# The geometry of the space of vortices on a two-sphere in the Bradlow limit

R.I. García Lara
Facultad de Matematicas, Universidad Autonoma de Yucatan
Merida, Mexico

J.M. Speight
School of Mathematics, University of Leeds
Leeds LS2 9JT, England E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>(corresponding author)

###### Abstract

It is proved that the normalized $L^{2}$ metric on the moduli space of
$n$-vortices on a two-sphere, endowed with any Riemannian metric, converges
uniformly in the Bradlow limit to the Fubini-Study metric. This establishes,
in a rigorous setting, a longstanding informal conjecture of Baptista and
Manton.

##  1 Introduction

Vortices are the simplest class of topological solitons arising in gauge
theory, making them interesting objects of mathematical study, independent of
their phenomenological applications in condensed matter physics and cosmology.
At critical coupling, they satisfy a self-duality (or Bogomol’nyi) type
condition which implies that static vortices exert no net force on one
another. The moduli space $M_{n}$ of static $n$-vortex solutions is thus
exceptionally large, forming a complex $n$-manifold. There is a well-developed
programme for studying their low energy dynamics in this regime, comprehending
classical, quantum and statistical mechanics, originally proposed by Manton.
The key object underpinning this programme is a natural Riemannian metric $g$
on $M_{n}$, called the $L^{2}$ metric, obtained by restricting the model’s
kinetic energy functional to $TM_{n}$. Vortex dynamics is modelled by geodesic
motion in $(M_{n},g)$. For a comprehensive review of the geodesic
approximation to vortex dynamics, see [15, ch. 3].

The $L^{2}$ metric on $M_{n}$ is, then, the object of strong and sustained
mathematical interest. There are very few situations in which $g$ can be
computed exactly, however. Other than the rather trivial case of $M_{1}$ when
physical space is homogeneous, Strachan has exactly computed the metric on
$M_{2}$ on the hyperbolic plane [17]. Samols [16] adapted this calculation to
obtain a semi-explicit localization formula for the metric on an arbitrary
surface (for example the Euclidean plane) which is useful for numerics, and
can be used to prove many interesting global and qualitative properties of
$g$, but it does not yield explicit formulae. Even the metric on $M_{1}$ is
nontrivial if physical space is not homogeneous [12].

This paper examines one case where explicit progress is possible, at least in
a certain parametric limit: vortices on a two-sphere. Bradlow [7] observed
that, on a compact Riemann surface $\Sigma$, there is a lower bound,
proportional to $n$, on the area $|\Sigma|$ required for the surface to
accommodate $n$-vortices. Above this bound, $M_{n}$ is biholomorphic to ${\rm
Sym}_{n}\Sigma$, the $n$-fold symmetric product of $\Sigma$, since vortices
are uniquely determined by the collection of $n$ points on $\Sigma$ at which
their Higgs field vanishes. As $|\Sigma|$ approaches the bound from above, the
vortices spread out and lose their spatial localization. Precisely at the
bound, they delocalize entirely: the Higgs field vanishes identically and the
magnetic field is uniform. Such vortex solutions are sometimes called
“dissolved” vortices, and the limit in which they appear called the
“dissolving” limit. The moduli space of dissolved vortices is precisely the
moduli space of constant curvature connexions on the line bundle of which the
Higgs field is a section. If the genus $r$ of $\Sigma$ is positive this is a
torus of dimension $2r$ which may naturally be identified with the Jacobian
variety of $\Sigma$. It also has an $L^{2}$ metric, to which the $L^{2}$
metric $g$ on $M_{n}$ conjecturally degenerates (more precisely, $g$
conjecturally degenerates to the pullback of the metric on the torus by the
Abel-Jacobi map ${\rm Sym}_{n}\Sigma\rightarrow{\rm Jac}(\Sigma)$), a scenario
studied in detail in [14]. If $r=0$, that is, $\Sigma=S^{2}$, the moduli space
of dissolved vortices is a point, while $M_{n}\equiv{\mathbb{C}}{{P}}^{n}$. To
understand the metric $g$ in the dissolving limit we must rescale it by (for
example) demanding that $(M_{n},g)$ has some normalized volume. In this
context, Baptista and Manton [4] made the remarkable conjecture (on the round
two-sphere), that $g$ converges in the dissolving limit to the Fubini-Study
metric on ${\mathbb{C}}{{P}}^{n}$, implying an enormous gain in symmetry in
this limit.

The purpose of this paper is to give a precise formulation and rigorous proof
of a slight generalization of Baptista and Manton’s conjecture: we will show
that, in the dissolving limit, the normalized $L^{2}$ metric on $M_{n}(S^{2})$
converges uniformly to the Fubini-Study metric, regardless of the metric on
$S^{2}$ (that is, we remove the assumption that the domain sphere is round).
The $L^{2}$ geometry in the dissolving limit is thus unreasonably simple: a
family of metrics with (generically) no nontrivial isometries at all converges
uniformly to a homogeneous metric of constant holomorphic sectional curvature.
The convergence established implies [2] that the spectrum of the Laplacian on
$M_{n}$ converges uniformly to the spectrum of the Fubini-Study Laplacian,
which is easily computed [5]. The theorem thus has immediate consequences for
the quantum dynamics of vortices on $S^{2}$ in the dissolving limit.

The paper is structured as follows. In section 2 we formulate the model and
give a precise statement of the theorem. In section 3 we review the notion of
pseudo-vortices, introduced by Baptista and Manton, and prove that vortices
converge uniformly to pseudo-vortices in the dissolving limit. In section 4,
we establish convergence of the metric, finishing the proof of the main
theorem, while in section 5 we study the convergence of the spectrum of the
Laplacian.

##  2 Statement of the theorem

Let $(\Sigma,g_{\Sigma})$ be an oriented Riemannian manifold diffeomorphic to
$S^{2}$ and $(L,h)$ be a hermitian line bundle over $\Sigma$ of degree $n\geq
0$. To any section $\varphi$ of $L$ and unitary connexion $A$ on $L$ we
associate the energy

$E(\varphi,A)=\int_{\Sigma}\left(\frac{1}{2}|\mathrm{d}_{A}\varphi|^{2}+\frac{1}{2}|F_{A}|^{2}+\frac{1}{8}(\tau-|\varphi|^{2})^{2}\right)$
(2.1)

where $\tau>0$ is a constant parameter and $F_{A}$ is the curvature of $A$. If
we choose a local section $\eta$ of $L$ with $|\eta|^{2}=h(\eta,\eta)=1$, we
may identify the connexion $A$ locally with the real one-form
$h(i\mathrm{d}_{A}\eta,\eta)$ which, in a slight abuse of notation, we also
denote $A$. Then for a general local section $f\eta$, where $f$ is smooth and
complex valued,

$\mathrm{d}_{A}(f\eta)=(\mathrm{d}f-iAf)\eta.$ (2.2)

We adopt the convention that the curvature $F_{A}$ is real, coinciding
(globally) with $\mathrm{d}A$. The energy functional is invariant under gauge
transformations,

$(\varphi,A)\mapsto(e^{i\chi}\varphi,A+\mathrm{d}\chi),$ (2.3)

where $\chi:\Sigma\rightarrow{\mathbb{R}}$ is smooth.

