# Solutions of the Ginzburg-Landau equations with vorticity concentrating near
a nondegenerate geodesic

Andrew<EMAIL_ADDRESS>Department of Mathematics,
University of Toronto, Toronto, ON M5S 2E4 Canada  Robert L.
<EMAIL_ADDRESS>Department of Mathematics, University of
Toronto, Toronto, ON M5S 2E4 Canada  Peter<EMAIL_ADDRESS>Department of Mathematics, Indiana University, Rawles Hall 831 East 3rd St.,
Bloomington, IN 47405

Abstract It is well-known that under suitable hypotheses, for a sequence of
solutions of the (simplified) Ginzburg-Landau equations $-\Delta
u_{\varepsilon}+\varepsilon^{-2}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon}=0$,
the energy and vorticity concentrate as $\varepsilon\to 0$ around a
codimension $2$ stationary varifold — a (measure theoretic) minimal surface.
Much less is known about the question of whether, given a codimension $2$
minimal surface, there exists a sequence of solutions for which the given
minimal surface is the limiting concentration set. The corresponding question
is very well-understood for minimal hypersurfaces and the scalar Allen-Cahn
equation, and for the Ginzburg-Landau equations when the minimal surface is
locally area-minimizing, but otherwise quite open.

We consider this question on a $3$-dimensional closed Riemannian manifold
$(M,g)$, and we prove that any embedded nondegenerate closed geodesic can be
realized as the asymptotic energy/vorticity concentration set of a sequence of
solutions of the Ginzburg-Landau equations.

##  1 Introduction

In this paper we construct certain geometrically meaningful solutions of the
Ginzburg-Landau equations

$-\Delta
u_{\varepsilon}+\frac{1}{\varepsilon^{2}}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon}=0$
(1.1)

for $u_{\varepsilon}:M\to\mathbb{C}$, where $(M,g)$ is a closed
$n$-dimensional Riemannian manifold, with $n=3$ in our main results. Such
solutions are critical points of the Ginzburg-Landau functional

$E_{\varepsilon}(u_{\varepsilon}):=\frac{1}{\pi{|\\!\log\varepsilon|}}\int_{M}e_{\varepsilon}(u_{\varepsilon})\,\mbox{vol}_{g},\qquad
e_{\varepsilon}(u_{\varepsilon}):=\frac{1}{2}|\nabla
u_{\varepsilon}|^{2}+\frac{1}{4\varepsilon^{2}}(|u_{\varepsilon}|^{2}-1)^{2}.$

If $M$ is simply connected, then given a sequence of solutions
$(u_{\varepsilon})$ of (1.1) satisfying the energy bound

$E_{\varepsilon}(u_{\varepsilon})\leq C,$ (1.2)

the rescaled energy density
${|\\!\log\varepsilon|}^{-1}e_{\varepsilon}(u_{\varepsilon})$ is known to
concentrate as $\varepsilon\to 0$, after possibly passing to a subsequence,
around an $(n-2)$-dimensional stationary varifold — a weak, measure-theoretic
minimal surface. This is proved in an appendix in [27], following earlier
results in simply-connected Euclidean domains, such as those in [4, 15, 17].
Similar but more complicated results hold when $M$ is not simply connected; in
this case, the limiting energy measure may have a diffuse part, but any
concentrated part must again be an $(n-2)$-dimensional stationary varifold.

In this paper we address a sort of converse question:

When can a given codimension $2$ minimal surface be realized as the energy
concentration set of a sequence of solutions of (1.1)?

A first answer is provided by Gamma-convergence results, see [10, 1], that
relate the Ginzburg-Landau functional and, roughly speaking, the
$(n-2)$-dimensional area (with multiplicity) of a limiting vorticity
concentration set, where the vorticity associated to a wave function $u$,
denoted $Ju$, is the $2$-form defined by

$Ju:=du^{1}\wedge{}du^{2},\qquad\mbox{where $u=u^{1}+iu^{2}$ and $u^{1},u^{2}$
are real-valued.}$ (1.3)

(We will also sometimes refer to $Ju$ as the Jacobian of $u$.) These results
imply as a general principle that one should be able to find solutions
$u_{\varepsilon}$ of (1.1) whose energy and vorticity concentrate around a
locally area-minimizing minimal surface of codimension 2. In the Euclidean
setting, specific instances of this general principle, for particular
compatible choices of boundary conditions on the minimal surface and the
solutions $u_{\varepsilon}$ of (1.1), have been established in [1, 25, 22].
However, arguments based on Gamma-convergence are of limited use for capturing
the behaviour of non-minimizing critical points.

The corresponding question is also very well-understood for minimal
hypersurfaces and the Allen-Cahn equation, i.e. the scalar counterpart of
(1.1), see for example [13, 23, 14, 7] among many others. Many of these
results are based on gluing techniques and elliptic PDE arguments, which can
be used to construct a great variety of solutions and establish detailed
descriptions of them. These techniques seem to be hard to implement for the
Ginzburg-Landau equation in $3$ or more dimensions.

A particularly basic case in which our question remains open concerns the
Ginzburg-Landau equation (1.1) on a smooth bounded domain
$\Omega\subset\mathbb{R}^{3}$ containing an unstable geodesic with respect to
natural boundary conditions, i.e. a line segment in $\Omega$ meeting
$\partial\Omega$ orthogonally at both ends, admitting perturbations that
decrease the arclength quadratically, and satisfying a natural nondegeneracy
condition.

In this situation one would like to prove the existence of a sequence
$(u_{\varepsilon})$ of solutions of the Ginzburg-Landau equations, also with
natural (Neumann) boundary conditions, whose energy and vorticity concentrate
around the given line segment. Such solutions would satisfy

$\lim_{\varepsilon\to 0}E_{\varepsilon}(u_{\varepsilon})=L=:\mbox{ the length
of the geodesic}.$ (1.4)

Partial progress toward this goal was achieved in [11], which develops a
general framework for using Gamma-convergence to study convergence, not of
critical points, but of critical values, then uses this framework to prove the
existence of solutions of (1.1), in the situation described above, that
satisfy (1.4), but without control over the limiting concentration set. An
example in the same paper (Remark 4.5) shows that the general framework is too
weak to characterize asymptotic behaviour of critical points — in this
context, to determine where the energy and vorticity concentrate. For this,
more detailed information about the sequence of solutions is needed.

The results of [11] were extended to the Riemannian setting in the Ph.D.
thesis of Jeffrey Mesaric in [20] which, starting with a nondegenerate
unstable closed geodesic on a closed, oriented $3$ dimensional Riemannian
manifold $(M,g)$, uses machinery from [11] to construct solutions to (1.1)
satisfying (1.4). Again, this result does not establish whether the energy of
the solutions concentrates along the geodesic.

In the main result of this paper, we fill in this gap in the Riemannian case.
Our main result is the following theorem.

######  Theorem 1.1.

Let $(M,g)$ be a closed oriented $3$-dimensional Riemannian manifold, and let
$\gamma$ be a closed, embedded, nondegenerate geodesic of length $L$. Assume
in addition that $\gamma=\partial S$ in the sense of Stokes’ Theorem for some
$2$-dimensional submanifold $S$ of $M$.

Then there exists $\varepsilon_{1}>0$ such that for every
$0<\varepsilon<\varepsilon_{1}$, there is a solution $u_{\varepsilon}$ of the
Ginzburg-Landau equation (1.1) such that

$\frac{1}{\pi}\int_{M}\varphi\wedge
Ju_{\varepsilon}\to\int_{\gamma}\varphi\qquad\mbox{ for every smooth $1$-form
$\varphi$ on $M$}$

and

$\frac{1}{\pi|\log\varepsilon|}\int_{M}\phi\,e_{\varepsilon}(u_{\varepsilon})\to\int_{\gamma}\phi\,d{\mathcal{H}}^{1}\qquad\mbox{
for every }\phi\in C^{\infty}(M)$

as $\varepsilon\to 0$, where ${\mathcal{H}}^{k}$ denotes $k$-dimensional
Hausdorff measure.

In fact we will prove a slightly stronger result; see Theorem 5.1 for the full
statement.

We briefly sketch the main ideas, not in the order in which they appear in the
body of the paper. Terminology such as “nondegenerate” and “stationary
varifold” are defined in Section 2 below.

  * •

In Section 4 we show that for any $\delta>0$, there exists $\varepsilon_{0}>0$
such that for $0<\varepsilon<\varepsilon_{0}$ and any $\tau>0$, one can find a
solution $u_{\varepsilon}$ of the Ginzburg-Landau heat flow whose vorticity is
initially concentrated near the geodesic $\Gamma:=\gamma([0,L))$, and such
that

$L-\delta\leq E_{\varepsilon}(u_{\varepsilon}(\cdot,t))\leq
L+\delta\qquad\mbox{ for all }t\in[0,\tau].$

See Proposition 4.1. This relies heavily on tools developed in the earlier
papers [11, 20].

The main point of the proof of Theorem 1.1 is to strengthen this by showing
that for such solutions, if $\varepsilon$ and $\delta$ are small enough, the
vorticity $\frac{1}{\pi}Ju_{\varepsilon}(\cdot,t)$ does not stray very far
from $\Gamma$ for any $t\in[0,\tau]$.

  * •

We carry this out in Section 5, using an argument by contradiction and passing
to limits to obtain a stationary $1$-dimensional varifold that is close, but
not equal, to the varifold associated to $\Gamma$. This argument requires,
among other ingredients, an extension to the Riemannian setting of an
important theorem of Bethuel, Orlandi, and Smets [5]. The extension we need is
stated in Theorem 2.3 and is proved in a companion paper, see [6]. The
stationary varifold satisfies additional good properties, notably including
lower density bounds.

  * •

To obtain a contradiction, we prove that this stationary varifold cannot
exist. This is the content of Proposition 3.1, which is a measure theoretic
strengthening of the classical fact that a nondegenerate closed geodesic is
isolated; it is the only closed geodesic in a tubular neighborhood of itself.
The proof relies, among other ingredients, on results from [2] about the
structure of stationary $1$-dimensional varifolds on Riemannian manifolds.

We believe that something like Theorem 1.1 should be valid in much greater
generality, including on higher-dimensional manifolds and on smooth, bounded
subsets of $\mathbb{R}^{n}$, $n\geq 3$, with natural boundary conditions both
for the geodesic $\Gamma$ (or codimension $2$ minimal surface, for $n\geq 4$)
and the Ginzburg-Landau equation. Our proof does not adapt in a
straightforward way to either of these settings.

  * •

Our strategy requires a sufficiently good version of Theorem 2.3. On a bounded
set $\Omega\subset\mathbb{R}^{n}$, even for $n=3$, such a result is not known.
If $\Omega$ is convex, a result of this type for the scalar parabolic Allen-
Cahn equation was proved several years ago in [21]. A similar strategy could
probably be pursued for the Ginzburg-Landau heat flow, but global convexity is
not a natural assumption for any analog of Theorem 1.1.

  * •

Our reliance on results from [2] about stationary $1$-dimensional varifolds
would seriously complicate any effort to adapt our argument to dimensions
$n\geq 4$, where one would confront stationary varifolds of dimension $n-2\geq
2$.

##  2 Background and notation

###  2.1 Geometric notions regarding a non-degenerate geodesic

Throughout this document we use $M$ or $(M,g)$ to denote a closed oriented
three dimensional Riemannian manifold where “closed” means compact and without
boundary. We let $TM$ be the bundle over $M$ whose fiber $T_{p}M$ at $p\in{}M$
is the tangent space to $M$ at $p$. We use the notation $(\cdot,\cdot)_{g}$ to
denote the inner product on $TM$ given by $g$. We also use
$\left|\cdot\right|_{g}$ to denote the corresponding norm, where we will omit
mention of $g$ when no confusion will arise. We write $\,\mbox{vol}_{g}$ to
denote the Riemannian volume form associated to the metric $g$.

We will write $r_{0}>0$ to be a number, fixed throughout this paper, such that

$r_{0}<\frac{1}{2}\,(\mbox{injectivity radius of $M$}).$ (2.1)

Throughout this paper, a central role will be played by a geodesic $\gamma$
that we take to be parametrized by arclength. That is, we will assume the
existence of an injective map $\gamma:\mathbb{R}/L{\mathbb{Z}}\to M$ whose
range consists of a simple closed curve
$\Gamma:=\\{\gamma(t):t\in\mathbb{R}/L{\mathbb{Z}}\\}$ of length $L$ such that

$|\gamma^{\prime}|=1,\qquad\nabla_{\gamma^{\prime}}\gamma^{\prime}=0\qquad\mbox{
everywhere in }\mathbb{R}/L{\mathbb{Z}}.$ (2.2)

We will insist that this curve $\Gamma$ bounds an orientable smooth surface
$S_{\Gamma}\subset M$, i.e.

$\Gamma=\partial S_{\Gamma}.$ (2.3)

We introduce here the notation

$d_{\Gamma}(x):=\operatorname{dist}(x,\Gamma):=\inf\left\\{\int_{0}^{1}|\lambda^{\prime}(t)|dt:\lambda\in
Lip([0,1];M),\lambda(0)=x,\lambda(1)\in\Gamma\right\\}$

as well as

$K_{r}:=\left\\{x\in{}M:d_{\Gamma}(x)<r\right\\}$ (2.4)

for a neighborhood of $\Gamma.$

For $t\in\mathbb{R}/L{\mathbb{Z}}$, we then let

$N_{\gamma(t)}\Gamma:=\\{u\in T_{\gamma(t)}M:(u,\gamma^{\prime}(t))_{g}=0\\}.$

A _normal vector field along $\gamma$_ is a map
$\xi:\mathbb{R}/L{\mathbb{Z}}\to TM$ such that $\xi(t)\in N_{\gamma(t)}\Gamma$
for every $t$. We also introduce the coordinates
$\psi:B_{r}(0)\times(0,L)\to{}K_{r}$ defined by

$\psi(y,t):=\exp_{\gamma(t)}\left(\sum_{i=1}^{2}y^{i}\Xi_{i}(t)\right)$ (2.5)

where $\Xi_{1},\Xi_{2}$ are fixed normal vector fields which are orthogonal
for each $t\in(0,L)$. We note for $r<r_{0}$, this map is smoothly invertible.
For future use, we will use the notation $\psi^{-1}(x)=(y(x),\tau(x))\in
B_{r}(0)\times(0,L)$, so that for $x\in K_{r}$,

$\psi(y,t)=x\qquad\iff\qquad y(x)=y\quad\mbox{ and }\quad\tau(x)=t.$ (2.6)

We observe that the mapping $\tau$ simply assigns to an $x\in K_{r}$ the
parameter value $t$ corresponding to the closest point on $\Gamma$ to $x$.

