., 1(3):223–239, 2008.
  * [Hir94] Morris W. Hirsch.  Differential topology, volume 33 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1994.  Corrected reprint of the 1976 original.
  * [Mil63] J. Milnor.  Morse theory.  Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton, N.J., 1963\.
  * [MM88] Mario J. Micallef and John Douglas Moore.  Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes.  Ann. of Math. (2), 127(1):199–227, 1988.
  * [MN16] Fernando C. Marques and André Neves.  Morse index and multiplicity of min-max minimal hypersurfaces.  Camb. J. Math., 4(4):463–511, 2016.
  * [MN18] Fernando C. Marques and André Neves.  Morse index of multiplicity one min-max minimal hypersurfaces.  arXiv e-prints, page arXiv:1803.04273, Mar 2018.
  * [Moo17] John Douglas Moore.  Introduction to global analysis, volume 187 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2017.  Minimal surfaces in Riemannian manifolds.
  * [Pit81] Jon T. Pitts.  Existence and regularity of minimal surfaces on Riemannian manifolds, volume 27 of Mathematical Notes.  Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981.
  * [PR20] Alessandro Pigati and Tristan Rivière.  A proof of the multiplicity 1 conjecture for min-max minimal surfaces in arbitrary codimension.  Duke Mathematical Journal, 169(11), Aug 2020.
  * [Riv17] Tristan Rivière.  A viscosity method in the min-max theory of minimal surfaces.  Publ. Math. Inst. Hautes Études Sci., 126:177–246, 2017.
  * [Sha17] Ben Sharp.  Compactness of minimal hypersurfaces with bounded index.  J. Differential Geom., 106(2):317–339, 2017.
  * [Sim96] Leon Simon.  Theorems on regularity and singularity of energy minimizing maps.  Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996\.  Based on lecture notes by Norbert Hungerbühler.
  * [SU81] J. Sacks and K. Uhlenbeck.  The existence of minimal immersions of $2$-spheres.  Ann. of Math. (2), 113(1):1–24, 1981.
  * [Sun20] Yuchin Sun.  Morse index bound for minimal two spheres, 2020.
  * [TW20] Chung-Jun Tsai and Mu-Tao Wang.  A strong stability condition on minimal submanifolds and its implications.  J. Reine Angew. Math., 764:111–156, 2020.
  * [Whi17] Brian White.  On the bumpy metrics theorem for minimal submanifolds.  Amer. J. Math., 139(4):1149–1155, 2017.
  * [Zho10] Xin Zhou.  On the existence of min-max minimal torus.  J. Geom. Anal., 20(4):1026–1055, 2010.
  * [Zho17] Xin Zhou.  On the existence of min-max minimal surface of genus $g\geq 2$.  Commun. Contemp. Math., 19(4):1750041, 36, 2017.
  * [Zho20] Xin Zhou.  On the multiplicity one conjecture in min-max theory.  Ann. of Math. (2), 192(3):767–820, 2020.