It is well known [6, 7] that $E(\varphi,A)\geq\tau\pi n$ with equality if and
only if

$\displaystyle\overline{\partial}_{A}\varphi$ $\displaystyle=$ $\displaystyle
0$ (2.4) $\displaystyle*F_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\tau-|\varphi|^{2}),$ (2.5)

where $\overline{\partial}_{A}$ denotes the $0,1$ part of $\mathrm{d}_{A}$
(with respect to the complex structure on $\Sigma$ defined by its orientation
and metric) and $*$ is the Hodge isomorphism defined by $g_{\Sigma}$.
Solutions of this system of equations are called vortices. Since they attain a
topological lower bound on $E$, they are automatically global minima of $E$.

A necessary condition for existence of vortices was obtained by Bradlow [7] by
integrating (2.5) over $\Sigma$:

$2\pi
n=\frac{1}{2}\tau|\Sigma|-\frac{1}{2}\int_{\Sigma}|\varphi|^{2}\leq\frac{1}{2}\tau|\Sigma|$
(2.6)

where $|\Sigma|$ denotes the area of $\Sigma$. Hence, if vortices exist,

$\varepsilon:=\tau|\Sigma|-4\pi n\geq 0.$ (2.7)

We refer to $\varepsilon\searrow 0$ as the Bradlow limit. More precisely, we
shall consider the limit where the metric $g_{\Sigma}$ (and hence $|\Sigma|$)
is fixed, and $\tau\searrow 4\pi n/|\Sigma|$. This is more convenient for our
purposes than the limit considered by Baptista and Manton ($\tau\equiv 1$ and
$g_{\Sigma}$ varying through round metrics, with $|\Sigma|$ approaching $4\pi
n$ from above) because it allows us to define Sobolev spaces with fixed norms
determined by $g_{\Sigma}$. That our limit includes theirs follows from the
observation that, if $(\varphi,A)$ solves the vortex equations on
$(\Sigma,g_{\Sigma})$ at coupling $\tau=\tau_{0}$, then
$(\tau_{0}^{-1/2}\varphi,A)$ satisfies the vortex equations on
$(\Sigma,\tau_{0}g_{\Sigma})$ at coupling $\tau=1$. Note that at
$\varepsilon=0$ any vortex must have $\varphi=0$ and $*F_{A}$ constant, so
that vortices completely lose their spatial localization, which is why this is
often called the dissolving limit. For $\varepsilon>0$, every vortex has
$\|\varphi\|_{L^{2}}^{2}=\varepsilon$.

The moduli space of vortices $M_{n}$ is the space of gauge equivalence classes
of solutions of (2.4), (2.5). This is a complex $n$-manifold canonically
diffeomorphic to ${\rm Sym}_{n}\Sigma$, the $n$-fold symmetric product of
$\Sigma$, the canonical diffeomorphism $F:M_{n}\rightarrow{\rm Sym}_{n}\Sigma$
being the map which sends the (gauge equivalence class of a) vortex solution
$(\varphi,A)$ to the set of points at which $\varphi$ vanishes, counted with
multiplicity. That is, we map the gauge equivalence class of solutions
$[(\varphi,A)]$ to the degree $n$ effective divisor
$F([(\varphi,A)])=(\varphi)$ in $\Sigma$. In this way we identify the
$\varepsilon$-dependent family of manifolds $M_{n}$ with the single fixed
manifold ${\rm Sym}_{n}\Sigma$ which, in a slight abuse of notation, we will
also denote $M_{n}$.

Since $\Sigma=S^{2}$, we may specify a degree $n$ effective divisor $D$ by
giving a polynomial

$a_{0}+a_{1}z+\cdots+a_{n}z^{n}$ (2.8)

vanishing on $D$, where $z$ is a stereographic coordinate. Two such
polynomials define the same divisor if and only if one is a constant multiple
of the other, so we identify $(\varphi,A)$, up to gauge, with
$[a_{0},a_{1},\ldots,a_{n}]\in{\mathbb{C}}{{P}}^{n}$. Hence
$M_{n}\equiv{\mathbb{C}}{{P}}^{n}$.

For our purposes, an alternative identification
$M_{n}\equiv{\mathbb{C}}{{P}}^{n}$ is more convenient, however. Let us choose
and fix, once and for all, a unitary connexion $\widehat{A}$ on $L$ of
constant curvature. This is unique up to gauge. We equip $L$ with the
holomorphic structure whose Dolbeault operator
$\overline{\partial}_{L}=\overline{\partial}_{\widehat{A}}$. Then $H^{0}(L)$,
the space of holomorphic sections of $L$, is a complex vector space of
dimension $n+1$. Each nonzero section in $H^{0}(L)$ defines an effective
divisor of degree $n$ (on which it vanishes), and two sections define the same
divisor if and only if one is a constant nonzero multiple of the other. Hence
we have an identification of $M_{n}$ with the complex projective space of
lines through $0$ in $H^{0}(L)$. In other words, we identify $M_{n}$ with
${\mathbb{C}}{{P}}^{n}$ via the bijective map

$f:M_{n}\rightarrow{\mathbb{P}}(H^{0}(L)),\qquad[(\varphi,A)]\mapsto\\{\psi\in
H^{0}(L):(\psi)=(\varphi)\\}$ (2.9)

which identifies $[(\varphi,A)]$ with the line in $H^{0}(L)$ consisting of
holomorphic sections vanishing on the same divisor as $\varphi$. This allows
us to equip $M_{n}$ with a Fubini-Study metric as follows. The $L^{2}$ inner
product on $\Gamma(L)$ defines a hermitian inner product on $H^{0}(L)$. Denote
by $S$ the unit sphere in $(H^{0}(L),\|\cdot\|_{L^{2}})$, and by
$\pi:S\rightarrow{\mathbb{P}}(H^{0}(L))$, $\pi(\psi)=[\psi]$, the Hopf
fibration. Then the Fubini-Study metric of constant holomorphic sectional
curvature $2$ is precisely the metric $g_{FS}$ on ${\mathbb{P}}(H^{0}(L))$
with respect to which $\pi$ is a Riemannian submersion. By the Fubini-Study
metric on $M_{n}$ we will always mean

$g_{0}:=f^{*}g_{FS}.$ (2.10)

A priori, there is no reason to expect $g_{0}$ to have any relevance to vortex
dynamics, which is controlled [15, ch. 3] by a natural Riemannian metric $g$
on $M_{n}$ called the $L^{2}$ metric. This is defined as follows. Any smooth
curve $(\varphi(t),A(t))$ of vortex solutions through $(\varphi(0),A(0))$
tautologically defines a tangent vector to $M_{n}$ at $[(\varphi(0),A(0))]$,
to which we associate the squared length

$\int_{\Sigma}\left(|P(\dot{\varphi}(0),\dot{A}(0))|^{2}\right)$ (2.11)

where $P$ denotes projection $L^{2}$ orthogonal to the gauge orbit through
$(\varphi(0),A(0))$. No explicit formula for the $L^{2}$ metric is known
(except in the trivial case that $n=1$ and $\Sigma$ is round), but it is known
to be kähler, and a formula for its kähler class was obtained by Baptista [3].
From this, it follows that the volume of $M_{n}$ is

$|M_{n}|=\frac{\pi^{n}}{n!}(\tau|\Sigma|-4\pi
n)^{n}=\frac{\pi^{n}\varepsilon^{n}}{n!},$ (2.12)

which vanishes in the Bradlow limit. Hence $g$ degenerates in this limit so,
to discuss its convergence, we must rescale. We define the normalized $L^{2}$
metric on $M_{n}$ to be

$g_{\varepsilon}:=\frac{1}{\varepsilon}g.$ (2.13)

This is a one-parameter family of metrics on $M_{n}$ of fixed volume
$\pi^{n}/n!$. The purpose of this paper is to establish

###### Theorem 1

In the limit $\varepsilon\searrow 0$, the normalized $L^{2}$ metric
${g}_{\varepsilon}$ converges in $C^{0}$ to ${g}_{0}$, the Fubini-Study metric
on $M_{n}$.