Given two normal vector fields along $\gamma$, denoted by $\xi,\tilde{\xi}$,
we can define the $L^{2}$ inner product in the natural way:

$(\xi,\tilde{\xi})_{L^{2}}:=\int_{\mathbb{R}/L{\mathbb{Z}}}(\xi(t),\tilde{\xi}(t))_{g}\,dt.$

We will write $L^{2}(N\Gamma)$ to denote the space of square integrable normal
vector fields, a Hilbert space with the above inner product.

For $\xi\in L^{2}(N\Gamma)$, we will use the notation

$\gamma_{\xi}(t):=\exp_{\gamma(t)}\xi(t),$ (2.7)

where exp denotes the exponential map.

We next recall the Jacobi operator $L_{J}$ which acts on smooth normal vector
fields $\xi$ along $\gamma$, and is defined by

$L_{J}\xi:=-\xi^{\prime\prime}+R(\xi,\gamma^{\prime})\gamma^{\prime},$ (2.8)

where $R$ denotes the curvature tensor. We say that a geodesic is
nondegenerate if $0$ is not an eigenvalue of $L_{J}$.

With this notion in hand, we add another crucial hypothesis on the geodesic by
assuming henceforth that

$\gamma:\mathbb{R}/L{\mathbb{Z}}\to M$ is a simple, closed, nondegenerate
geodesic with $|\gamma^{\prime}|\equiv 1$,  (2.9)

One says that $\gamma$ has finite index if the total number (algebraic
multiplicity) of negative eigenvalues of $L_{J}$ is finite. Since $M$ is
closed, this is always true, as a consequence of standard Sturm-Liouville
theory. Our standing assumption (2.9) that $\gamma$ is nondegenerate then
imples there exists some $\ell\geq 0$ and a nondecreasing sequence of
eigenvalues

$\lambda_{1}\leq\ldots\leq\lambda_{\ell}<0<\lambda_{\ell+1}\leq\ldots$ (2.10)

of $L_{J}$, together with an associated orthonormal basis of $L^{2}(N\Gamma)$
consisting of (smooth) eigensections $\\{\xi_{j}\\}_{j=1}^{\infty}$. We will
always assume that $\ell>0$, since otherwise the results presented here admit
much simpler proofs. We define

$H_{-}:=\mbox{span}\\{\xi_{1},\ldots\xi_{\ell}\\},\qquad
H_{+}:=H_{-}^{\perp}.$ (2.11)

We will say that $\xi$ is Lipschitz, and we will write $\xi\in Lip$, if
$\gamma_{\xi}$ is Lipschitz continuous. It is clear that

$H_{-}(r_{0}):=\\{\xi\in H_{-}:\|\xi\|_{L^{\infty}}\leq r_{0}\\}\subset Lip$

for $r_{0}$ and $H_{-}$ from (2.1) and (2.11) respectively.

The standard fact that the Jacobi operator, cf. (2.8), is the second variation
of arclength, together with the definition (2.11) of $H_{-}$, implies that
there exist $c_{0},r_{0}>0$ such that for all $\|\xi\|_{L^{\infty}}\leq r_{0}$
one has

$\displaystyle\int_{\mathbb{R}/L{\mathbb{Z}}}|\gamma_{\xi}^{\prime}(t)|\,dt$
$\displaystyle\leq L-c_{0}\|\xi\|_{L_{2}}^{2}\quad\mbox{ if }\xi\in
H_{-}(r_{0})$ (2.12)
$\displaystyle\int_{\mathbb{R}/L{\mathbb{Z}}}|\gamma_{\xi}^{\prime}(t)|\,dt$
$\displaystyle\geq L+c_{0}\|\xi\|_{L_{2}}^{2}\quad\ \,\ \mbox{ if }\xi\in
H_{+}\cap Lip.$

###  2.2 Forms and currents

We denote, for $k\in{\mathbb{N}}\cup\left\\{0\right\\}$, the space of smooth
$k$-forms on $M$ by

${\mathcal{D}}^{k}(M):=\left\\{\phi\in{}C^{\infty}(M;\wedge^{k}M)\right\\}$

where $\wedge^{k}M$ is an abbreviated notation for $\wedge^{k}T^{*}M$. We
denote the dual space of ${\mathcal{D}}^{k}(M)$, for
$k\in{\mathbb{N}}\cup\left\\{0\right\\}$, by

${\mathcal{D}}_{k}(M):=\\{\mbox{$k$-currents on $M$}\\}.$

We refer to the elements of ${\mathcal{D}}_{k}(M)$ as _$k$ -currents_. For a
$k$-current $T$, we define

$\mbox{ the {\em mass} of $T$}={\bf
M}(T):=\sup\\{T(\phi):\|\phi\|_{\infty}\leq 1\\}\in[0,+\infty].$

We will be most interested in $1$-currents. A basic class of examples consists
of $1$-currents we shall write as $T_{\lambda}$ whose action on
$\phi\in{\mathcal{D}}^{1}(M)$ takes the form

$T_{\lambda}(\phi):=\int_{\lambda}\phi,\qquad\mbox{ where }\lambda:(a,b)\to
M\mbox{ is a Lipschitz curve}.$ (2.13)

We will say a $1$-current is integer multiplicity rectifiable if it can be
written as a finite or countable sum of currents of the form (2.13). We will
write

${\mathcal{R}}_{1}(M):=\\{T\in{\mathcal{D}}_{1}(M)\ :\ {\bf M}(T)<\infty,\ \
\mbox{$T$ is integer multiplicity rectifiable }\\}.$

For a $1$-current $J$, we write $\|J\|$ to denote the associated total
variation measure, defined through its action on continuous, nonnegative
functions $f:M\to\mathbb{R}$ via

$\int f\,d\|J\|:=\sup\\{J(\phi):\phi\in{\mathcal{D}}^{1}(M),\;|\phi|_{g}\leq
f\\}.$ (2.14)

For a $k$-current $S$, the boundary of $S$ is the $(k-1)$-current $\partial S$
defined by

$\partial S(\phi):=S(d\phi),\qquad\mbox{for
all}\;\phi\in{\mathcal{D}}^{k-1}(M).$

We define

$\mathcal{F}_{1}^{\prime}(M):=\\{T\in{\mathcal{D}}_{1}(M):\ \ T=\partial
S\mbox{ for some }S\in{\mathcal{D}}_{2}(M),\,{\bf M}(S)<\infty\\}.$

and for $T\in\mathcal{F}_{1}^{\prime}(M)$, we will write

$\|T\|_{\mathcal{F}}:=\inf\\{{\bf M}(S):T=\partial S\\}.$

We also define

${\mathcal{R}}_{1}^{\prime}(M):={\mathcal{R}}_{1}(M)\cap\mathcal{F}_{1}^{\prime}(M).$

We note that the $1$-current $T_{\gamma}$ associated with the geodesic
$\gamma$ via (2.13), in particular, bounds a finite mass $2$-current; that is,

$T_{\gamma}\in{\mathcal{R}}_{1}^{\prime}(M),$ (2.15)

in light of the assumption (2.3).

Lastly, we will at times wish to identify the Jacobian (i.e. vorticity) of a
map $u\in H^{1}(M;\mathbb{C})$ with an element of ${\mathcal{D}}_{1}(M)$,
which we denote $\star J(u)$, and which is defined through its action on
$1$-forms $\phi$ by

$\star J(u)(\phi)=\int\phi\wedge J(u),$ (2.16)

where $J(u)=du^{(1)}\wedge du^{(2)}$ for $u=u^{(1)}+iu^{(2)}$ where
$u^{(1)},u^{(2)}$ are real-valued.

###  2.3 Gamma-limit of the Ginzburg-Landau functional

Below we state the version we will need of standard Gamma-convegence results
for the Ginzburg-Landau functional.

We first fix the notation $V=\mathcal{F}_{1}^{\prime}(M)$, with the flat norm
$\|v\|_{V}:=\|v\|_{\mathcal{F}}$. We also define the functional

$E_{V}(T):=\begin{cases}{\bf M}(T)&\mbox{ if
}T\in{\mathcal{R}}_{1}^{\prime}(M)\\\ +\infty&\mbox{ if not}.\end{cases}$
(2.17)

Thus $E_{V}$ is an extension to $V$ of the “arclength functional” in the sense
that if $\lambda:(a,b)\to M$ is an injective Lipschitz continuous curve and
$T_{\lambda}$ is the corresponding current, then $E_{V}(T_{\lambda})=\mbox{
arclength of }Image(\lambda)$.

The following result is deduced in [20], Theorem 5.1 from corresponding
Euclidean results, cf. [1, 10].

######  Theorem 2.1.

Let $(M,g)$ be a closed $3$-dimensional Riemannian manifold.

1. Let $(u_{\varepsilon})_{0<\varepsilon<\varepsilon_{0}}$ be a sequence in $H^{1}(M;\mathbb{C})$. If there exists $C>0$ such that $E_{\varepsilon}(u_{\varepsilon})\leq C$ for all $\varepsilon\in(0,\varepsilon_{0})$, then $(\frac{1}{\pi}\star Ju_{\varepsilon})_{0<\varepsilon<\varepsilon_{0}}$ is precompact in V, and any limit as $\varepsilon\to 0$ belongs to ${\mathcal{R}}_{1}^{\prime}(M)$

2. Let $(u_{\varepsilon})_{0<\varepsilon<\varepsilon_{0}}$ be a sequence in $H^{1}(M;\mathbb{C})$. If $T\in V$ and
$\|\frac{1}{\pi}\star Ju_{\varepsilon}-T\|_{\mathcal{F}}\to 0$ as
$\varepsilon\to 0$, then $\liminf_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon})\geq E_{V}(T)$.

3. For any $T\in V$, there exists a sequence $(u_{\varepsilon})_{0<\varepsilon<\varepsilon_{0}}$ in $H^{1}(M;\mathbb{C})$ such that $\|\frac{1}{\pi}\star Ju_{\varepsilon}-T\|_{\mathcal{F}}\to 0$ and $\limsup_{\varepsilon\to 0}E_{\varepsilon}(u_{\varepsilon})\leq E_{V}(T)$.

The geodesic $\gamma$ is a saddle point of the arclength with respect to
smooth perturbations, as reflected in (2.12). For use in combination with
Theorem (2.1), one needs to identify a sense in which the corresponding
current $T_{\gamma}$ is a saddle point of $E_{V}$. We defer a discussion of
this and related issues to Section 4.

###  2.4 Varifolds

We briefly recall the definition of a rectifiable $1$-varifold and introduce
some notation that will be used later. After doing this we will introduce the
definition of a general $1$-varifold. We note that the general definition will
only be used in the proof of Proposition 3.1. For general varifolds we will
follow [2] with some terminology from [26].

For any $1$-dimensional rectifiable set $\Sigma$, basic theory (see for
example [26], Lemma 11.1) shows that there exists a countable family of
$C^{1}$ curves $(\Lambda_{j})_{j\in{\mathbb{N}}}$ in $M$ such that

$\Sigma\subset N_{0}\cup\left(\cup_{j\in N}\ \Lambda_{j}\right),\qquad\mbox{
and }{\mathcal{H}}^{1}(N_{0})=0,$

and every point in $\Sigma\setminus N_{0}$ is contained in exactly one
$\Lambda_{j}$. We then define, for $x\in\Sigma\setminus N_{0}$

$\mbox{ap}T_{x}\Sigma=T_{x}\Lambda_{j}\mbox{ for the unique $j$ such that
}x\in\Lambda_{j}.$

We will write $\tau_{\Sigma}(x)$ to denote a unit vector in
$\mbox{ap}T_{x}\Sigma$.

First, we recall that if ${\mathcal{S}}$ is a countably $1$-rectifiable,
${\mathcal{H}}^{1}$ measurable subset of $M$ and
$\Theta:{\mathcal{S}}\to(0,\infty)$ is a locally
${\mathcal{H}}^{1}$-integrable function on ${\mathcal{S}}$ then we can use the
pair $({\mathcal{S}},\Theta)$ to form the measure ${\mathcal{H}}^{1}\hskip
1.0pt{\hbox to10.8pt{\hfill\vrule
height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}\Theta$, where we have extended
$\Theta$ to be zero outside of ${\mathcal{S}}$. We refer to such a measure as
a _rectifiable $1$-varifold_. We also refer to the function $\Theta$ as the
_multiplicity function_ of this rectifiable $1$-varifold and, at times, we
will write $\Theta_{{\mathcal{S}}}$ to emphasize the association. We will also
sometimes use the alternate notation $\Theta{\mathcal{H}}^{1}\hskip
1.0pt{\hbox to10.8pt{\hfill\vrule
height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}{\mathcal{S}}$ for
${\mathcal{H}}^{1}\hskip 1.0pt{\hbox to10.8pt{\hfill\vrule
height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}\Theta_{\mathcal{S}}$. If
$\Theta$ happens to be integer-valued ${\mathcal{H}}^{1}$-almost everywhere
then we will say this rectifiable varifold is of _integer multiplicity_.
Finally, if there is a $\lambda>0$ such that $\Theta\geq\lambda$ at
${\mathcal{H}}^{1}$-almost every point then we say that the rectifiable
varifold has _density bounded below_. A particular example of an integer
multiplicity rectifiable $1$-varifold that we will be interested in will be
integration over a countable collection of geodesics.

Next, for a smooth Riemannian manifold, $M$, we let $PM$ be the bundle whose
fiber $P_{a}M$ at $a\in{}M$ consists of the lines through the origin in
$T_{a}M$. If $x\in M$ and $\xi$ is a unit vector in $T_{x}M$, we will
sometimes abuse notation slightly and write $(x,\xi)$ to denote the element of
$PM$

$(x,\xi)\sim\\{s\xi:s\in\mathbb{R}\\}\subset T_{x}M.$ (2.18)

Thus $(x,\xi)$ and $(x,-\xi)$ correspond to the same element of $PM$. Suppose
that $\eta$ is a smooth function on $M$. When representing points in $PM$ as
described above, a mapping such as $(x,\xi)\in
PM\mapsto|\nabla_{\xi}\eta(x)|^{2}$ is well-defined as a function
$PM\to\mathbb{R}$, since it is independent of the choice of sign for the unit
vector $\xi$.