A result of this kind was conjectured by Baptista and Manton in the case that
$\Sigma$ is round, motivated by the approximation of vortices by what they
called pseudo-vortices [4]. These are pairs $(\varphi,\widehat{A})$ consisting
of a connexion of constant curvature, and a section holomorphic with respect
to $\overline{\partial}_{\widehat{A}}$ satisfying a normalization condition.
Our proof proceeds by generalizing this approximation to arbitrary spheres,
and rigorously controlling its errors.

##  3 Convergence of vortices to pseudo-vortices

Since we are concerned with the limit $\varepsilon\searrow 0$, we assume
henceforth that

$\frac{4\pi n}{|\Sigma|}<\tau<\frac{1+4\pi n}{|\Sigma|},$ (3.1)

so $\varepsilon\in(0,1)$. Recall we have fixed a connexion $\widehat{A}$ on
$L$ of constant curvature $*F_{\widehat{A}}=2\pi n/|\Sigma|$, and used this to
equip $L$ with a holomorphic structure. Given a divisor $D\in{\rm
Sym}_{n}\Sigma$, let us denote by $\widehat{\varphi}_{D}\in H^{0}(L)$ a
holomorphic section of $L$ with $\|\widehat{\varphi}_{D}\|_{L^{2}}=1$ and
$\widehat{\varphi}^{-1}(0)=D$. This is unique up to multiplication by a
constant in $U(1)$. Let us define the pseudo-vortex corresponding to the
divisor $D$ to be the pair
$(\sqrt{\varepsilon}\widehat{\varphi}_{D},\widehat{A})$, unique up to
$\widehat{\varphi}_{D}\mapsto e^{ic}\widehat{\varphi}_{D}$ where
$c\in{\mathbb{R}}$ is constant. This pair satisfies the first vortex equation
(2.4), but not the second (2.5). It does satisfy (2.5) “on average,” however,
that is, it satisfies the constraint (2.6) obtained by integrating (2.5) over
$\Sigma$. This observation led Baptista and Manton to conjecture that, for
small $\varepsilon$, the actual vortex with divisor $D$ should be well
approximated by its corresponding pseudo-vortex [4]. In this section we will
prove that this intuition is correct in the following precise sense:

###### Proposition 2

There exists a constant $C>0$, depending only on $(L,h)$ and $g_{\Sigma}$,
such that, for all vortex solutions $[(\varphi,A)]\in M_{n}$,

$\|\varepsilon^{-1/2}|\varphi|-|\widehat{\varphi}_{D}|\|_{C^{0}}\leq
C\varepsilon,\qquad\|F_{A}-F_{\widehat{A}}\|_{C^{0}}\leq C\varepsilon,$

where $D=(\varphi)$ and $\|\cdot\|_{C^{0}}$ denotes the $\sup$ norm.

The proof relies on the following observation. Choose and fix $D$, some
effective divisor of degree $n$ on $\Sigma$. Then any section $\varphi$ of $L$
with $(\varphi)=D$ has a unique representative in its gauge equivalence class
of the form

$\varphi=\sqrt{\varepsilon}e^{u/2}\widehat{\varphi}_{D}$ (3.2)

where $u:\Sigma\rightarrow{\mathbb{R}}$ is a smooth function. If we define the
connexion $A$ to be

$A=\widehat{A}-\frac{1}{2}*\mathrm{d}u,$ (3.3)

then the pair $(\varphi,A)$ satisfies the first vortex equation (2.4) [9]. The
connexion $A$ has curvature

$*F_{A}=*F_{\widehat{A}}-\frac{1}{2}*\mathrm{d}*\mathrm{d}u=\frac{2\pi
n}{|\Sigma|}+\frac{1}{2}\Delta u.$ (3.4)

Hence, $(\varphi,A)$ satisfies the second vortex equation (2.5) if and only if

$\Delta
u-\frac{\varepsilon}{|\Sigma|}+\varepsilon|\widehat{\varphi}_{D}|^{2}e^{u}=0.$
(3.5)

It follows from the main theorem of [7], or by direct appeal to [11], that the
solution to (3.5) exists and is unique and smooth. In this way we reduce the
problem of constructing the vortex with divisor $D$ to a semilinear elliptic
PDE for $u$.

The error in the approximation

$(\varphi,A)=(\sqrt{\varepsilon}e^{u/2}\widehat{\varphi}_{D},\widehat{A}-\frac{1}{2}*\mathrm{d}u)\approx(\sqrt{\varepsilon}\widehat{\varphi}_{D},\widehat{A})$
(3.6)

is controlled by $u$, and hence we seek bounds on this function. We will use
four different norms on $C^{\infty}(\Sigma)$, namely,

$\displaystyle\|f\|^{2}_{L^{2}}$ $\displaystyle:=$
$\displaystyle\int_{\Sigma}|f|^{2}$ $\displaystyle\|f\|^{2}_{H^{1}}$
$\displaystyle:=$ $\displaystyle\int_{\Sigma}(|f|^{2}+|\mathrm{d}f|^{2})$
$\displaystyle\|f\|_{H^{2}}^{2}$ $\displaystyle:=$
$\displaystyle\int_{\Sigma}(|f|^{2}+|\mathrm{d}f|^{2}+|\nabla\mathrm{d}f|^{2})$
$\displaystyle\|f\|_{C^{0}}$ $\displaystyle:=$
$\displaystyle\sup\\{|f(x)|:x\in\Sigma\\}$ (3.7)

and will denote by $\langle\cdot,\cdot\rangle_{L^{2}}$ the $L^{2}$ inner
product. In the defintion of the $H^{2}$ norm, $\nabla$ denotes the Levi-
Civita connexion associated to $g_{\Sigma}$. We will also make frequent use of
two standard analytic facts which, for convenience, we collect here [8, 1]:

###### Proposition 3

There exists a constant $c>0$, depending only on $g_{\Sigma}$, such that

  1. 1.

For all $f\in C^{\infty}(\Sigma)$, $\|f\|_{C^{0}}\leq c\|f\|_{H^{2}}$ (Sobolev
embedding);

  2. 2.

For all $f\in C^{\infty}(\Sigma)$ with $\int_{\Sigma}f=0$, $\|f\|_{H^{2}}\leq
c\|\Delta f\|_{L^{2}}$ (standard elliptic estimate for $\Delta$).

Proposition 2 will quickly follow once we have shown that $\|u\|_{C^{0}}\leq
C\varepsilon$ uniformly in $D$. We begin by bounding the average of $u$ in
terms of its zero-average component.

###### Lemma 4

Let $u$ be a solution of (3.5) and $u_{0}=u-\overline{u}$, where

$\overline{u}:=\frac{1}{|\Sigma|}\int_{\Sigma}u.$

Then $|\overline{u}|\leq\|u_{0}\|_{C^{0}}$.

###### Proof.

Substituting $u=\overline{u}+u_{0}$ into (3.5) and integrating over $\Sigma$
we find (by the Divergence Theorem)

$0-\varepsilon+\varepsilon\int_{\Sigma}|\widehat{\varphi}_{D}|^{2}e^{\overline{u}}e^{u}_{0}=0,$
(3.8)

which may be solved for $\overline{u}$,

$\overline{u}=-\log\left(\int_{\Sigma}|\widehat{\varphi}_{D}|^{2}e^{u_{0}}\right).$
(3.9)

Now $-\|u_{0}\|_{C^{0}}\leq u_{0}\leq\|u_{0}\|_{C^{0}}$ and $\exp$ is
increasing, so

$e^{-\|u_{0}\|_{C^{0}}}=e^{-\|u_{0}\|_{C^{0}}}\int_{\Sigma}|\widehat{\varphi}_{D}|^{2}\leq\int_{\Sigma}|\widehat{\varphi}_{D}|^{2}e^{u_{0}}\leq
e^{\|u_{0}\|_{C^{0}}}\int_{\Sigma}|\widehat{\varphi}_{D}|^{2}=e^{\|u\|_{C^{0}}},$
(3.10)

where we have used the fact that $\|\widehat{\varphi}_{D}\|_{L^{2}}=1$. But
$\log$ is also increasing, so

$-\|u_{0}\|_{C^{0}}\leq\log\left(\int_{\Sigma}|\widehat{\varphi}_{D}|^{2}e^{u_{0}}\right)\leq\|u_{0}\|_{C^{0}}$
(3.11)

whence the claim immediately follows. ∎

This observation quickly yields a crude bound on $\|u\|_{C^{0}}$.

###### Lemma 5

There exists $C>0$, depending only on $n$ and $g_{\Sigma}$, such that, for all
$D$, the solution $u$ of (3.5) satisfies $\|u\|_{C^{0}}\leq C$.

###### Proof.

As before, let $u=\overline{u}+u_{0}$ with
$\overline{u}=|\Sigma|^{-1}\int_{\Sigma}u$. Now, by (3.4),

$\displaystyle\frac{1}{4}\|\Delta u_{0}\|_{L^{2}}^{2}$ $\displaystyle=$
$\displaystyle\|*F_{A}-2\pi n|\Sigma|^{-1}\|_{L^{2}}^{2}$ (3.12)
$\displaystyle=$ $\displaystyle\|F_{A}\|_{L^{2}}^{2}-\frac{(2\pi
n)^{2}}{|\Sigma|}$ $\displaystyle<$ $\displaystyle\|F_{A}\|_{L^{2}}^{2}$
$\displaystyle\leq$ $\displaystyle 2E(\varphi,A)$ $\displaystyle=$
$\displaystyle 2\pi\tau n\leq 2\pi n(1+4\pi n)/|\Sigma|.$

Hence, $\|\Delta u_{0}\|_{L^{2}}\leq C_{1}$, some constant depending only on
$n$ and $g_{\Sigma}$. Now $\int_{\Sigma}u_{0}=0$ so, by the standard elliptic
estimate for $\Delta$ (Proposition 3),

$\|u_{0}\|_{H^{2}}\leq c\|\Delta u_{0}\|_{L^{2}}\leq cC_{1},$ (3.13)

and hence, by the Sobolev embedding (Proposition 3),

$\|u_{0}\|_{C^{0}}\leq c^{2}C_{1}.$ (3.14)

Hence, by Lemma 4,

$\|u\|_{C^{0}}\leq|\overline{u}|+\|u_{0}\|_{C^{0}}\leq 2\|u_{0}\|_{C^{0}}\leq
2c^{2}C_{1}.$ (3.15)

∎

Equation (3.5) allows us to improve this crude bound.

###### Lemma 6

There exists $C>0$, depending only on $(L,h)$ and $g_{\Sigma}$, such that, for
all $D$, the solution $u$ of (3.5) satisfies $\|u\|_{C^{0}}\leq C\varepsilon$.

###### Proof.

Once again, by Proposition 3 and Lemma 4, it suffices to show that

$\|\Delta u_{0}\|_{L^{2}}\leq C\varepsilon,$ (3.16)

where $u_{0}$ is the zero-average part of $u$. It is convenient to define

$\alpha:=\sup\\{|\widehat{\varphi}_{D}(p)|:(p,D)\in\Sigma\times{\rm
Sym}_{n}\Sigma\\},$ (3.17)

noting that this number certainly exists, by continuity and compactness of
$\Sigma$, and is, by definition, dependent only on $(L,h)$ and $g_{\Sigma}$.
Now, by equation (3.5),

$\|\Delta
u_{0}\|_{L^{2}}\leq\frac{\varepsilon}{|\Sigma|^{1/2}}+\varepsilon\alpha^{2}e^{\|u\|_{C^{0}}}|\Sigma|^{1/2}\leq
C\epsilon,$ (3.18)

by Lemma 5. ∎

Proof of Proposition 2:  By definition of $u$,

$\frac{\varphi}{\sqrt{\varepsilon}}-\widehat{\varphi}_{D}=\widehat{\varphi}_{D}(1-e^{u/2}).$
(3.19)

By the mean value theorem, for each $x\in\Sigma$, there exists $q(x)$ between
$u(x)$ and $0$ such that

$e^{u(x)/2}-e^{0/2}=\frac{1}{2}e^{q(x)/2}(u(x)-0).$ (3.20)

Hence

$\|1-e^{u/2}\|_{C^{0}}\leq\frac{1}{2}e^{\|u\|_{C^{0}}/2}\|u\|_{C^{0}},$ (3.21)

and so,

$\left\|\frac{\varphi}{\sqrt{\varepsilon}}-\widehat{\varphi}_{D}\right\|_{C^{0}}\leq\frac{1}{2}\alpha\|u\|_{C^{0}}e^{\|u\|_{C^{0}}/2},$
(3.22)

where $\alpha$ is the constant defined in (3.17). Similarly,
$*(F_{A}-F_{\widehat{A}})=\Delta u/2$, so by (3.5),

$\|F_{A}-F_{\widehat{A}}\|_{C^{0}}\leq\frac{\varepsilon}{2}\left(\frac{1}{|\Sigma|}+\alpha^{2}e^{\|u\|_{C^{0}}}\right).$
(3.23)

The claimed inequalities now follow immediately from Lemma 6. $\Box$

##  4 Convergence of the metric

Consider a smooth curve of solutions of the vortex equations,
$(\varphi(t),A(t))$, and denote by $v$ the tangent vector to $M_{n}$ at
$[(\varphi(0),A(0))]$ that it generates. Given any such curve, there exists a
smooth curve $\widehat{\varphi}(t)$ in the unit sphere
$S\subset(H^{0}(L),\|\cdot\|_{L^{2}})$ such that

$\displaystyle\varphi(t)$ $\displaystyle=$
$\displaystyle\sqrt{\varepsilon}\widehat{\varphi}(t)e^{\frac{1}{2}u(t)+i\chi(t)}$
(4.1) $\displaystyle A(t)$ $\displaystyle=$
$\displaystyle\widehat{A}+\frac{1}{2}*du(t)+\mathrm{d}\chi(t)$ (4.2)

where $u(t)$, $\chi(t)$ are smooth curves in $C^{\infty}(\Sigma)$ and, at each
time $t$, $u(t)$ satisfies the PDE (3.5). We may choose $\chi(t)$ freely:
changing it merely gauge transforms each solution $(\varphi(t),A(t))$. Note
that $\|\widehat{\varphi}(t)\|_{L^{2}}^{2}\equiv 1$, so

$\langle\widehat{\varphi}(t),\dot{\widehat{\varphi}}(t)\rangle_{L^{2}}\equiv
0,$ (4.3)

where here and henceforth an overdot denotes differentiation with respect to
$t$. Without loss of generality, we may also assume that

$\langle i\widehat{\varphi}(t),\dot{\widehat{\varphi}}(t)\rangle_{L^{2}}\equiv
0.$ (4.4)