We let $\pi:PM\to{}M$ be the bundle projection.

We refer to a measure $\mathcal{V}\in{\mathcal{M}}(PM)$ as a _$1$ -varifold_.

Observe that to a rectifiable $1$-varifold $V={\mathcal{H}}^{1}\hskip
1.0pt{\hbox to10.8pt{\hfill\vrule
height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}\Theta_{\mathcal{S}}$, we may
associate a $1$-varifold $\mathcal{V}$ defined by

$\mathcal{V}(A):=V\left(\\{a\in M:\mbox{ap}T_{a}{\mathcal{S}}\in
A\\}\right)=\int_{\\{a\in{\mathcal{S}}\,:\,{\rm{ap}}T_{a}{\mathcal{S}}\in
A\\}}\Theta_{\mathcal{S}}(a)\,d{\mathcal{H}}^{1}.$ (2.19)

Roughly speaking, the difference between a rectifiable $1$-varifold and the
associated general $1$-varifold is that the latter explicitly records
information about the approximate tangent spaces to the set ${\mathcal{S}}$ on
which the former lives.

###  2.5 Definitions: first variation, stationarity, Brakke flow

For a rectifiable 1-varifold $\nu$ given by $\nu={\mathcal{H}}^{1}\hskip
1.0pt{\hbox to10.8pt{\hfill\vrule
height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}\Theta_{\Sigma}$, where $\Sigma$
is a 1-rectifiable set, the first variation of $\nu$ is a distribution,
denoted $\delta\nu$, whose action on smooth vector fields $X$ is defined by

$\delta\nu(X):=\int_{\Sigma}(\tau_{\Sigma}(x),\nabla_{\tau_{\Sigma}(x)}X(x))_{g}\
\Theta(x)\ d{\mathcal{H}}^{1}.$ (2.20)

(Note that since $\tau_{\Sigma}$ appears quadratically, the choice of unit
vector in $\mbox{ap}T_{x}\Sigma$ does not matter.) A $1$-d varifold $\nu$ of
the given form is stationary if

$\delta\nu=0.$ (2.21)

We remark that in light of (2.2), of course it follows from an integration by
parts that one can associate a multiplicity-one stationary varifold with the
geodesic $\gamma$. Properties of stationary varifolds will be recalled later
as needed.

For simplicity, we discuss Brakke flows and related notions from geometric
measure theory only in the case of $1$-dimensional varifolds in the
$3$-dimensional manifold $(M,g)$.

Let

$\nu_{*}^{t}=\Theta_{*}(x,t){\mathcal{H}}^{1}\hskip 1.0pt{\hbox
to10.8pt{\hfill\vrule height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}\Sigma^{t}_{\nu},\qquad t\geq 0$
(2.22)

be a family of rectifiable $1$-varifolds in $M$. To say that
$(\nu_{*}^{t})_{t>0}$ is a Brakke flow means that for $t>0$ there exists a
$\nu^{t}_{*}$-integrable vector field $H(\cdot,t)$ along $\Sigma^{t}_{\nu}$
(that is, $\vec{H}(x,t)\in T_{x}M$ for $\nu^{t}_{*}$ almost every
$x\in\Sigma^{t}_{\nu}$) such that the following hold. First,

$\delta\nu^{t}_{*}(X)=\int_{M}(X,H)_{g}\,d\nu^{t}_{*}=\int_{\Sigma^{t}_{\nu}}(X,H)_{g}\,\Theta_{*}(x,t)\,d{\mathcal{H}}^{1}$

for all $C^{1}$ vector fields $X$. Second, for every $t>0$ and every
nonnegative $\chi\in C^{2}(M)$,

$\limsup_{s\to
t}\frac{\nu^{t}_{*}(\chi)-\nu^{s}_{*}(\chi)}{t-s}\leq-\int_{M}\chi|H|^{2}d\nu^{t}_{*}+\int_{M}(\nabla\chi,P^{\perp}(H))_{g}d\nu^{t}_{*}$
(2.23)

where at a point $x\in\Sigma^{t}_{\nu}$ at which $T_{x}\Sigma^{t}_{\nu}$
exists, we write $P^{\perp}(\cdot)$ to denote orthogonal projection onto
$(T_{x}\Sigma^{t}_{\nu})^{\perp}\subset T_{x}M$.

For $(\nu^{t}_{*})_{t>0}$ a Brakke flow in $M$ of the form (2.22), it is an
immediate consequence of (2.23) that

$t\mapsto\nu^{t}_{*}(M)\mbox{ is nonincreasing}.$ (2.24)

Another simple fact we will need is the following.

######  Lemma 2.2.

If there exist numbers $0\leq a<b$ such that

$t\mapsto\nu^{t}_{*}(M)\mbox{ is constant for $a<t<b$}$ (2.25)

then

$\exists$ a stationary varifold $V_{*}$ in $M$ such that $\nu^{t}_{*}=V_{*}$
for all $a<t<b$. (2.26)

###### Proof.

Clearly, if (2.25) holds, then by taking $\chi=1$ in (2.23), we find that
$H=0$ a.e. in $\Sigma^{t}_{\nu}$ for every $t\in(a,b)$. It follows that
$\nu^{t}_{*}$ is stationary for such $t$. It is also easy to see that
$t\mapsto\nu^{t}_{*}$ is constant for $t\in(a,b)$. Indeed, given any
nonnegative $\chi_{1}\in C^{2}(M)$, choose $\chi_{2}\in C^{2}(M)$ such that
$\chi_{1}+\chi_{2}$ is constant on $M$. Then it follows from (2.25) that

$\int_{M}(\chi_{1}+\chi_{2})\,d\nu^{t}_{*}=\int_{M}\chi_{1}\,d\nu^{t}_{*}+\int_{M}\chi_{2}\,d\nu^{t}_{*}=c\nu^{t}_{*}(M)\mbox{
is constant }\mbox{ for }t\in(a,b).$

On the other hand, since $H=0$, it follows from (2.23) that

$\int_{M}\chi_{j}\,d\nu^{t}_{*}\mbox{ is nonincreasing for $j=1,2$, for
$t\in(a,b)$.}$

These together imply that $t\mapsto\int_{M}\chi_{j}\,d\nu^{t}_{*}$ is constant
for $j=1,2$. Since this holds for all nonnegative $\chi_{1}\in C^{2}(M)$, it
easily follows that $\nu^{t}_{*}$ does not depend on $t\in(a,b)$, proving
(2.26). ∎

We will make heavy use of results from a paper of Allard and Almgren [2] on
stationary $1$-dimensional varifolds with positive density in a Riemannian
manifold. Among other results, they prove that a stationary $1$-d varifold
with density bounded away from $0$ is supported on a finite or countable union
of geodesic segments terminating at singular points. From these singular
points multiple segments emanate, with a balance condition on the weighted
sum, at each singular point, of the tangent vectors generating the geodesics
that meet there. Other results from [2] will be cited as the need arises.

###  2.6 Asymptotic analysis of the Ginzburg-Landau heat flow

As a last preliminary, we state a recently established extension to the
Riemannian setting of a theorem of Bethuel, Orlandi and Smets [5], who built
on prior work of a number of authors, including [9, 3, 16].

The theorem quoted below is proved in [6].

######  Theorem 2.3.

Assume that $(N,h)$ is a closed Riemannian manifold of dimension $n\geq 3$.
Let $u_{\varepsilon}:N\times[0,\infty)\to\mathbb{C}$ solve the Ginzburg-Landau
heat flow

$\partial_{t}u_{\varepsilon}-\Delta
u_{\varepsilon}+\frac{1}{\varepsilon^{2}}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon}=0\qquad\mbox{
on }N\times(0,\infty)$

with initial data $u_{\varepsilon}(x,0)=u_{\varepsilon}^{0}(x)$. Assume that
there exists $M_{0}>0$ such that

$E_{\varepsilon}(u_{\varepsilon}^{0})\leq M_{0}.$

For every $t\geq 0$, let $\mu_{\varepsilon}^{t}$ be the measure on $N$ defined
by

$\mu_{\varepsilon}^{t}(A)=\int_{A}\frac{e_{\varepsilon}(u_{\varepsilon}(\cdot,t))}{{|\\!\log\varepsilon|}}\,\emph{\,\mbox{vol}}\qquad\mbox{
for every Borel }A\subset N.$

Then after passing to a subsequence (still denoted simply by $\varepsilon$),
there exist measures $\mu_{*}^{t}\in{\mathcal{M}}(N)$ for every $t>0$ such
that

$\displaystyle\mu_{\varepsilon}^{t}\rightharpoonup\mu_{*}^{t}$
$\displaystyle\mbox{weakly as measures for every }t>0.$

Moreover, there exists a smooth function $\Phi_{*}$ solving the heat equation
on $N\times(0,\infty)$ and a family of measures $(\nu^{t}_{*})_{t>0}$ on $N$,
such that for every $t>0$

$\mu_{*}^{t}=\frac{1}{2}|\nabla\Phi_{*}(x,t)|^{2}\,\emph{\,\mbox{vol}}+\nu^{t}_{*}$

with $\nu^{t}_{*}$ taking the form

$\nu_{*}^{t}=\Theta_{*}(x,t){\mathcal{H}}^{n-2}\hskip 1.0pt{\hbox
to10.8pt{\hfill\vrule height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}\Sigma^{t}_{\nu}$ (2.27)

where $\Sigma_{\nu}^{t}$ is an $(n-2)$-dimensional rectifiable subset of $N$
and $\Theta_{*}$ is a bounded measurable function. In addition, there exists a
function $\eta:(0,\infty)\to(0,\infty)$ such that

$\Theta_{*}(x,t)=\lim_{r\to
0}\frac{\nu^{t}_{*}(B_{r}(x))}{\omega_{n}r^{n-2}}\geq\eta(t)$ (2.28)

for ${\mathcal{H}}^{n-2}$ a.e. $x\in\Sigma^{t}_{\nu}$, for a.e. $t>0$.
Finally, the family $(\nu_{*}^{t})_{t>0}$ is a Brakke flow.

##  3 A non-existence result for stationary $1$-varifolds near a non-
degenerate geodesic

The proof of our main result hinges crucially on showing there is no
stationary varifold sitting over a $1$-current that is nearby $T_{\gamma}$,
the $1$-current associated with the non-degenerate geodesic $\gamma$. While
the non-degeneracy assumption (2.9) easily precludes the existence of another
nearby smooth geodesic, it is the need to rule out proximity in the weaker
sense of (3.2), (3.3) below and within the larger class of varifolds that
makes the result below much more challenging to establish.

######  Proposition 3.1.

Let $T_{\gamma}$ be the $1$-current in $M$ corresponding to integration over
the nondegenerate geodesic $\gamma$, and let $\eta>0$ be given. Then there
exists $r_{0}>0$ depending on $M,\gamma$, and $\eta$, such that for
$0<r<r_{0}$ there is no stationary $1$-dimensional rectifiable varifold
$V_{*}$ and $1$-current
$J_{1}\in{\mathcal{R}}_{1}\cap\mathcal{F}_{1}^{\prime}(M)$ satisfying the
conditions

$V_{*}=\Theta_{*}(x){\mathcal{H}}^{1}\hskip 1.0pt{\hbox to10.8pt{\hfill\vrule
height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}\Sigma_{*},$ (3.1)

for $\Sigma_{*}\;1$-rectifiable and $\Theta_{*}\geq\eta>0\;{\mathcal{H}}^{1}$
a.e. in $\Sigma_{*}$,

$\|J_{1}-T_{\gamma}\|_{\mathcal{F}}=r,$ (3.2)

and

$\qquad V_{*}\geq\|J_{1}\|,\qquad V_{*}(M)\leq L.$ (3.3)

The starting point of the proof is provided by the following lemma,
established by Mesaric [20].

######  Lemma 3.2.

For $T_{\gamma}$ as above, let
$J_{1}\in{\mathcal{R}}_{1}\cap\mathcal{F}_{1}^{\prime}(M)$ be a current
satisfying (3.2), and such that

$\partial J_{1}=0\qquad\mbox{and }\qquad{\bf M}(J_{1})\leq L.$

Then provided $r$ is taken sufficiently small, there is a $1$-current
$J_{1}^{*}\in{\mathcal{R}}_{1}(M)$ such that the support of $J_{1}^{*}$,
denoted by $\Gamma^{*}$, consists of a single Lipschitz curve with no boundary
satisfying

$\Gamma^{*}\subset{}K_{2\sqrt[3]{r}}\cap\operatorname{spt}(J_{1}),$ (3.4)

and

$\Gamma^{*}\cap\tau^{-1}(t)\neq\varnothing\text{ for all
}t\in\mathbb{R}/{}L{\mathbb{Z}},$ (3.5)

(cf. (2.6) for the definition of $\tau$).

In addition,

${\bf M}(J_{1}-J_{1}^{*})={\bf M}(J_{1})-{\bf M}(J_{1}^{*}),$ (3.6)

there exists a constant $C_{1}>0$ such that

${\bf M}(J_{1}^{*})\geq L-C_{1}\sqrt[3]{r},$ (3.7)

and

$\displaystyle J_{1}^{*}-$ $\displaystyle T_{\gamma}=\partial{}S^{*}\text{ for
some }2\text{-current }S^{*}\text{ with}$ (3.8)
$\displaystyle\operatorname{spt}(S^{*})\subset\overline{K_{5\sqrt[3]{r}}}\text{
and }{\bf M}(S^{*})<\infty.$

This is demonstrated in Lemma $4.4$ and comments following Lemma $4.6$ of
[20]. The proof is an adaptation to the Riemannian setting of arguments from
[11], Lemma 5.5. The idea is that (3.2) and the definition of the flat norm
imply that a large set of transverse slices to $\Gamma$ must intersect $J_{1}$
at exactly one point, and this point must be close to $\Gamma$. Behavior of
$J_{1}$ on other slices is constrained by the assumption that $\partial
J_{1}=0$ and the mass bound. The proof also uses Federer’s decomposition of
integral $1$-currents, [8], 4.2.25.

Proof of Proposition 3.1
Step 1: First we show that the rectifiable varifold $V_{*}$ does not have any
mass outside of $K_{\sqrt[4]{r}}$. The idea is that if this fails, then the
monotonicity formula and (3.7) would contradict the assumption ${\bf
M}(V_{*})\leq L$. This argument relies crucially on the uniform lower density
bound for $V_{*}$.

We recall that the Hessian Comparison Theorem, see Theorem $6.6.1$ of [12],
gives us that if $\mu>0$ is an upper bound on the absolute value of the scalar
curvature over $M$ and $r>0$ is chosen so that

$r<\frac{1}{2}\min\left\\{r_{0},\frac{\pi}{2\sqrt{\mu}}\right\\},$

then for each $p\in{}M$ and all $x\in{}B_{r}(p)$ and $v\in{}T_{x}M$ we have

$\sqrt{\mu}\rho(x)\cot\left(\sqrt{\mu}\rho(x)\right)\left|v\right|^{2}\leq\text{Hess}_{g}\left(\frac{\rho^{2}}{2}\right)(v,v)\leq\sqrt{\mu}\rho(x)\coth\left(\sqrt{\mu}\rho(x)\right)\left|v\right|^{2}$
(3.9)

where $\rho(x)=d(x,p)$. In view of the lower density bound
$\Theta_{*}\geq\eta$, $V_{*}$ almost everywhere, it follows from a Riemannian
version of the Monotonicity Formula (established with different notation in
[2] in item $(5)$ of Theorem $1$ on pg. $87$) that there exists $r_{con}(M)>0$
such that for every $0<s<r_{con}$,

$\eta\leq\frac{1}{2s}\int_{B_{s}(p)}\\!{}\text{Hess}_{g}\left(\frac{\rho^{2}(x)}{2}\right)(v,v)dV_{*}$
(3.10)

for $V_{*}$-almost every $p\in M$, where $v$ is a unit vector in
$\mbox{ap}T_{x}\Sigma_{*}$. (Clearly the value of
$\text{Hess}_{g}\left(\frac{\rho^{2}(x)}{2}\right)(v,v)$ does not depend on
which unit tangent is chosen.)

It follows from $\eqref{HessComp}$ that

$2s\eta\leq(1+\mu{}s^{2})V_{*}\left(B_{s}(p)\right)$

for all
$0<s<\frac{1}{2}\min\left\\{r_{0},r_{con},\frac{\pi}{2\sqrt{\mu}}\right\\}$
and $V_{*}$-almost every $p\in M$. We conclude that for all
$0<s<\frac{1}{2}\min\left\\{r_{0},r_{con},\frac{\pi}{2\sqrt{\mu}},1\right\\}$
we have

$V_{*}\left(B_{s}(p)\right)\geq\frac{2s\eta}{1+\mu},\qquad\mbox{ for
}V_{*}\mbox{ {\em a.e. }}p\in M.$ (3.11)

We now use this to prove that if $r$ is chosen sufficiently small, then

$V_{*}\left(M\setminus{}K_{\sqrt[4]{r}}\right)=0.$ (3.12)

To verify (3.12), suppose to the contrary; then
$V_{*}\left(M\setminus{}K_{\sqrt[4]{r}}\right)>0$ and so there is a point $p$
of $\operatorname{spt}(V_{*})$ in $M\setminus{}K_{\sqrt[4]{r}}$ for which
$\eqref{LowerBoundBall}$ holds. By $\eqref{LowerBoundBall}$ we have that

$V_{*}\left(B_{\frac{1}{2}\sqrt[4]{r}}(p)\right)\geq\frac{\eta}{1+\mu}\sqrt[4]{r},\quad\qquad\mbox{
if }r<\min\left\\{r_{con},r_{0},\frac{\pi}{2\sqrt{\mu}},1\right\\}^{4}.$

By shrinking $r_{0}$ if necessary, we may assume that
$B_{\frac{1}{2}\sqrt[4]{r}}(p)\cap K_{2\sqrt[3]{r}}=\varnothing$. Hence,
appealing to (3.7), we find that

$L\geq{}V_{*}(M)\geq{}V_{*}(K_{2\sqrt[3]{r}})+V_{*}\left(B_{\frac{1}{2}\sqrt[4]{r}}(p)\right)>L-C_{1}\sqrt[3]{r}+\frac{\eta}{1+\mu}\sqrt[4]{r}.$

Choosing $r$ smaller if necessary, depending on $C_{1},\eta,\mu$, this yields
a contradiction. We conclude that (3.12) holds. Since
$\left\|J_{1}\right\|\leq{}V_{*}$, we remark that $J_{1}$ is also supported in
$K_{\sqrt[4]{r}}$.

Step 2: Next we demonstrate that

$\left|\nabla\tau(x)\right|=1+O(\sqrt[4]{r}),\qquad\left[\text{Hess}_{g}(\tau)(x)\right]\left(v,v\right)=O(\sqrt[4]{r})$
(3.13)

where $v\in{}T_{x}M$ is a unit vector, $x\in{}K_{\sqrt[4]{r}}$, and $\tau$ is
the mapping defined in (2.6).

We prove only the statement about the Hessian, as the gradient estimate
follows by similar arguments. In coordinates introduced by
$\psi:B_{\sqrt[4]{r}}(0)\times(0,L)\to{}K_{\sqrt[4]{r}}$ as defined in (2.5),
we can write $\tau$ as

$\tilde{\tau}(y,t)=t$

where $\tilde{\tau}=\tau\circ\psi$. These are what are called Fermi
coordinates, and a basic fact, proved for example in Section V of [18], is
that the vectors $\Xi_{1},\Xi_{2}$ in (2.5) can be chosen so that all
Christoffel symbols vanish along the central geodesic, that is, when $y=0$:

$\Gamma^{k}_{ij}(0,t)=0\quad\mbox{ for }i,j,k=1,\ldots,3$

where $x^{k}=y^{k}$ for $k=1,2$, and $x^{3}=t$. In general, the expression for
the Hessian in coordinates is

$\text{Hess}_{g}(\tau)(\psi(0,t))=\sum_{i=1}^{3}\sum_{j=1}^{3}\left(\frac{\partial^{2}\tilde{\tau}}{\partial{}x^{i}\partial{}x^{j}}(0,t)-\sum_{k=1}^{3}\Gamma_{ij}^{k}(0,t)\frac{\partial\tilde{\tau}}{\partial{}x^{k}}(0,t)\right)\mathrm{d}x^{i}\otimes\mathrm{d}x^{j},$

see for example [12], Definition 4.3.5. By combining these, we readily deduce
that

$\text{Hess}_{g}(\tau)(\psi(0,t))=0$

and thus that $\text{Hess}_{g}(\tau)(\psi(y,t))_{ij}=O_{t}(\left|y\right|)$
for $1\leq i,j\leq 3$. The Hessian estimate in (3.13) follows directly.

Step 3: Let $\mathcal{V}_{*}$ be the $1$-varifold associated as in (2.19) to
the rectifiable $1$-varifold $V_{*}$. We next demonstrate that for each
$\delta>0$ there is $r_{1}>0$ such that if $0<r<r_{1}$ in (3.2) and hence in
(3.12), then

$\mathcal{V}_{*}\left(\left\\{(x,\xi)\in{}PM:|\nabla_{\xi}\tau(x)|^{2}\leq(1-\delta)^{2}\right\\}\right)<\delta,$
(3.14)

where we recall our convention that a generic element of $PM$ — that is, a
line in $T_{x}M$ for some $x\in M$ — is represented by a pair $(x,\xi)$, where
$\xi$ is a unit vector in $T_{x}M$ spanning the given line, see (2.18). This
will establish that most tangent vectors to the support of $V_{*}$ are,
according to the measure $\mathcal{V}_{*}$, nearly parallel to $\nabla\tau$.

We suppose toward a contradiction that there is a $\delta>0$, a sequence
$\left(r_{k}\right)_{k\in{\mathbb{N}}}$ tending to $0$ from the right, and a
sequence of stationary rectifiable varifolds
$\left(V_{k}\right)_{k\in{\mathbb{N}}}$ on $M$ satisfying the hypotheses of
Proposition 3.1 with $r$ replaced by $r_{k}$ in (3.2), and such that the
associated $1$-varifolds $\mathcal{V}_{k}$ satisfy

$\mathcal{V}_{k}\left(\left\\{(x,\xi)\in{}PM:|\nabla_{\xi}\tau(x)|^{2}\leq(1-\delta)^{2}\right\\}\right)\geq\delta,$
(3.15)

for all $k\in{\mathbb{N}}$. In particular we have

$V_{k}(M)\leq{}L,\hskip
10.0pt\operatorname{spt}(V_{k})\subset{}K_{\sqrt[4]{r_{k}}},\hskip
10.0pt\text{and }\;\Theta_{V_{k}}(x)\geq\eta\text{ for
}x\in\operatorname{spt}(V_{k}).$ (3.16)

Since $\left(\mathcal{V}_{k}\right)_{k\in{\mathbb{N}}}$ is a sequence of
stationary rectifiable varifolds we may combine
$\eqref{VarifoldSequenceProperties}$ with the compactness result of Theorem
$42.7$, pg. 247 of [26] to conclude that there is a subsequence
$\left(\mathcal{V}_{k_{j}}\right)_{j\in{\mathbb{N}}}$ and a rectifiable
varifold $V$ with associated multiplicity $\Theta_{V}$ such that

  1. 1.

$\mathcal{V}_{k_{j}}\rightharpoonup\mathcal{V}$ weakly as measures

  2. 2.

$\Theta_{V}(x)\geq\eta$ on $\operatorname{spt}(V)$

  3. 3.

$\delta{}V(W)\leq\liminf_{j\to\infty}\delta{}V_{k_{j}}(W)$ for all
$W\subset\subset{}M$

It follows from $\eqref{VarifoldSemicontinuity}$ and the fact that each
$\mathcal{V}_{k_{j}}$ is stationary that $\mathcal{V}$ is also stationary.
Then from $\eqref{WeakConvergence}$ and the fact that
$\operatorname{spt}(V_{k_{j}})\subset{}K_{\sqrt[4]{r_{k_{j}}}}$ we conclude
that $\operatorname{spt}(V)\subset\Gamma$. Next, we observe that, due to
$\eqref{EachHeight}$, each $V_{k_{j}}$ has support that contains a closed
curve that meets every level set of $\tau$. Hence,
$\operatorname{spt}(V)=\Gamma$ as a result of $\eqref{WeakConvergence}$.
Observe that since $\mathcal{V}$ is a stationary varifold with density bounded
below and $\operatorname{spt}(V)=\Gamma$, then by the Theorem on page $89$ of
[2] we have that $V$ is simply a constant multiplicity multiple of the
stationary varifold $V_{\Gamma}$ associated with the geodesic $\gamma$.
Applying the weak convergence (1) to (3.15), however, we see that

$\mathcal{V}\left(\left\\{(x,\xi)\in{}PM:|\nabla_{\xi}\tau(x)|^{2}\leq(1-\delta)^{2}\right\\}\right)\geq\delta,$

an impossibility given that all tangent vectors $\xi$ along $\Gamma$ coincide
with $\pm\nabla\tau$. Step 4: Next we introduce three sets corresponding to
slices normal to the central geodesic that are in some sense bad. We will
argue that two of them correspond to sets of $t$-values of measure zero while
the third is of small measure.

We introduce the first such set, ${\mathcal{B}}_{1}$, through the function
$h:(0,L)\to\mathbb{R}$ given by

$h(t)=\int_{\pi^{-1}\left(\left\\{0\leq\tau\leq{}t\right\\}\cap{}K_{\sqrt[4]{r}}\right)}\\!{}\left|\nabla_{\xi}\tau(x)\right|^{2}d\mathcal{V}_{*}(x,\xi),$

with ${\mathcal{B}}_{1}$ defined by

${\mathcal{B}}_{1}:=\left\\{a\in(0,L):\;h\;\text{is not differentiable
at}\;a\right\\}.$ (3.17)

Since $h$ is non-decreasing, it is differentiable ${\mathcal{L}}^{1}$-almost
everywhere and consequently ${\mathcal{L}}^{1}({\mathcal{B}}_{1})=0$.

Now we recall that the singular set $S_{V_{*}}$, as defined in [2], is the set
of points of $M$ near which $\Theta_{V_{*}}$, restricted to
$\operatorname{spt}(V_{*})$, is not constant. Then we introduce the set
${\mathcal{B}}_{2}$ as the set of slices meeting the singular set:

${\mathcal{B}}_{2}:=\left\\{t\in(0,L):\left\\{\tau=t\right\\}\cap{}S_{V_{*}}\neq\varnothing\right\\}=\\{\tau(x):x\in
S_{V_{*}}\cap K_{\sqrt[4]{r}}\\}.$

We claim that ${\mathcal{L}}^{1}\left({\mathcal{B}}_{2}\right)=0$ as well.

To this end, we note that in the remark following the theorem on page $89$ of
[2], it is stated that

$V_{*}\left(S_{V_{*}}\right)=0,$ (3.18)

and so by $\eqref{ldb}$ and $\eqref{VarifoldMeasureSingularSet}$ we have that

$0=\int_{S_{V_{*}}}\\!{}\Theta(x)d{\mathcal{H}}^{1}(x)\geq\eta\,{\mathcal{H}}^{1}\left(S_{V_{*}}\right).$

We conclude that

${\mathcal{H}}^{1}\left(S_{V_{*}}\right)=0.$ (3.19)

Since ${\mathcal{B}}_{2}$ is the image of a subset of $S_{V_{*}}$ by the
Lipschitz map $\tau:K_{\sqrt[4]{r}}\to(0,L)$, it follows that
${\mathcal{L}}^{1}\left({\mathcal{B}}_{2}\right)=0$ as claimed.

The final ‘bad’ set of slices is ${\mathcal{B}}_{3}$ defined by

${\mathcal{B}}_{3}:=\left\\{t\in(0,L):\;\exists(x,\xi)\in\operatorname{spt}(\mathcal{V}_{*})\;s.t.\;|\nabla_{\xi}\tau(x)|^{2}<(1-\delta)^{2},\tau(x)=t\right\\}.$

Replacing the role of the equality (3.18) by the inequality (3.14) in the
argument above, the same line of reasoning goes to show that there is a
constant $C_{2}>0$ such that

${\mathcal{L}}^{1}\left({\mathcal{B}}_{3}\right)<\frac{C_{2}\delta}{\eta},$

where $r$ is chosen sufficiently small.

Step 5: We now use the results obtained in Steps $1$ and $2$ to show that, for
$r$ and $\delta$ chosen sufficiently small, and
$a,b\in(0,L)\setminus{\mathcal{B}}_{1}$ we have

$h^{\prime}(b)=h^{\prime}(a)+O(\sqrt[4]{r})$

where $h$ is as defined in Step 4.