If not, we gauge transform

$\widehat{\varphi}(t)\mapsto e^{ic(t)}\widehat{\varphi}(t)$ (4.5)

by a suitable curve of space-independent functions $c(t)$ and redefine
$\chi(t)\mapsto\chi(t)-c(t)$. This is convenient because
$\widehat{\varphi}(t)$ then moves orthogonal to the fibres of the Hopf
fibration, so by definition of the Fubini-Study metric,

$g_{0}(v,v)=g_{FS}([\dot{\widehat{\varphi}}(0)],[\dot{\widehat{\varphi}}(0)])=\|\dot{\widehat{\varphi}}(0)\|_{L^{2}}^{2}.$
(4.6)

This, then, is the squared length assigned to $v$ by the Fubini-Study metric
$g_{0}$. We wish to compare this with the squared length of the same vector
with respect to the $L^{2}$ metric. For this purpose, it is convenient to
choose $\chi(t)$ so that $(\dot{\varphi}(0),\dot{A}(0))$ is $L^{2}$ orthogonal
to the gauge orbit through $(\varphi(0),A(0))$, and hence
$P(\dot{\varphi}(0),\dot{A}(0))=(\dot{\varphi}(0),\dot{A}(0))$. We may take
$\chi(0)=0$. A general gauge transform is

$(\varphi,A)\mapsto(e^{is}\varphi,A+\mathrm{d}s),\qquad s\in
C^{\infty}(\Sigma),$ (4.7)

so a general tangent vector to the gauge orbit through $(\varphi,A)$ takes the
form

$v=(is\varphi,\mathrm{d}s)\in\Gamma(L)\oplus\Omega^{1}(\Sigma),\qquad s\in
C^{\infty}(\Sigma).$ (4.8)

The tangent vector to our curve is

$(\dot{\varphi}(0),\dot{A}(0))=(\sqrt{\varepsilon}(\dot{\widehat{\varphi}}(0)+\widehat{\varphi}(0)(\frac{1}{2}\dot{u}(0)+i\dot{\chi}(0)))e^{u(0)/2},\frac{1}{2}*\mathrm{d}\dot{u}(0)+\mathrm{d}\dot{\chi}(0)).$
(4.9)

Hence we require that, for all $s\in C^{\infty}(\Sigma)$,

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\langle
is\sqrt{\varepsilon}\widehat{\varphi}(0)e^{u(0)/2},\sqrt{\varepsilon}(\dot{\widehat{\varphi}}(0)+\widehat{\varphi}(0)(\frac{1}{2}\dot{u}(0)+i\dot{\chi}(0))e^{u(0)/2})\rangle_{L^{2}}+\langle\mathrm{d}s,\frac{1}{2}*\mathrm{d}\dot{u}(0)+\mathrm{d}\dot{\chi}(0)\rangle_{L^{2}}$
(4.10) $\displaystyle=$ $\displaystyle\int_{\Sigma}s\varepsilon
e^{u(0)}h(i\widehat{\varphi}(0),\dot{\widehat{\varphi}}(0)+\frac{1}{2}\dot{u}(0)\widehat{\varphi}(0)+i\dot{\chi}(0)\widehat{\varphi}(0))+\langle
s,\frac{1}{2}\delta*\mathrm{d}\dot{u}(0)+\delta\mathrm{d}\dot{\chi}(0)\rangle_{L^{2}}$
$\displaystyle=$ $\displaystyle\langle
s,\Delta\dot{\chi}(0)+\varepsilon|\widehat{\varphi}(0)|^{2}\dot{\chi}(0)+\varepsilon
e^{u(0)}h(i\widehat{\varphi}(0),\dot{\widehat{\varphi}}(0))\rangle_{L^{2}}.$

Thus, the gauge orthogonality condition is

$\Delta\dot{\chi}(0)+\varepsilon
e^{u(0)}|\widehat{\varphi}(0)|^{2}\dot{\chi}(0)=-\varepsilon
e^{u(0)}h(i\widehat{\varphi}(0),\dot{\widehat{\varphi}}(0)).$ (4.11)

Since $u(t)$ satisfies (3.5) at each $t$, we find, by differentiating with
respect to $t$, that

$\Delta\dot{u}(0)+\varepsilon
e^{u(0)}|\widehat{\varphi}(0)|^{2}\dot{u}(0)=-2\varepsilon
e^{u(0)}h(\widehat{\varphi}(0),\dot{\widehat{\varphi}}(0)).$ (4.12)

The conditions on our curve of solutions imply that $\dot{\chi}(0)$ and
$\dot{u}(0)$ satisfy the driven linear PDEs (4.11) and (4.12), and,
conversely, given $(\widehat{\varphi}(0),\dot{\widehat{\varphi}}(0))$, these
PDEs uniquely determine $\dot{\chi}(0)$ and $\dot{u}(0)$. From now on, we
restrict attention to the time $t=0$ and drop the parameter $t$ from our
notation.

The squared length of the velocity vector with respect to the $L^{2}$ metric
is

$\displaystyle g(v,v)$ $\displaystyle=$
$\displaystyle\|\dot{\varphi}\|_{L^{2}}^{2}+\|\dot{A}\|_{L^{2}}^{2}$ (4.13)
$\displaystyle=$
$\displaystyle\varepsilon\int_{\Sigma}e^{u}\left\\{|\dot{\widehat{\varphi}}|^{2}+h(\dot{\widehat{\varphi}},\widehat{\varphi})\dot{u}+2h(\dot{\widehat{\varphi}},i\widehat{\varphi})\dot{\chi}+|\widehat{\varphi}|^{2}\left(\frac{\dot{u}^{2}}{4}+\dot{\chi}^{2}\right)\right\\}$
$\displaystyle+\frac{1}{4}\|\mathrm{d}\dot{u}\|_{L^{2}}^{2}+\|\mathrm{d}\dot{\chi}\|_{L^{2}}^{2}.$

Our aim is to show that the normalized $L^{2}$ metric
$g_{\varepsilon}=\varepsilon^{-1}g$ satisfies a bound of the form

$|g_{\varepsilon}(v,v)-\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2}|\leq
C\varepsilon\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2},$ (4.14)

where $C$ is independent of $v$. Comparing with (4.13) it is clear that we
must control suitable norms of $\dot{u}$ and $\dot{\chi}$.

Note that both $v=\dot{\chi}$ and $v=\dot{u}/2$ satisfy a PDE of the form

$\Delta v+av=b$ (4.15)

where $a,b\in C^{\infty}(\Sigma)$,

$a=\varepsilon|\widehat{\varphi}|^{2}e^{u},\qquad
b=\left\\{\begin{array}[]{cc}-\varepsilon
e^{u}h(i\widehat{\varphi},\dot{\widehat{\varphi}}),&v=\dot{\chi},\\\
-\varepsilon
e^{u}h(\widehat{\varphi},\dot{\widehat{\varphi}}),&v=\dot{u}/2.\end{array}\right.$
(4.16)

An obvious strategy at this point would be to use the standard elliptic
estimate for the operator $\Delta+a$ to bound $\|v\|_{H^{2}}$ in terms of
$\|b\|_{L^{2}}$. Indeed, since $a\geq 0$ and vanishes only on a finite set, it
is clear that the kernel of $\Delta+a$ is trivial, so a bound of the form
$\|v\|_{H^{2}}\leq C(a)\|b\|_{L^{2}}$ is immediately available. The problem
with this approach is that the dependence of the coercivity constant $C(a)$ on
the function $a$ (hence on $\widehat{\varphi}$) is unknown, so it does not
yield a uniform bound as we require.