For $s$ small and positive, we define the function

$H_{a,b;s}(t)=\begin{cases}1&\text{if }a+s<t<b-s\\\ 0&\text{if }0<t<a\text{ or
}b<t<L\\\ \frac{t-a}{s}&\text{if }a\leq{}t\leq{}a+s\\\ \frac{b-t}{s}&\text{if
}b-s\leq{}t\leq{}b,\end{cases}$

and we let $X$ be a smooth vector field on $M$ such that
$X(x)=H_{a,b;s}(\tau(x))\nabla\tau(x)$ for $x\in K_{r_{0}/2}$. The fact that
$\mathcal{V}_{*}$ is stationary implies that $\delta V_{*}(X)=0$, cf. (2.20),
and this means that

$\displaystyle\int_{\pi^{-1}\left(K_{\sqrt[4]{r}}\right)}$
$\displaystyle\\!{}H_{a,b;s}^{\prime}(\tau)\left|\nabla_{\xi}\tau\right|^{2}d\mathcal{V}_{*}(x,\xi)+$
(3.20)
$\displaystyle\int_{\pi^{-1}\left(K_{\sqrt[4]{r}}\right)}\\!{}H_{a,b;s}(\tau)\text{Hess}_{g}(\tau)\left(\xi,\xi\right)d\mathcal{V}_{*}(x,\xi)=0,$

We have used Step $1$ to obtain that $V_{*}$ is concentrated on
$K_{\sqrt[4]{r}}$. Next, we use Step $2$, $V_{*}(M)\leq{}L$, and the fact that
$\left\|H_{a,b;s}\right\|_{L^{\infty}(\mathbb{R})}=1$ to conclude that

$\int_{\pi^{-1}\left(K_{\sqrt[4]{r}}\right)}\\!{}H_{a,b;s}(\tau)\text{Hess}(\tau)\left(\xi,\xi\right)d\mathcal{V}_{*}(x,\xi)=O(\sqrt[4]{r}).$
(3.21)

Then we observe that, by the definition of $H_{a,b;s}$, we have

$\displaystyle\int_{\pi^{-1}\left(K_{\sqrt[4]{r}}\right)}\\!{}H_{a,b;s}^{\prime}\left|\nabla_{\xi}\tau\right|^{2}d\mathcal{V}_{*}(x,\xi)$
$\displaystyle=\frac{1}{s}\int_{\pi^{-1}\left(\left\\{a\leq\tau\leq{}a+s\right\\}\cap{}K_{\sqrt[4]{r}}\right)}\\!{}\left|\nabla_{\xi}\tau\right|^{2}d\mathcal{V}_{*}(x,\xi)$
$\displaystyle-\frac{1}{s}\int_{\pi^{-1}\left(\left\\{b-s\leq\tau\leq{}b\right\\}\cap{}K_{\sqrt[4]{r}}\right)}\\!{}\left|\nabla_{\xi}\tau\right|^{2}d\mathcal{V}_{*}(x,\xi).$

Combing this with $\eqref{VariationCondition}$ and (3.21) yields

$\frac{h(b)-h(b-s)}{s}=\frac{h(a+s)-h(a)}{s}+O(\sqrt[4]{r}).$

Letting $s$ tend to zero, we find that

$h^{\prime}(b)=h^{\prime}(a)+O(\sqrt[4]{r}).$ (3.22)

Step 6: We next introduce a set of ‘good’ slices via

${\mathcal{G}}:=\left\\{a\in(0,L):{\mathcal{H}}^{0}\left(\left\\{\tau=a\right\\}\cap\operatorname{spt}(V_{*})\right)=1\right\\},$

and in this step we will demonstrate that

$(0,L)\setminus\left({\mathcal{B}}_{1}\cup{\mathcal{B}}_{2}\cup{\mathcal{B}}_{3}\right)\subset{\mathcal{G}}.$
(3.23)

From this and Step 4 it will follow that

${\mathcal{L}}^{1}\left({\mathcal{G}}\right)\geq{}L-\frac{C_{2}\delta}{\eta}$
(3.24)

provided that $r$ and $\delta$ are chosen sufficiently small.

Suppose by way of contradiction that there exists a value $a$ such that

$a\in(0,L)\setminus\left({\mathcal{B}}_{1}\cup{\mathcal{B}}_{2}\cup{\mathcal{B}}_{3}\cup{\mathcal{G}}\right).$
(3.25)

Then in light of (3.5) we have that

${\mathcal{H}}^{0}\left(\left\\{\tau=a\right\\}\cap\operatorname{spt}(V_{*})\right)\geq
2$

for some
$a\in(0,L)\setminus\left({\mathcal{B}}_{1}\cup{\mathcal{B}}_{2}\cup{\mathcal{B}}_{3}\right)$.
We first argue that for such an $a$ and every $c,\delta\in(0,1)$ it must hold
that

$h^{\prime}(a)\geq(1-c)(1-\delta)^{2}(1+\eta)$ (3.26)

provided $r$ is chosen sufficiently small, depending on $c$. Fix
$0<c,\delta<1$, and choose $1<\alpha<\frac{1}{\delta}$. Note that since
$a\not\in{\mathcal{B}}_{2}$ then by item $(3)$ of the theorem on page $89$ of
[2] we have that $\left\\{\tau=a\right\\}\cap\operatorname{spt}(V_{*})$
consists only of interior points of the constituent geodesics (or “intervals”
as they are referred to in [2]) making up $\mathcal{V}_{*}$. Moreover, the
endpoints of these geodesics cannot accumulate at $a$. We conclude by
compactness of $\operatorname{spt}(V_{*})$ that
$\left\\{\tau=a\right\\}\cap\operatorname{spt}(V_{*})$ can intersect only
finitely many of these constituents of $\operatorname{spt}(V_{*})$. Also since
$a\not\in{\mathcal{B}}_{3}$, it follows that this slice must intersect
$\operatorname{spt}(V_{*})$ transversally, so that
$\left\\{\tau=a\right\\}\cap\operatorname{spt}(V_{*})$ consists of finitely
many points, say $x_{1},\ldots,x_{K}$, where $K\geq 2$ by the choice of $a$.

It follows from the gradient estimate $|\nabla\tau|=1+O(\sqrt[4]{r})$ in
$K_{\sqrt[4]{r}}$, established in (3.13), that for $0<\alpha<\beta<L$, the
geodesic distance between the sets $\\{\tau=\alpha\\}\cap K_{\sqrt[4]{r}}$ and
$\\{\tau=\beta\\}\cap K_{\sqrt[4]{r}}$ is at least
$(\beta-\alpha)/(1+O(\sqrt[4]{r}))$, if $\beta-\alpha<L/2$. Hence if $r$ is
small enough, then

$B_{(1-c)s}(x_{i})\cap K_{\sqrt[4]{r}}\subset\\{x\in
K_{\sqrt[4]{r}}:a-s<\tau(x)<a+s\\}.$

(Here and below, we tacitly assume that $0<a-s<a+s<L$ and $s<L/2$.)

Next we again use that $a\notin{\mathcal{B}}_{3}$ to choose $s_{0}>0$ small
enough so that

$\left|\nabla_{\xi}\tau(y)\right|^{2}\geq(1-\alpha\delta)^{2}$ (3.27)

if $y\in{}\cup_{i=1}^{K}B_{s_{0}}(x_{i})$ and
$(y,\xi)\in\operatorname{spt}(\mathcal{V}_{*})$. Combining these facts, for
each $0<s\leq s_{0}$ we estimate

$\frac{h(a+s)-h(a-s)}{2s}\geq(1-\alpha\delta)^{2}\sum_{i=1}^{K}\frac{V_{*}\left(B_{(1-c)s}(x_{i})\right)}{2s}$

We now apply item $(5)$ of the theorem on page $87$ of [2] and let $s\to
0^{+}$, using the differentiability of $h$ at $a$ guaranteed by the assumption
$a\notin{\mathcal{B}}_{1}$, to find

$h^{\prime}(a)\geq{}(1-c)(1-\alpha\delta)^{2}\sum_{i=1}^{K}\Theta_{*}(x_{i})\geq{}(1-c)(1-\alpha\delta)^{2}(1+\eta)$

Here we have used Lemma 3.2 to assert that $\Gamma^{*}$ intersects each level
set of $\tau$ with $\Theta_{*}(x)\geq 1$ for $x\in\Gamma^{*}$ by (3.3), and
that $\Theta_{*}\geq\eta$ in general by $\eqref{formofVstar}$. Since
$\alpha>1$ was arbitrary we may let $\alpha\to 1^{+}$ to obtain (3.26).

In light of (3.22), it then follows from (3.26) that for any
$b\notin{\mathcal{B}}_{1}$ we obtain

$h^{\prime}(b)\geq{}(1-c)(1-\delta)^{2}(1+\eta)+O(\sqrt[4]{r})$

Thus, choosing $c$, $r$, and $\delta$ sufficiently small and recalling that
${\mathcal{L}}^{1}({\mathcal{B}}_{1})=0$, we deduce that

$h^{\prime}(b)\geq 1+\frac{\eta}{2}\qquad\mbox{ for {\em a.e.} }b\in(0,L).$
(3.28)

Thus, if there were a value $a\in(0,L)$ satisfying (3.25), then

$\displaystyle\left[1+C_{3}\sqrt[4]{r}\right]L\geq{}\left[1+C_{3}\sqrt[4]{r}\right]V_{*}\left(\pi^{-1}\left\\{0\leq\tau\leq{}L\right\\}\right)$
$\displaystyle\geq
h(L)-h(0)=\int_{0}^{L}\\!{}h^{\prime}(s)ds\geq\left(1+\frac{\eta}{2}\right)L.$

Here we use (3.13) in the second inequality and the constant $C_{3}$ depends
only on $M$ and $\Gamma$. If we choose $r$ sufficiently small, the
contradiction is reached, establishing (3.23) and (3.24). Step 7: In this step
we show that

$S_{V_{*}}=\varnothing.$

It will immediately follow that the Lipschitz curve $\Gamma^{*}$ guaranteed by
Lemma 3.2 represents the entire rectifiable varifold and is in fact a closed
geodesic.

Crucial use in this step will be made of the following general property of
stationary $1$-varifolds (cf. [2], pg. 88.):

Every point $p\in M$ is contained in an open set $U_{p}$ such that if $V$ is
any stationary varifold on $M$ with support in $U_{p}$, and if the support of
$\delta V$ consists of exactly two points, then $V$ is the varifold
corresponding to a constant multiple of the geodesic joining these two points.
This result is proved in [2] for possibly noncompact manifolds. Since $M$ is
compact, we may invoke the Lebesgue Number Lemma to conclude that there exists
$\lambda>0$ such that for any $p\in M$, the geodesic ball $B_{\lambda}(p)$ has
the stated property. For any $A\subset(0,L)$, we will write

$K_{\sqrt[4]{r}}(A):=\\{x\in K_{\sqrt[4]{r}}:\tau(x)\in
A\\}=\\{\psi(y,t):|y|\leq\sqrt[4]{r},\ \ t\in A\\}.$

By extending $\psi$ to be periodic with respect to the $t$ variable in the
natural way, we can define $K_{\sqrt[4]{r}}(A)$ for any $A\subset\mathbb{R}$.
By shrinking $r$ and $\delta$, we may arrange that if $I$ is any interval of
length at most $2C_{2}\delta/\eta$, where $C_{2}$ is the constant appearing in
(3.24), then $K_{\sqrt[4]{r}}(I)$ is contained in a ball of radius $\lambda$.
We will prove the claim by showing that

$\mbox{ for any $t\in[0,L)$, }\qquad K_{\sqrt[4]{r}}(\\{t\\})\cap
S_{V_{*}}=\varnothing.$

Indeed, for any $t$ we can appeal to (3.24) to find some
$s_{1},s_{2}\in{\mathcal{G}}$ such that

$s_{1}<t<s_{2},\qquad s_{2}-s_{1}<\frac{2C_{2}\delta}{\eta}.$

We now apply the result stated at the outset of this step to the varifold

$\tilde{V}=V_{*}\hskip 1.0pt{\hbox to10.8pt{\hfill\vrule
height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}K_{\sqrt[4]{r}}([s_{1},s_{1}]),$

in an open ball $B_{\lambda}(p)$ that contains
$K_{\sqrt[4]{r}}([s_{1},s_{1}])$. The definition of ${\mathcal{G}}$ implies
that $V_{*}$ intersects $\\{\tau=s_{j}\\}$ in exactly one point, say $x_{j}$,
and that $\delta\tilde{V}$ is supported in $\\{x_{1},x_{2}\\}$. Hence, this
restriction of $V_{*}$ consists of a multiple of the geodesic joining these
two points. This immediately implies the claim. Since $S_{V_{*}}=\varnothing$,
$V_{*}$ must simply be the rectifiable varifold associated with a single,
closed, smooth geodesic.

By perhaps shrinking $r$ one more time and applying the Morse-Palais Lemma,
cf. [24], pg. 307, we may conclude that the central geodesic $\Gamma$, being a
nondegenerate critical point of length, is isolated and so necessarily
$\operatorname{spt}(V_{*})=\Gamma$. However, this contradicts $\eqref{J0.p1}$
since $r>0$, and the proof of Proposition 3.1 complete.

∎

##  4 Finding good trajectories

The critical points of Ginzburg-Landau that we seek will be obtained as limits
of certain carefully chosen trajectories of the Ginzburg-Landau heat flow. In
this section we identify these trajectories.

######  Proposition 4.1.

Given $\delta>0$, there exists $\varepsilon_{0}(\delta)>0$ such that for every
$\varepsilon\in(0,\varepsilon_{0})$ and every $\tau>0$, there is a solution
$u_{\varepsilon}=u_{\varepsilon}(x,t;\delta,\tau)$ of the Ginzburg-Landau heat
flow

$\displaystyle\partial_{t}u_{\varepsilon}=\Delta
u_{\varepsilon}-\frac{1}{\varepsilon^{2}}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon}\qquad\mbox{
in }M\times(0,\infty),$ (4.1) $\displaystyle
u_{\varepsilon}(x,0)=u_{\varepsilon}^{0}(x)\qquad\mbox{ for }x\in M,$

such that $|u_{\varepsilon}(x,t)|\leq 1$ for every $(x,t)\in
M\times[0,\infty)$, and

$\|\frac{1}{\pi}\star
Ju_{\varepsilon}^{0}-T_{\gamma}\|_{\mathcal{F}}<\delta,\qquad
E_{\varepsilon}(u_{\varepsilon}^{0})\leq L+\delta,\qquad
E_{\varepsilon}(u_{\varepsilon}(t))\geq L-\delta\ \mbox{ for all
}t\in[0,\tau].$ (4.2)

The proposition follows from small modifications of the asymptotic minmax
theory developed in [11, 20]. Indeed, in the end the proof amounts to this:

###### short proof of Proposition 4.1.

This follows from arguments in [11, 20]. ∎

In the remainder of this section we expand on this, aiming to provide enough
detail to convey the main ideas, to explain where we depart from [11, 20], and
to make it possible, in principle, to check the terse proof given above.

We remark that the main difference between [11, 20] and our present treatment
is that those earlier works use a pseudo-gradient flow for the energy
$E_{\varepsilon}$, whereas we employ a small modification of the Ginzburg-
Landau heat flow (4.1) for similar purposes. The use of (4.1) is necessary for
our approach, due to our reliance in Section 5 below on Theorem 2.3.

###  4.1 Saddle point property of $E_{V}$

Our assumptions about $\gamma$ imply, roughly speaking, that the “arclength
functional” has a local minmax geometry near $\gamma$, as reflected in (2.12),
with respect to smooth perturbations. In particular, there is an
$\ell$-parameter family of arclength-decreasing perturbations of $\gamma$, and
arclength increases for sufficiently transverse smooth perturbations. Here and
below, $\ell$ is the index of $\gamma$, see (2.10).

The result below states that the “generalized arclength functional” $E_{V}$
defined in (2.17) has a a saddle point, in a suitable weak sense, at the
current $T_{\gamma}\in V$ corresponding to $\gamma$. The relevant notion of
saddle point was first introduced in [11].

######  Lemma 4.2 (cf. Theorem 4.1, [20]).

For the geodesic $\gamma$ satisfying (2.9) and (2.15), the associated current
$T_{\gamma}$ is a saddle point of $E_{V}$ in the sense that there exist
$R,\delta_{0}>0$ and continuous functions

$P_{WV}:V\to\mathbb{R}^{\ell},\qquad Q_{VW}:W\to V$

for

$W=B_{R}^{\ell}=\\{w\in\mathbb{R}^{\ell}:|w|<R\\}$ (4.3)

such that $P_{WV}(T_{\gamma})=0$, and the following conditions are satisfied:

$\displaystyle E_{V}(T_{\gamma})<E_{V}(v)\mbox{ for }\\{v\in
V:0<\|v-T_{\gamma}\|_{\mathcal{F}}\leq\delta_{0},P_{WV}(v)=0\\},$ (4.4)
$\displaystyle Q_{VW}(0)=T_{\gamma},$ (4.5) $\displaystyle P_{WV}\circ
Q_{VW}(w)=w\quad\mbox{ for all }w\in W,$ (4.6) $\displaystyle\mbox{ and for
every $r>0$,}\sup_{\\{w\in W,|w|\geq r\\}}E_{V}(Q_{VW}(w))<E_{V}(T_{\gamma}).$
(4.7)

We sketch the proof from [20], although we note that this will not play any
role in what follows, except that the notation $\gamma_{w}$ for the curve
defined in (4.9) and $T_{\gamma_{w}}$ for the associated $1$-current, cf.
(2.13), will be used below.

To start, for $w\in W$ we define

$\xi(w)=w_{1}\xi_{1}+\ldots+w_{\ell}\xi_{\ell}$ (4.8)

where $\xi_{j}$ denote eigenfunctions of the Jacobi operator, see in
particular (2.10). We then define the curve $\gamma_{w}$ via

$\gamma_{w}(t):=\exp_{\gamma(t)}\xi(w)(t),$ (4.9)

cf. (2.7). We will always assume that $R$ is small enough that

$\|\xi(w)\|_{L^{\infty}}\leq r_{0},\qquad\mbox{ in other words, }\
\gamma_{w}\in H_{-}(r_{0}),\quad\mbox{ for }w\in W.$

With this in hand, we define
$Q_{VW}:B_{R}^{\ell}\to{\mathcal{R}}_{1}^{\prime}$ via

$Q_{VW}(w):=T_{\gamma_{w}}\quad\mbox{for}\;w\in W.$

Then (4.5) is immediate, and (4.7) follows directly from (2.12).

The construction of $P_{WV}$ is carried out in [20], Lemma 4.3, by designing
an $\mathbb{R}^{\ell}$-valued $1$-form $\Phi$ such that

$T_{\gamma_{\xi}}(\Phi)=((\xi,\xi_{1})_{L^{2}},\ldots,(\xi,\xi_{\ell})_{L^{2}})$
(4.10)

for $\xi\in L^{2}(N\Gamma)\cap Lip$, as long as $\|\xi\|_{L^{\infty}}\leq
r_{0}$, a condition that can be guaranteed by a suitable choice of $R$. Here
we recall that $\gamma_{\xi}$ is the curve given by (2.7) and
$T_{\gamma_{\xi}}$ is its associated 1-current. We then simply define
$P_{WV}(T)=T(\Phi)$. With this choice, (4.6) follows directly from (4.8),
(4.9), and (4.10).

The hard part of the proof of Lemma 4.2 is the verification of (4.4). This is
carried out in Proposition 4.1 of [20], to which we refer for the details. We
remark that the main ideas in this proof, including the construction of
$\Phi$, are similar to elements in the proof of Theorem 5 in [28].

Note that we may shrink at will the parameter $R$ in the definition (4.3) of
$W$, and the conclusions of the lemma remain valid.

###  4.2 An $\ell$-parameter family of solutions of (4.1)

To prove Proposition 4.1, we will define an $\ell$-parameter family of
solutions of (4.1) for every sufficiently small $\varepsilon>0$. In the final
step, given $\varepsilon$ and $\tau$ (where ultimately we will take
$\varepsilon<\varepsilon_{0}(\delta)$), we will choose from this family one
solution $u_{\varepsilon}$ such that $E_{\varepsilon}(u_{\varepsilon}(t))\geq
L-o(1)$ for all times $t\in[0,\tau]$.

The initial data for this family of solutions is provided by the following
result.

######  Lemma 4.3.

There exist $R,\varepsilon_{1}>0$ such that for every
$\varepsilon\in(0,\varepsilon_{1})$ and $w\in B^{\ell}_{R}$, there exists a
function $U^{\varepsilon,0}_{w}\in H^{2}(M)$ satisfying the conditions:

  1. 1.

$w\mapsto U^{\varepsilon,0}_{w}\mbox{ is Lipschitz continuous from
}B^{\ell}_{R}\mbox{ into }H^{2}(M)$

  2. 2.

$\|U^{\varepsilon,0}_{w}\|_{L^{\infty}}\leq 1$ and
$\|U^{\varepsilon,0}_{w}\|_{H^{2}}\leq C_{\varepsilon}\qquad\mbox{ for all
}\varepsilon\in(0,\varepsilon_{1}),w\in B^{\ell}_{R},$

  3. 3.

$E_{\varepsilon}(U^{\varepsilon,0}_{w})\leq L-c_{0}|w|^{2}+o(1)\quad\mbox{ as
}\varepsilon\to 0$

  4. 4.

$\|\frac{1}{\pi}\star JU^{\varepsilon,0}_{w}-T_{\gamma_{w}}\|_{V}\to 0$

uniformly for $w\in B^{\ell}_{R}$.

For fixed $w$, in view of (2.12) and the construction of $\gamma_{w}$,
conclusions (3) and (4) hold if $U^{\varepsilon,0}_{w}$ is a recovery sequence
for the current $T_{\gamma_{w}}$ the Gamma-limit in Theorem 2.1. Such
constructions are rather standard. It is easy to arrange that
$\|U^{\varepsilon,0}_{w}\|_{L^{\infty}}\leq 1$. The only points requiring
attention are that the construction has to be carried out so that it depends
continuously on $w$, in the $H^{1}$ norm, and with some control over the
$H^{2}$ norm. The former point is carried out in [20], and the latter can be
achieved by a small modification of the construction of [20]. We defer a more
detailed discussion to Appendix A.

The $H^{2}$ estimate facilitates the proof of Lemma 4.4 below, whose need
arises because we require the Ginzburg-Landau heat flow rather than the
pseudo-gradient flows employed in [11, 20].

Having constructed appropriate initial data for the Ginzburg-Landau flow, we
are now ready to define the flow that we will use in our arguments below.

######  Lemma 4.4.

For $\varepsilon\in(0,\varepsilon_{1})$ and $w\in W$, let
$U^{\varepsilon,1}_{w}(x,t)$ solve the Ginzburg-Landau heat flow with initial
data $U^{\varepsilon,0}_{w}$. Then

$(t,w)\in[0,\infty)\times W\mapsto U^{\varepsilon,1}_{w}(\cdot,t)\in
H^{1}(M;\mathbb{C})\qquad\mbox{ is continuous}.$

See Appendix A for the proof, which involves rather standard parabolic
estimates.

Finally, we define $U_{\varepsilon}:[0,\infty)\times W\to H^{1}(M;\mathbb{C})$
by

$U_{\varepsilon}(t,w):=U^{\varepsilon,1}_{w}(\cdot,\chi(w)t).$ (4.11)

for a smooth, compactly supported $\chi:B^{\ell}_{R}\to[0,1]$ such that
$\chi=1$ in $B^{\ell}_{R/2}$. We point out that

$\mbox{ if }|w|=R,\qquad\mbox{ then }\quad
U_{\varepsilon}(t,w)=U^{\varepsilon,0}_{w}\quad\mbox{ for all }t\geq 0.$
(4.12)

We can guarantee that
$E^{\varepsilon}_{U}(U_{\varepsilon}(0,w))<c_{\varepsilon}-\delta$ whenever
$|w|>R/2$ and $\varepsilon$ is small enough, for suitable $\delta$.

###  4.3 Choosing a good trajectory

We finally make use of the asymptotic saddle point geometry of
$E_{\varepsilon}$, inherited from $E_{V}$ via the Gamma-convergence Theorem
2.1, to complete the proof of Proposition 4.1.

We will use the notation

$P_{WU}(u)=P_{WV}\big{(}\frac{1}{\pi}\star Ju\big{)}\qquad\mbox{for}\;u\in
H^{1}(M;\mathbb{C}).$

######  Lemma 4.5.

There exist $\delta_{0}>0$ and $R_{0}>0$ such that for every $R\in(0,R_{0})$,
there is some $\delta=\delta(R)>0$ such that if we define

$\displaystyle a_{\varepsilon}$
$\displaystyle:=\max\\{E_{\varepsilon}(U^{\varepsilon,0}_{w}):|w|=R\\}$
$\displaystyle c_{\varepsilon}$
$\displaystyle:=\min\\{E_{\varepsilon}(u):P_{WU}(u)=0,\|\frac{1}{\pi}\star Ju-
T_{\gamma}\|_{V}\leq\delta_{0}\\}$ $\displaystyle d_{\varepsilon}$
$\displaystyle:=\max\\{E_{\varepsilon}(U^{\varepsilon,0}_{w}):|w|\leq R\\},$

then

$a_{\varepsilon}\to L-2\delta,\qquad\liminf_{\varepsilon\to
0}c_{\varepsilon}\geq L,\qquad d_{\varepsilon}\to L$ (4.13)

as $\varepsilon\to 0$, where $L$ is the length of the geodesic $\gamma$.

The assertions about $a_{\varepsilon}$ and $d_{\varepsilon}$ follow directly
from (2.12) and Lemma 4.3, and Step 3 of the proof of Theorem 4.4 of [11]
shows exactly that $c_{\varepsilon}\to L$. The proof uses only ingredients
that we have collected in Theorem 2.1 and Lemma 4.2 below.

Below we will not refer explicitly to the assertion about
$\lim_{\varepsilon\to 0}a_{\varepsilon}$, but it plays a role in the proof of
Lemma 4.6, and together with the lower bound for $\liminf c_{\varepsilon}$, it
reflects the asymptotic minmax geometry of $E_{\varepsilon}$.

Proposition 4.1 will essentially follow from the next fact.

######  Lemma 4.6.

For each $r>0$ there exists $\varepsilon_{0}>0$ and $R>0$ such that for every
$0<\varepsilon<\varepsilon_{0}$ and every $\tau>0$, there exists
$w=w(\varepsilon,\tau)$ such that

$P_{WU}(U_{\varepsilon}(\tau,w))=0,\qquad\|\frac{1}{\pi}\star
JU_{\varepsilon}(\tau,w)-J_{\gamma}\|_{V}\leq r.$ (4.14)

As a result, $w=w(\varepsilon,\tau)$ satisfies

$d_{\varepsilon}\geq E_{\varepsilon}(U_{\varepsilon}(t,w))\geq
E_{\varepsilon}(U_{\varepsilon}(\tau,w))\geq c_{\varepsilon}\quad\mbox{ for
all }t\in[0,\tau]\mbox{ and }\varepsilon\in(0,\varepsilon_{0}).$ (4.15)

Finally, $w(\varepsilon,\tau)\to 0$ as $\varepsilon\to 0$.

###### Proof.

One may prove (4.14) by simply repeating the arguments from Steps 5-8 of the
proof of Theorem $4.4$ in [11], for $r>0$ such that $0<r<\delta_{0}$, where
$\delta_{0}$ is the constant from item $\eqref{saddlepoint1}$ of Lemma 4.2.
Some comments are in order:

First, the argument in [11] is stated for a pseudo-gradient flow (see Lemma
4.8, [11]) with certain properties that our flow $(t,w)\to
U_{\varepsilon}(t,w)$ does not possess. These are in fact not needed for the
proof of (4.14), and some of them may appear in [11] only because there Lemma
4.8 is quoted directly from a standard text, which provides more than is
actually needed. These are the only properties of the flow that are required
for the proof of (4.14):

  * •

$t\mapsto E_{\varepsilon}(U_{\varepsilon}(t,w))$ is nonincreasing.

  * •

$t\mapsto U_{\varepsilon}(t,w)$ is constant for $w\in\partial W$, see (4.12).

  * •

continuity properties of the flow, as summarized in Lemma 4.4.

All of these are available here.