To proceed further, we appeal to an estimate obtained from the Lax-Milgram
Lemma which gives a bound on $\|v\|_{H^{1}}$ (not $\|v\|_{H^{2}}$) in terms of
$a$ and $b$. This estimate is likely to be useful in a wide variety of
contexts, so we state and prove it in some generality. At first sight, the
bound looks (perhaps needlessly) elaborate, and one is tempted to replace the
right hand side by a simpler but less sharp expression. Our argument will
require the full detail of the bound as stated, however.

###### Lemma 7

Let $M$ be a compact Riemannian manifold. Then there exists a constant $C>0$,
depending only on $M$, such that, for all $a,b,v\in C^{\infty}(M)$ with $a\geq
0$ and $\int_{M}a>0$, if

$\Delta v+av=b,$

then

$\|v\|_{H^{1}}\leq
C\left\\{\left(1+\frac{\|a\|_{L^{2}}}{\int_{M}a}\right)\left(\|b\|_{L^{2}}+\frac{\|a\|_{L^{2}}}{\int_{M}a}\left|\int_{M}b\right|\right)+\frac{|\int_{M}b|}{\int_{M}a}\right\\}.$

###### Proof.

Decompose $v=\overline{v}+v_{0}$ where

$\overline{v}=\frac{1}{|M|}\int_{M}v,$ (4.17)

so $v_{0}\in{\mathscr{X}}=\\{f\in H^{1}:\int_{M}f=0\\}$. We note that
${\mathscr{X}}$ is a Hilbert space with respect to the $H^{1}$ inner product.
Substituting (4.17) into the PDE for $v$ we find that

$\Delta v_{0}+av_{0}=b-a\overline{v},$ (4.18)

which, on integration over $M$ yields,

$\overline{v}=\frac{\int_{M}b-\langle a,v_{0}\rangle_{L^{2}}}{\int_{M}a}.$
(4.19)

Substituting (4.19) back into (4.18), one finds that

$\Delta v_{0}+a\left\\{v_{0}-\frac{\langle
a,v_{0}\rangle_{L^{2}}}{\int_{M}a}\right\\}=\left\\{b-\frac{\int_{M}b}{\int_{M}a}a\right\\}.$
(4.20)

It is to (4.20) that we apply the Lax-Milgram Lemma.

Define the bilinear and linear forms

$\displaystyle{\cal
A}:{\mathscr{X}}\times{\mathscr{X}}\rightarrow{\mathbb{R}},\qquad{\cal
A}(p,q)$ $\displaystyle=$
$\displaystyle\langle\mathrm{d}p,\mathrm{d}q\rangle_{L^{2}}+\langle
ap,q\rangle_{L^{2}}-\frac{1}{\int_{M}a}\langle a,p\rangle_{L^{2}}\langle
a,q\rangle_{L^{2}}$ (4.21) $\displaystyle{\cal
B}:{\mathscr{X}}\rightarrow{\mathbb{R}},\qquad{\cal B}(q)$ $\displaystyle=$
$\displaystyle\langle b-\frac{\int_{M}b}{\int_{M}a}a,q\rangle_{L^{2}}.$ (4.22)

Then, since $v_{0}$ satisfies (4.20), for all $q\in{\mathscr{X}}$,

${\cal A}(v_{0},q)={\cal B}(q).$ (4.23)

But ${\cal A}$ is coercive (with respect to the $H^{1}$ norm on
${\mathscr{X}}$) with coercivity constant $\lambda_{1}/(1+\lambda_{1})$, where
$\lambda_{1}>0$ is the smallest positive eigenvalue of $\Delta$, that is, for
all $q\in{\mathscr{X}}$,

${\cal A}(q,q)\geq\frac{\lambda_{1}}{1+\lambda_{1}}\|q\|_{H^{1}}^{2}.$ (4.24)

To see this, note that

$\displaystyle{\cal A}(q,q)$ $\displaystyle=$
$\displaystyle\|\mathrm{d}q\|_{L^{2}}^{2}+\|\sqrt{a}q\|_{L^{2}}^{2}-\frac{1}{\int_{M}a}\langle\sqrt{a},\sqrt{a}q\rangle_{L^{2}}^{2}$
(4.25) $\displaystyle\geq$
$\displaystyle\|\mathrm{d}q\|_{L^{2}}^{2}+\|\sqrt{a}q\|_{L^{2}}^{2}-\frac{1}{\int_{M}a}\|\sqrt{a}\|_{L^{2}}^{2}\|\sqrt{a}q\|_{L^{2}}^{2}$
$\displaystyle=$ $\displaystyle\|\mathrm{d}q\|_{L^{2}}$

by Cauchy-Schwarz. But
$\|\mathrm{d}q\|_{L^{2}}^{2}\geq\lambda_{1}\|q\|_{L^{2}}^{2}$ by the Poincaré
inequality, so

$\|q\|_{H^{1}}^{2}=\|\mathrm{d}q\|_{L^{2}}^{2}+\|q\|_{L^{2}}^{2}\leq(1+\lambda_{1}^{-1})\|\mathrm{d}q\|_{L^{2}}^{2},$
(4.26)

and (4.24) immediately follows.

Hence, by the Lax-Milgram Lemma [10, p. 78],

$\|v_{0}\|_{H^{1}}\leq\frac{1+\lambda_{1}}{\lambda_{1}}\|{\cal
B}\|_{{\mathscr{X}}^{*}}$ (4.27)

where

$\displaystyle\|{\cal B}\|_{{\mathscr{X}}^{*}}$ $\displaystyle=$
$\displaystyle\sup_{q\in{\mathscr{X}},\|q\|_{H^{1}}=1}|{\cal B}(q)|$ (4.28)
$\displaystyle=$
$\displaystyle\sup_{q\in{\mathscr{X}},\|q\|_{H^{1}}=1}\left|\langle
b-\frac{\int_{M}b}{\int_{M}a}a,q\rangle_{L^{2}}\right|$ $\displaystyle\leq$
$\displaystyle\sup_{q\in{\mathscr{X}},\|q\|_{H^{1}}=1}\|q\|_{L^{2}}\left\|b-\frac{\int_{M}b}{\int_{M}a}a\right\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\left\|b-\frac{\int_{M}b}{\int_{M}a}a\right\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\|b\|_{L^{2}}+\frac{\|a\|_{L^{2}}}{\int_{M}a}\left|\int_{M}b\right|.$

But then, by (4.19),

$\displaystyle|\overline{v}|$ $\displaystyle\leq$
$\displaystyle\frac{|\int_{M}b|}{\int_{M}a}+\frac{\|a\|_{L^{2}}}{\int_{M}a}\|v_{0}\|_{L^{2}},$
(4.29)

so

$\displaystyle\|v\|_{H^{1}}$ $\displaystyle=$
$\displaystyle\|v_{0}+\overline{v}\|_{H^{1}}\leq\|v_{0}\|_{H^{1}}+\sqrt{|M|}|\overline{v}|$
(4.30) $\displaystyle\leq$
$\displaystyle\left(1+\sqrt{|M|}\frac{\|a\|_{L^{2}}}{\int_{M}a}\right)\|v_{0}\|_{H^{1}}+\sqrt{|M|}\frac{|\int_{M}b|}{\int_{M}a}.$

Defining $C:=(1+\lambda_{1}^{-1})\max\\{1,\sqrt{|M|}\\}$, the claimed estimate
now follows from inequalities (4.27), (4.28) and (4.30). ∎

We next apply Lemma 7 to the PDEs for $\dot{\chi}$, $\dot{u}$, (4.11), (4.12),
to obtain order $\varepsilon$ bounds on their $H^{1}$ norms. The last term in
the inequality in Lemma 7, namely $|\int_{\Sigma}b|/\int_{\Sigma}a$, looks
fatal to this endeavour, since both $a$ and $b$ are of order $\varepsilon$. It
will turn out, however, that $\int_{\Sigma}b$ is actually of order
$\varepsilon^{2}$ due to subtle cancellations in the integral.