Without going into detail, we remark that the basic strategy of the proof is
to apply degree theory arguments to the maps $w\mapsto
P_{WU}(U_{\varepsilon}(t,w)):W\to W$ as $t$ varies from $0$ to $\tau$ (where
$\tau=1$ in [11], a harmless normalization).

Next, (4.15) follows directly from (4.14) and Lemma 4.5. Finally, we deduce
from (4.15) and Lemma 4.5 that

$E_{\varepsilon}\left(U_{\varepsilon}(0,w)\right)=E_{\varepsilon}\left(U^{\varepsilon,0}_{w}\right)\to
L\qquad\mbox{ as $\varepsilon\to 0$}.$

Then conclusion (3) of Lemma 4.3 implies that $w=w(\varepsilon,\tau)\to 0$ as
$\varepsilon\to 0$. ∎

We are now in a position to present:

###### Proof of Proposition 4.1.

Let $0<\delta<\delta_{0}$ be given where $\delta_{0}$ is as in item
$\eqref{saddlepoint1}$ of Lemma 4.2. We take $\varepsilon_{0}(\delta)$ and
$R(\delta)$ as defined in Lemma 4.6 and set
$u_{\varepsilon}(x,t;\delta,\tau)=U_{\varepsilon}(t,w(\varepsilon,\tau))$. By
shrinking $\varepsilon_{0}$ if necessary, we may ensure that
$\left|w(\varepsilon,\tau)\right|<\frac{R(\delta)}{2}$, as shown in Lemma 4.6.
Thus, $u_{\varepsilon}(x,t;\delta,\tau)$ solves $\eqref{glh}$ since

$U_{\varepsilon}(t,w(\varepsilon,\tau))=U_{w}^{\varepsilon,1}(t,x)$

for such $\varepsilon>0$.

Since $\|U^{\varepsilon,0}_{w}\|_{L\infty}\leq 1$, it is clear that
$|u_{\varepsilon}(x,t)|\leq 1$ everywhere, and all other conclusions of the
Proposition follow directly from Lemmas 4.5 and 4.6.

∎

##  5 Proof of the main result

The main result of this paper, stated more informally in the introduction as
Theorem 1.1, can now be phrased precisely as

######  Theorem 5.1.

Let $(M,g)$ be a closed oriented $3$-dimensional Riemannian manifold, and let
$\gamma$ be a closed, embedded, nondegenerate geodesic of length $L$. Assume
in addition that $\gamma=\partial S$ (in the sense of Stokes’ Theorem) for
some $2$-dimensional submanifold $S$ of $M$.

Then for every $r>0$, there exists $\varepsilon_{1}(r)>0$ such that if
$0<\varepsilon<\varepsilon_{1}(r)$, then there is a solution $u_{\varepsilon}$
of the Ginzburg-Landau equations

$-\Delta
u_{\varepsilon}+\frac{1}{\varepsilon^{2}}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon}=0\qquad\mbox{
on }M$ (5.1)

such that

$\|\frac{1}{\pi}\star Ju_{\varepsilon}-T_{\gamma}\|_{\mathcal{F}}\leq r$

and

$\left|E_{\varepsilon}(u_{\varepsilon})-L\right|<r.$

As a result, there exists a sequence $(u_{\varepsilon})_{\varepsilon>0}\subset
H^{1}(M;\mathbb{C})$ of solutions of the Ginzburg-Landau equations such that

$\|\frac{1}{\pi}\star Ju_{\varepsilon}-T_{\gamma}\|_{\mathcal{F}}\to 0,\qquad
E_{\varepsilon}(u_{\varepsilon})\to L\qquad\qquad\mbox{ as $\varepsilon\to
0$.}$ (5.2)

We remark that standard Gamma-convergence results (see Theorem 2.1) imply that
the sequence of solutions in (5.2) satisfies

$\frac{e_{\varepsilon}(u_{\varepsilon})}{\pi{|\\!\log\varepsilon|}}\to\|T_{\gamma}\|={\mathcal{H}}^{1}\hskip
1.0pt{\hbox to10.8pt{\hfill\vrule
height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}\Gamma\quad\mbox{ weakly as
measures},$ (5.3)

which is the last conclusion of Theorem 1.1. Indeed, since
$E_{\varepsilon}(u_{\varepsilon})$ is uniformly bounded, there exists some
measure $\mu$ such that

$\frac{e_{\varepsilon}(u_{\varepsilon})}{\pi{|\\!\log\varepsilon|}}\to\mu\quad\mbox{
weakly as measures},$

after perhaps passing to a subsequence, and $\mu(M)=\lim_{\varepsilon\to
0}E_{\varepsilon}(u_{\varepsilon})=L$. Standard Gamma-convergence results and
(5.2) imply

$\mu\geq\|T_{\gamma}\|,$

and since $\|T_{\gamma}\|(M)=L=\mu(M)$, it follows that $\mu=\|T_{\gamma}\|$,
proving (5.3).

###### Proof.

The proof relies on an improvement on the properties of the flow defined in
the previous section. The assertion is that the trajectory solving the
Ginzburg-Landau flow identified in Proposition 4.1 remains close to
$T_{\gamma}$ in the flat norm. More precisely, we will show: Claim:
For every $r>0$, there exist positive constants $\delta_{1}(r)$ and
$\varepsilon_{1}(r)<\varepsilon_{0}(\delta_{1})$, depending on $r$, such that
for every $\varepsilon\in(0,\varepsilon_{1})$ and $\delta\in(0,\delta_{1})$,
and for every $\tau>0$, if $u_{\varepsilon}=u_{\varepsilon}(x,t;\delta,\tau)$
is the solution of (4.1) found in Proposition 4.1, then

$\|\frac{1}{\pi}\star
Ju_{\varepsilon}(t)-T_{\gamma}\|_{\mathcal{F}}<r\qquad\mbox{ for all
}t\in[0,\tau].$ (5.4)

To establish this, we assume toward a contradiction that there exists $r>0$
and sequences $\delta_{k},\varepsilon_{k}\to 0$ and $0<t_{k}\leq\tau_{k}$ such
that $u_{k}(x,t):=u_{\varepsilon_{k}}(x,t;\delta_{k},\tau_{k})$ satisfies

$\|\frac{1}{\pi}\star Ju_{k}(\cdot,t_{k})-T_{\gamma}\|_{\mathcal{F}}=r.$ (5.5)

Clearly, we may assume that $r<r_{0}$ for some $r_{0}$ to be chosen below.

Step 1: Under the assumption (5.5), we will argue that necessarily
$t_{k}\to\infty$ as $k\to\infty$. (In fact, below we only need to know that
$t_{k}$ is bounded away from $0$.)

Assume toward a contradiction that $\liminf_{k}\ t_{k}<K$, for some $K>0$. By
passing to subsequences, relabelling, and invoking standard compactness,
continuity, and Gamma-convergence results (i.e. Theorem 2.1), we may assume
that the following hold.

First, $t_{k}\leq K$ for all $k$.

Second, there exists a $1$-current
$J_{1}\in{\mathcal{R}}_{1}\cap\mathcal{F}_{1}^{\prime}(M)$ such that

$\|\frac{1}{\pi}\star Ju_{k}(\cdot,t_{k})-J_{1}\|_{\mathcal{F}}\to 0.$

Hence, by (5.5),

$\|J_{1}-T_{\gamma}\|_{\mathcal{F}}=r.$ (5.6)

Third, there exists a Radon measure $\mu_{1}$ such that

$\frac{e_{\varepsilon_{k}}(u_{k}(\cdot,t_{k}))}{\pi{|\\!\log\varepsilon|}}\rightharpoonup\mu_{1},\qquad\mbox{weakly
as measures.}$

and in addition

$\mu_{1}\geq\|J_{1}\|,$ (5.7)

cf. (2.14).

Finally, since $\delta_{k}\to 0$ and $M$ is compact, it follows from
Proposition 4.1 that

$\mu_{1}(M)=L=\|T_{\gamma}\|(M).$ (5.8)

We next claim that

$\frac{e_{\varepsilon_{k}}(u_{k}^{0}(\cdot))}{\pi{|\\!\log\varepsilon|}}\rightharpoonup\mu_{0}=\|T_{\gamma}\|,\qquad\mbox{weakly
as measures.}$

Indeed, we may assume that
$\frac{e_{\varepsilon_{k}}(u_{k}^{0}(\cdot))}{\pi{|\\!\log\varepsilon|}}$
converges weakly as measures to a limit $\mu_{0}$ as $k\to\infty$. Then
recalling that $\|\frac{1}{\pi}\star
Ju_{k}^{0}-T_{\gamma}\|_{\mathcal{F}}\leq\delta_{k}\to 0$, standard Gamma-
convergence results as in (5.7) imply that $\mu_{0}\geq\|T_{\gamma}\|$. On the
other hand, as in (5.8),

$\mu_{0}(M)=L=\mbox{length}(\gamma)=\|T_{\gamma}\|(M),$

so it follows that in fact $\mu_{0}=\|T_{\gamma}\|$ as claimed. It then
follows from (5.6) and (5.7) that $\mu_{1}\neq\mu_{0}=\|T_{\gamma}\|$.

We will obtain a contradiction, completing Step 1, by showing that under our
assumptions $\mu_{0}$ and $\mu_{1}$ must be equal. Indeed, after taking the
inner product of (4.1) with $\partial_{t}u_{k}$, standard computations show
that

$\partial_{t}e_{\varepsilon_{k}}(u_{k})=-|\partial_{t}u_{k}|^{2}+\mbox{div}(\partial_{t}u_{k}\cdot\nabla
u_{k}).$

Multiplying by a function $\phi\in C^{1}(M)$ and integrating by parts,

$-\left.\int_{M}\phi
e_{\varepsilon_{k}}(u_{k})\,\mbox{vol}\right|_{0}^{t_{k}}=\int_{0}^{t_{k}}\int_{M}\phi|\partial_{t}u_{k}|^{2}-(\nabla\phi,\partial_{t}u_{k}\cdot\nabla
u_{k})_{g}\,\,\mbox{vol}\,dt$ (5.9)

Clearly

$\frac{1}{\pi|\log\varepsilon_{k}|}\left.\int_{M}\phi
e_{\varepsilon_{k}}(u_{k})\,\mbox{vol}\right|_{0}^{t_{k}}\rightarrow\int_{M}\phi\
(\mu_{0}-\mu_{1}).$

On the other hand, it is not hard to see that after dividing by
$\pi|\log\varepsilon_{k}|$, the right-hand side of (5.9) tends to $0$ as
$k\to\infty$. First, taking $\phi=1$ in (5.9), we see that

$\frac{1}{\pi|\log\varepsilon_{k}|}\int_{0}^{t_{k}}\int_{M}\,|\partial_{t}u_{k}|^{2}\,\mbox{vol}\,dt=E_{\varepsilon_{k}}(u_{k}^{0})-E_{\varepsilon_{k}}(u_{k}(\cdot,t_{k}))\leq
2\delta_{k}\to 0$

as $k\to\infty$. It immediately follows that

$\displaystyle\left|\int_{0}^{t_{k}}\int_{M}\phi\frac{|\partial_{t}u_{k}|^{2}}{\pi|\log\varepsilon|}\,\mbox{vol}\,dt\right|\to
0\qquad\mbox{ as }k\to\infty$

Similarly, from Cauchy-Schwarz, the fact that

$\int_{M}\frac{|\nabla
u_{k}(\cdot,t)|^{2}}{\pi|\log\varepsilon_{k}|}\,\mbox{vol}\,dt\leq
E_{\varepsilon_{k}}(u_{k}(\cdot,t))\leq L+\delta_{k}\qquad\mbox{ for all
$t\geq 0$}$

and the assumption that $t_{k}\leq K$, we easily see that

$\frac{1}{\pi|\log\varepsilon_{k}|}\left|\int_{0}^{t_{k}}\int_{M}(\nabla\phi,\partial_{t}u_{k}\cdot\nabla
u_{k})_{g}\,\,\mbox{vol}\,dt\right|\to 0\qquad\mbox{ as }k\to\infty.$

Combining these, we conclude that $\int_{M}\phi(\mu_{1}-\mu_{0})=0$ for all
$\phi\in C^{1}(M)$, and hence that $\mu_{0}=\mu_{1}$. This contradiction
yields the conclusion that if (5.5) holds, then $t_{k}\to\infty$, completing
Step 1. Step 2: We will now reach a contradiction to (5.5), and so obtain
Claim 5.4.

To this end, we apply Theorem 2.3 to the sequence of functions

$\tilde{u}_{k}(x,t)=u_{k}(x,t+t_{k}-1),$

which, in particular, satisfies (4.1) on $M\times[0,\infty)$ with
$\varepsilon=\varepsilon_{k}\to 0$. This yields a function
$\Phi_{*}:M\times(0,1]\to\mathbb{R}$ that solves the heat equation, as well as
measures $(\mu^{t}_{*})_{0<t\leq 1}$ and $(\nu^{t}_{*})_{0<t\leq 1}$ such that

$\frac{e_{\varepsilon_{k}}(\tilde{u}_{k}(\cdot,t))}{\pi|\log\varepsilon_{k}|}\,\mbox{vol}\rightharpoonup\mu^{t}_{*}=\frac{1}{2}|\nabla\Phi_{*}(\cdot,t)|^{2}\,\mbox{vol}+\nu^{t}_{*}$

weakly as measures and $(\nu^{t}_{*})_{0<t\leq 1}$ is a $1$\- dimensional
Brakke flow satisfying (2.27) and (2.28) (with $n=2$).

As with (5.8), and using Claim 1, it follows that

$\mu^{t}_{*}(M)=L\qquad\mbox{ for all }t\in(0,1].$ (5.10)

We recall the standard estimate

$\frac{d}{dt}\int_{M}\frac{|\nabla\Phi_{*}(\cdot,t)|^{2}}{2}\,\mbox{vol}=-\int_{M}|\partial_{t}\Phi_{*}|^{2}\,dx\leq
0$ (5.11)

(the counterpart for the linear heat equation of (5.9)). Since
$t\mapsto\nu^{t}_{*}(M)$ is also nonincreasing, as noted in (2.24), we
conclude from (5.10) that both

$\nu^{t}_{*}(M)\quad\mbox{ and
}\quad\int_{M}\frac{|\nabla\Phi_{*}(\cdot,t)|^{2}}{2}\,\mbox{vol}\qquad\mbox{
are both independent of $t\in(0,1]$}.$

It then follows from (5.11) that $\partial_{t}\Phi_{*}=0$ and hence that
$\Phi_{*}=\Phi_{*}(x)$ is independent of $t$ and harmonic. Similarly, it
follows from (2.25), (2.26) that there exists a stationary $1$-dimensional
varifold $V_{*}$ such that

$\nu^{t}_{*}=V_{*}\qquad\mbox{ for all }\;t\in(0,1],$

and then by continuity at $t=1$ as well. Also, (2.27) and (2.28) imply that
there exists a $1$-rectifiable set $\Sigma_{*}\subset M$ and a function
$\Theta_{*}$ such that

$V_{*}=\Theta_{*}(x){\mathcal{H}}^{1}\hskip 1.0pt{\hbox to10.8pt{\hfill\vrule
height=7.0pt,width=0.4pt,depth=0.0pt\hbox{\vrule
height=0.4pt,width=7.6pt,depth=0.0pt}\hfill}}\Sigma_{*},\qquad\qquad\Theta_{*}\geq\eta>0\quad{\mathcal{H}}^{1}\
\mbox{{\em a.e.} in }\Sigma_{*}.$ (5.12)

Moreover, as in the proof of Claim 1, there exists a $1$-current
$J_{1}\in{\mathcal{R}}_{1}\cap\mathcal{F}_{1}^{\prime}(M)$ such that

$\|\frac{1}{\pi}\star
J\tilde{u}(\cdot,1)-J_{1}\|_{\mathcal{F}}=\|\frac{1}{\pi}\star
Ju(\cdot,t_{k})-J_{1}\|_{\mathcal{F}}\to 0$

and thus

$\|J_{1}-T_{\gamma}\|_{\mathcal{F}}=r.$ (5.13)

We claim that in addition

$V_{*}\geq\|J_{1}\|,\qquad V_{*}(M)\leq L.$ (5.14)

The second assertion follows from (5.10), and the first assertion is a
consequence of standard Gamma-convergence results, which imply that

$\mu^{1}_{*}=\frac{1}{2}|\nabla\Phi_{*}(x)|^{2}\,\mbox{vol}+V_{*}\geq\|J_{1}\|.$

Since $\frac{1}{2}|\nabla\Phi_{*}(x)|^{2}\,\mbox{vol}$ is absolutely
continuous with respect to vol and $\|J_{1}\|$ is concentrated on a
$1$-rectifiable set, this implies that $V_{*}\geq\|J_{1}\|$, as claimed.

However, through an appeal to Proposition 3.1, we see that no such varifold
can exist. Claim 5.4 is established. Step 3: Fix $r>0$, and let
$\delta_{1}(r)$ and $\varepsilon_{1}(r)$ be as provided in Claim 5.4. We may
assume that $\delta_{1}(r)\leq r$. For $\varepsilon\in(0,\varepsilon_{1}(r))$,
let

$u_{k}=u_{\varepsilon}(\cdot,\cdot;\frac{1}{2}\delta_{1}(r),2^{k}).$

It follows from Proposition 4.1 that

$\displaystyle\delta_{1}(r)$ $\displaystyle\geq
E^{\varepsilon}_{U}(u_{k}(\cdot,0))-E^{\varepsilon}_{U}(u_{k}(\cdot,2^{k}))=\frac{1}{\pi|\log\varepsilon|}\int_{0}^{2^{k}}\int_{M}|\partial_{t}u_{k}|^{2}\,\mbox{vol}\,dt,$

so there exists $\sigma_{k}\in(0,2^{k})$ such that
$w_{k}:=u_{k}(\cdot,\sigma_{k})$ satisfies

$\int_{M}|\Delta
w_{k}-\frac{1}{\varepsilon^{2}}(|w_{k}|^{2}-1)w_{k}|^{2}\,\,\mbox{vol}=\left.\int_{M}|\partial_{t}u_{k}|^{2}\,\,\mbox{vol}\right|_{t=\sigma_{k}}\leq\delta_{1}\pi|\log\varepsilon|2^{-k}.$

Also, it follows from Claim 5.4 that

$\|\frac{1}{\pi}\star Jw_{k}-T_{\gamma}\|_{\mathcal{F}}=\|\frac{1}{\pi}\star
Ju_{k}(\cdot,\sigma_{k})-T_{\gamma}\|_{\mathcal{F}}<r.$

Since $|u_{k}|\leq 1$ everywhere, we have that $\|\Delta
w_{k}\|_{L^{2}(M)}\leq C_{\varepsilon}$, and hence by elliptic regularity,
$\|w_{k}\|_{H^{2}}\leq C_{\varepsilon}$. One may thus extract a subsequence
and a function $u_{\varepsilon}\in H^{2}(M;\mathbb{C})$ such that $w_{k}\to
u_{\varepsilon}$ weakly in $H^{2}$, and it easily follows from the above that

$-\Delta
u_{\varepsilon}+\frac{1}{\varepsilon^{2}}(|u_{\varepsilon}|^{2}-1)u_{\varepsilon}=0\qquad\mbox{
and }\quad\|\frac{1}{\pi}\star Ju_{\varepsilon}-T_{\gamma}\|_{\mathcal{F}}\leq
r.$

Finally, we may insist that $\delta<r$, and then it follows from Proposition
4.1 that $|E_{\varepsilon}(u_{\varepsilon})-L|<r$. ∎

##  Appendix A Appendix

###  A.1 On the proof of Lemma 4.3

As remarked above, this lemma is essentially proved in [20]. We describe the
proof given there and the extremely small modifications that we need.

The idea of the proof is first to construct $U^{\varepsilon,0}_{0}$, with its
vorticity concentrating around the central geodesic $\Gamma$, then for $w\in
W$, to define

$U^{\varepsilon,0}_{w}:=U^{\varepsilon,0}_{0}\circ{}O_{w}^{-1}$ (A.1)

where $O_{w}:M\to M$ is a suitable family of diffeomorphisms indexed by $w\in
W$ such that $(w,x)\to O_{w}(x)$ is smooth, described below. Construction of
$U^{\varepsilon,0}_{0}$ Recall that in (2.6) we defined a map
$y:K_{r_{0}}\to\mathbb{R}^{2}$, smooth and nonvanishing away from $\Gamma$.
Let $y^{0}:M\setminus K_{r_{0}/2}\to S^{1}$ be any smooth function such that
$y^{0}(x)=y(x)/\left|y(x)\right|$ in $K_{r_{0}}\setminus K_{r_{0}/2}$. The
existence of such a function is a consequence of the topological assumption
(2.3).

Then we set $\tilde{v}^{\varepsilon}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ by
$\tilde{v}^{\varepsilon}(p)=f\left(\frac{\left|p\right|}{\varepsilon}\right)\frac{p}{\left|p\right|}$
where $f:[0,\infty)\to[0,1]$ is a smooth nondecreasing function such that
$f(s)=s$ for $s\in[0,1/2]$ and $f(s)=1$ for $s\geq 1$. Finally we define

$U^{\varepsilon,0}_{0}(x):=\begin{cases}\tilde{v}^{\varepsilon}(y(x))&\text{for
}x\in{}K_{r_{0}},\\\ y^{0}(x)&\text{for
}x\in{}M\setminus{}K_{r_{0}}.\end{cases}$

The only way in which this construction differs from that in [20] is that
there, $f$ is chosen to be $f(s)=\min(s,1)$, which is Lipschitz continuous but
not smooth. With this change, $U^{\varepsilon,0}_{0}$ is smooth.

Construction of $O_{w}$. We take $O_{w}$ in (A.1) to be exactly the same map
as in [20], see pg. 62. The construction easily implies that $(w,x)\mapsto
U^{\varepsilon,0}_{w}(x)$ is smooth and hence that
$\|U^{\varepsilon,0}_{w}\|_{H^{2}}\leq C_{\varepsilon}$ for all $w\in W$. All
other conclusions are proved in [20], and some are obvious anyway, such as
that $\|U^{\varepsilon,0}_{w}\|_{L^{\infty}}\leq 1.$ In particular, $(3)$,
which follows from a Gamma-limsup type estimate together with (2.12), is
verified in Lemma $5.5$ from [20]. Finally, $(4)$ follows from Lemma $5.4$ of
[20].

###  A.2 Proof of Lemma 4.4

###### Proof.

The maximum principle and standard energy estimates imply that for every
$t>0$,

$\displaystyle\|U^{\varepsilon,1}_{w}(\cdot,t)\|_{L^{\infty}(M)}$
$\displaystyle\leq 1,$ $\displaystyle
E_{\varepsilon}(U^{\varepsilon,1}_{w}(\cdot,t))+\frac{1}{\pi{|\\!\log\varepsilon|}}\int_{0}^{t}\int_{M}|\partial_{t}U^{\varepsilon,1}_{w}|^{2}\,d\,\mbox{vol}\,dx$
$\displaystyle\leq E_{\varepsilon}(U^{\varepsilon,1}_{w}(\cdot,0))\leq L+1$

for all $|w|\leq R$, provided $\varepsilon$ and $R$ are small enough. We next
claim that for every $t>0$, there exists $C=C_{\varepsilon,\tau}$ such that

$\|U^{\varepsilon,1}_{w}(\cdot,t)\|_{H^{2}}\leq
C_{\varepsilon,\tau}\qquad\mbox{ for all }t\in[0,\tau]\mbox{ and }|w|\leq R.$
(A.2)

To specify the norm, we fix an open cover $\\{U_{j}\\}_{j\in J}$ of $M$, with
local coordinates $\varphi_{j}:U_{j}\to V_{j}\subset\mathbb{R}^{3}$ on each
patch, and a finite partition of unity $\\{\eta_{j}\\}$ subordinate to
$\\{U_{j}\\}$. We then define

$\|u\|_{H^{2}}^{2}=\|u\|_{L^{2}}^{2}+\||\nabla u|_{g}\|_{L^{2}}^{2}+\sum_{j\in
J}\sum_{k=1}^{3}\|\sqrt{\eta_{j}}|\nabla\partial_{k}(u\circ\varphi_{j}^{-1})|_{g}\|^{2}_{L^{2}(V_{j})},$

where $\partial_{k}$ denotes differentiation with respect to local coordinates
on $V_{j}$. To prove (A.2), we write (4.1) in local coordinates on each patch,
apply $\partial_{k}$ to derive an equation for
$V_{k}:=\partial_{k}U^{\varepsilon,1}_{w}$ of the form

$\partial_{t}V_{k}-\Delta_{g}V_{k}=\mbox{ terms involving
$U^{\varepsilon,1}_{w},\nabla U^{\varepsilon,1}_{w}$}.$

Multiplying by $\partial_{t}V_{k}$, using the fact from Lemma 4.3 point (2)
that $\|\nabla V_{k}(\cdot,0)\|\leq C_{\varepsilon}$, integrating by parts,
and carrying out rather standard estimates leads to (A.2).

It follows from the above estimates and the equation that
$\|\partial_{t}U^{\varepsilon,1}_{w}(\cdot,t)\|_{L^{2}}\leq
C_{\varepsilon,\tau}$ for $0<s\leq\tau$. Thus, for $0\leq t_{1}<t_{2}\leq\tau$
and any $w\in{}W$, we have

$\displaystyle\|U^{\varepsilon,1}_{w}(\cdot,t_{2})-U^{\varepsilon,1}_{w}(\cdot,t_{2})\|_{L^{2}}^{2}\
$
$\displaystyle\leq(t_{2}-t_{1})\int_{M\times[t_{1},t_{2}]}\\!{}\left|\partial_{t}U_{w}^{\varepsilon,1}(x,t)\right|^{2}\,\mbox{vol}\
{}dt$ $\displaystyle\leq C_{\varepsilon,\tau}(t_{2}-t_{1}).$

Then the interpolation estimate $\|u\|_{H^{1}}\leq
C\|u\|_{L^{2}}^{1/2}\|u\|_{H^{2}}^{1/2}$ and (A.2) imply that

$\left\|U_{w}^{\varepsilon,1}(\cdot,t_{2})-U_{w}^{\varepsilon,1}(\cdot,t_{1})\right\|_{H^{1}}\leq
C_{\varepsilon,\tau}\sqrt{t_{2}-t_{1}}\quad\mbox{ for }w\in W,\ 0\leq
t_{1}<t_{2}\leq\tau.$

Now consider $w_{1},w_{2}\in{}W$. Writing
$f_{\varepsilon}(u)=\frac{1}{\varepsilon^{2}}(1-|u|^{2})u$ and using the
identity

$f_{\varepsilon}(b)-f_{\varepsilon}(a)=\int_{0}^{1}\frac{d}{d\sigma}f_{\varepsilon}(\sigma
b+(1-\sigma)a)d\sigma=\int_{0}^{1}f_{\varepsilon}^{\prime}(\sigma
b+(1-\sigma)a)d\sigma\ (b-a)\ ,$

we find that $V:=U_{w_{2}}^{\varepsilon,1}-U_{w_{1}}^{\varepsilon,1}$
satisfies the equation

$\partial_{t}V-\Delta V=gV,$

where $\|g(\cdot,t)\|_{L^{\infty}}\leq C$ for every $t$. In addition, it
follows from Lemma 4.3 that $\|V(\cdot,0)\|_{H^{1}}\leq C|w_{2}-w_{1}|$. Thus
carrying out further standard parabolic estimates (multiplying by $V$ or
$\partial_{t}V$, integrating by parts …) leads to

$\|U^{\varepsilon,1}_{w_{2}}(\cdot,t)-U^{\varepsilon,1}_{w_{1}}(\cdot,t)\|_{H^{1}}\leq
C_{\varepsilon,\tau}|w_{2}-w_{1}|\qquad\mbox{ for }0\leq t\leq\tau.$

We conclude that the map
$(t,w)\in[0,\tau]\times{}W\mapsto{}U_{w}^{\varepsilon,1}(\cdot,t)\in{}H^{1}(M;\mathbb{C})$
is continuous, since it is separately uniformly continuous in $t$ and $w$. A
more detailed reference for such parabolic estimates on manifolds can be
found, e.g. in appendix A of [19]. ∎

Acknowledgements. The work of A.C. and R.J. on this project was partly
supported by the Natural Sciences and Engineering Research Council of Canada
under Operating Grant 261955. P.S. gratefully acknowledges that this research
was supported by the Fields Institute for Research in Mathematical Sciences
and by a Simons Collaboration grant 585520. The contents of this paper are
solely the responsibility of the authors and do not necessarily represent the
official views of any of the organizations mentioned above.

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