###### Proposition 8

There exists a constant $C>0$, depending only on $(L,h)$ and $g_{\Sigma}$,
such that

$\|\dot{\chi}\|_{H^{1}},\,\|\dot{u}\|_{H^{1}}\leq
C\varepsilon\|\dot{\widehat{\varphi}}\|_{L^{2}}.$

###### Proof.

Throughout this proof, $c$ will denote a positive constant depending (at most)
on $g_{\Sigma}$ and $(L,h)$. The value of $c$ may vary from line to line. Let
$a=\varepsilon|\widehat{\varphi}|^{2}e^{u}\geq 0$, as defined in (4.16). Since
$\widehat{\varphi}^{-1}(0)$ is finite, it is clear that $\int_{\Sigma}a>0$. By
Lemmas 5 and 6 and the Mean Value Theorem,

$e^{u}=1+U$ (4.31)

where $\|U\|_{C^{0}}\leq\|u\|_{C^{0}}e^{\|u\|_{C^{0}}}\leq c\varepsilon$.
Hence

$\displaystyle\int_{\Sigma}a$ $\displaystyle=$
$\displaystyle\varepsilon\int_{\Sigma}|\widehat{\varphi}|^{2}(1+U)$ (4.32)
$\displaystyle=$
$\displaystyle\varepsilon+\varepsilon\int_{\Sigma}|\widehat{\varphi}|^{2}U$
$\displaystyle\geq$
$\displaystyle\varepsilon-\varepsilon\|U\|_{C^{0}}\int_{\Sigma}|\widehat{\varphi}|^{2}$
$\displaystyle\geq$ $\displaystyle\varepsilon-c\varepsilon^{2}\geq
c\varepsilon.$

Furthermore,

$\displaystyle\|a\|_{L^{2}}^{2}$ $\displaystyle=$
$\displaystyle\varepsilon^{2}\int_{\Sigma}|\widehat{\varphi}|^{4}e^{2u}$
(4.33) $\displaystyle\leq$ $\displaystyle
2\varepsilon^{2}\int_{\Sigma}|\widehat{\varphi}|^{4}(1+U^{2})$
$\displaystyle\leq$ $\displaystyle
2\varepsilon^{2}\alpha^{2}(1+\|U\|_{C^{0}}^{2})$ $\displaystyle\leq$
$\displaystyle c\varepsilon^{2}.$

Consider now the function $b$ defined in (4.16). Note that

$b=-\varepsilon e^{u}k,\qquad
k:=\left\\{\begin{array}[]{cc}h(i\widehat{\varphi},\dot{\widehat{\varphi}}),&v=\dot{\chi},\\\
h(\widehat{\varphi},\dot{\widehat{\varphi}}),&v=\dot{u}/2,\end{array}\right.$
(4.34)

and, crucially, in either case $\int_{\Sigma}k=0$ by (4.3) or (4.4). Hence

$\int_{\Sigma}b=\varepsilon\int_{\Sigma}(1+U)k=\varepsilon\int_{\Sigma}Uk,$
(4.35)

and so

$\displaystyle\left|\int_{\Sigma}b\right|$ $\displaystyle\leq$
$\displaystyle\varepsilon\|U\|_{C^{0}}\int_{\Sigma}|k|$ (4.36)
$\displaystyle\leq$
$\displaystyle\varepsilon\|U\|_{C^{0}}\int_{\Sigma}|\widehat{\varphi}||\dot{\widehat{\varphi}}|$
$\displaystyle\leq$
$\displaystyle\varepsilon\|U\|_{C^{0}}\|\widehat{\varphi}\|_{L^{2}}\|\dot{\widehat{\varphi}}\|_{L^{2}}$
$\displaystyle=$
$\displaystyle\varepsilon\|U\|_{C^{0}}\|\dot{\widehat{\varphi}}\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
c\varepsilon^{2}\|\dot{\widehat{\varphi}}\|_{L^{2}}.$

Furthermore

$\displaystyle\|b\|_{L^{2}}$ $\displaystyle=$
$\displaystyle\varepsilon\|e^{u}f\|_{L^{2}}$ (4.37) $\displaystyle\leq$
$\displaystyle\varepsilon
e^{\|u\|_{C^{0}}}\|\widehat{\varphi}\|_{C^{0}}\|\dot{\widehat{\varphi}}\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
c\varepsilon\|\dot{\widehat{\varphi}}\|_{L^{2}}$

by Lemma 5. Hence

$\frac{\|a\|_{L^{2}}}{\int_{\Sigma}a}\leq
c,\qquad\mbox{and}\qquad\frac{|\int_{\Sigma}b|}{\int_{\Sigma}a}\leq
c\varepsilon\|\dot{\widehat{\varphi}}\|_{L^{2}}.$ (4.38)

Hence, by Lemma 7, both $v=\dot{\chi}$ and $v=\dot{u}/2$ satisfy

$\displaystyle\|v\|_{H^{1}}$ $\displaystyle\leq$ $\displaystyle
c\left(\|b\|_{L^{2}}+|\int_{\Sigma}b|+\frac{|\int_{\Sigma}b|}{\int_{\Sigma}a}\right)$
(4.39) $\displaystyle\leq$ $\displaystyle
c\varepsilon\|\dot{\widehat{\varphi}}\|_{L^{2}},$

whence the claim follows. ∎

###### Proposition 9

There exists a constant $C>0$, depending only on $(L,h)$ and $g_{\Sigma}$,
such that

$\left|\|\dot{\varphi}\|_{L^{2}}^{2}+\|\dot{A}\|_{L^{2}}^{2}-\varepsilon\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2}\right|\leq
C\varepsilon^{2}\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2}.$

###### Proof.

Again, $c$ will denote a positive constant depending (at most) on $g_{\Sigma}$
and $(L,h)$, whose value may change from line to line. Differentiating (4.1)
and (4.2) with respect to $t$, we see that

$\displaystyle\|\dot{\varphi}\|_{L^{2}}^{2}$ $\displaystyle=$
$\displaystyle\varepsilon\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2}+\varepsilon\int_{\Sigma}U|\dot{\widehat{\varphi}}|^{2}+\varepsilon\int_{\Sigma}\left\\{h(\dot{\widehat{\varphi}},(\dot{u}+2i\dot{\chi})\widehat{\varphi})+\left(\frac{1}{4}\dot{u}^{2}+\dot{\chi}^{2}\right)|\widehat{\varphi}|^{2}\right\\}e^{u}$
$\displaystyle\|\dot{A}\|_{L^{2}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\|\mathrm{d}\dot{u}\|_{L^{2}}^{2}+\|\mathrm{d}\dot{\chi}\|_{L^{2}}^{2},$
(4.40)

where, as before, $e^{u}=:1+U$ with $\|U\|_{C^{0}}\leq c\varepsilon$. Hence

$\displaystyle\left|\|\dot{\varphi}\|_{L^{2}}^{2}+\|\dot{A}\|_{L^{2}}^{2}-\varepsilon\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2}\right|$
$\displaystyle\leq$
$\displaystyle\varepsilon\|U\|_{C^{0}}\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2}+\varepsilon\|\widehat{\varphi}\|_{C^{0}}e^{\|u\|_{C^{0}}}\|\dot{\widehat{\varphi}}\|_{L^{2}}(\|\dot{u}\|_{L^{2}}+2\|\dot{\chi}\|_{L^{2}})$
(4.41) $\displaystyle+\varepsilon
e^{\|u\|_{C^{0}}}\|\widehat{\varphi}\|_{C^{0}}^{2}\left(\frac{1}{4}\|\dot{u}\|_{L^{2}}^{2}+\|\dot{\chi}\|_{L^{2}}^{2}\right)+\frac{1}{4}\|\dot{u}\|_{H^{1}}^{2}+\|\dot{\chi}\|_{H^{1}}^{2}$
$\displaystyle\leq$ $\displaystyle c\varepsilon^{2}+\varepsilon\alpha
c\|\dot{\widehat{\varphi}}\|_{L^{2}}3c\varepsilon\|\dot{\widehat{\varphi}}\|_{L^{2}}+\varepsilon
c\alpha^{2}3C^{2}\varepsilon^{2}\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2}+\frac{3}{4}c^{2}\varepsilon^{2}\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2}$
$\displaystyle=$ $\displaystyle
c\varepsilon^{2}\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2},$

where we have used Lemma 5 to bound $e^{\|u\|_{C^{0}}}$ and Proposition 8.

∎

Our main theorem immediately follows.

Proof of Theorem 1:  We must show that the symmetric bilinear form
$\sigma_{\varepsilon}:=g_{\varepsilon}-g_{0}$ converges uniformly to zero, as
$\varepsilon\searrow 0$, on the unit tangent bundle of any fixed reference
metric for $M_{n}$.

Given any tangent vector $X\in T_{[(\varphi,A)]}M_{n}$ we may choose a
representative curve $(\varphi(t),A(t))$ of the form (4.1), (4.2) to compute

$g_{\varepsilon}(X,X)=\frac{1}{\varepsilon}(\|\dot{\varphi}\|_{L^{2}}^{2}+\|\dot{A}\|_{L^{2}}^{2}),$
(4.42)

and

$g_{0}(X,X)=\|\dot{\widehat{\varphi}}\|_{L^{2}}^{2}.$ (4.43)

Proposition 9 implies that

$|g_{\varepsilon}(X,X)-g_{0}(X,X)|\leq C\varepsilon g_{0}(X,X)$ (4.44)

where the constant $C$ is independent of $X$ and $\varepsilon$. Hence, for all
unit length vectors ${X},{Y}$ tangent to $(M_{n},g_{0})$,

$\displaystyle|\sigma_{\varepsilon}({X},{Y})|$ $\displaystyle=$
$\displaystyle\frac{1}{4}|\sigma_{\varepsilon}({X}+{Y},{X}+{Y})-\sigma_{\varepsilon}({X}-{Y},{X}-{Y})|$
(4.45) $\displaystyle\leq$
$\displaystyle\frac{1}{4}C\varepsilon(g_{0}({X}+{Y},{X}+{Y})+g_{0}({X}-{Y},{X}-{Y}))$
$\displaystyle=$ $\displaystyle C\varepsilon.$

This establishes uniform convergence with respect to the reference metric
$g_{0}$.

$\Box$

##  5 Convergence of the spectrum

Recall that the spectrum of a closed Riemannian manifold $(M,g)$ is the
spectrum of its Laplace-Beltrami operator $\Delta_{g}=\delta\mathrm{d}$,
arranged in non-decreasing order,

$0=\lambda_{0}(g)<\lambda_{1}(g)\leq\lambda_{2}(g)\leq\lambda_{3}(g)\leq\cdots,$
(5.1)

a sequence of numbers that diverges to infinity. In the case of
$(M_{n},g_{\varepsilon})$, this spectrum has a direct physical interpretation
in terms of the quantum dynamics of $n$-vortices. The Hamiltonian operator
governing this dynamics is (conjecturally) $H=\frac{1}{2}\Delta_{g}$, where
$g=\varepsilon g_{\varepsilon}$ is the (non-normalized) $L^{2}$ metric. Hence,
the energy spectrum of $n$ quantum vortices moving on $\Sigma$ is

$E_{k}(\varepsilon)=\frac{\lambda_{k}(g_{\varepsilon})}{2\varepsilon},\qquad
k=0,1,2,\ldots.$ (5.2)

Note that, in local coordinates,

$\Delta_{g_{\varepsilon}}=-g^{ij}_{\varepsilon}\left(\frac{\partial\>}{\partial
x_{i}\partial x_{j}}-\Gamma^{\varepsilon,k}_{ij}\frac{\partial\>}{\partial
x_{k}}\right),$ (5.3)

where $\Gamma^{\varepsilon}$ denotes the Christoffel symbols of the metric
$g_{\varepsilon}$. Since $\Gamma^{\varepsilon}$ involves derivatives of
$g_{\varepsilon}$, the $C^{0}$ convergence proved in Theorem 1 does not imply
that $\Delta_{g_{\varepsilon}}$ converges to $\Delta_{g_{0}}$. Rather
remarkably, we can still conclude, by work of Bando and Urakawa [2], that the
spectrum of $\Delta_{g_{\varepsilon}}$ converges uniformly to the spectrum of
$\Delta_{g_{0}}$, in the sense that

$\frac{\lambda_{k}(g_{\varepsilon})}{\lambda_{k}(g_{0})}\stackrel{{\scriptstyle\varepsilon\rightarrow
0}}{{\longrightarrow}}1$ (5.4)

uniformly in $k$. More precisely:

###### Corollary 10

Let $C>0$ be the constant, depending only on $(L,h)$ and $g_{\Sigma}$, whose
existence is established in Proposition 9. Then for all
$\varepsilon\in[0,1/C)$ and all $k\geq 1$,

$\frac{(1-C\varepsilon)^{n}}{(1+C\varepsilon)^{n+1}}\leq\frac{\lambda_{k}(g_{\varepsilon})}{\lambda_{k}(g_{0})}\leq\frac{(1+C\varepsilon)^{n}}{(1-C\varepsilon)^{n+1}}.$

###### Proof.

Given a Riemannian metric $g$ on $M_{n}$, and a finite dimensional subspace
$V$ of $C^{\infty}(M_{n})$, let

$\Lambda_{g}(V):=\sup\\{\|\mathrm{d}f\|_{L^{2},g}^{2}/\|f\|_{L^{2},g}^{2}\>:\>f\in
V\backslash\\{0\\}\\}.$ (5.5)

Then, by Proposition 2.1 of [2],

$\lambda_{k}(g)=\inf\\{\Lambda_{g}(V)\>:\>V<C^{\infty}(M_{n}),\,\dim
V=k+1\\},$ (5.6)

the infimum of $\Lambda_{g}$ over all $(k+1)$-dimensional subspaces of
$C^{\infty}(M_{n})$.

At a given point $p\in M_{n}$, let $E_{1},\ldots,E_{2n}$ be an orthonormal
frame for $(T_{p}M_{n},g_{0})$, $G_{\varepsilon}$ be the matrix with entries
$g_{\varepsilon}(E_{i},E_{j})$, $S_{\varepsilon}$ the matrix with entries
$\sigma_{\varepsilon}(E_{i},E_{j})$ (where, as before,
$\sigma_{\varepsilon}=g_{\varepsilon}-g_{0}$) and $H_{\varepsilon}$ be the
inverse of $G_{\varepsilon}$. Then the volume form defined by
$g_{\varepsilon}$ is

${\rm vol}_{g_{\varepsilon}}=v_{\varepsilon}{\rm vol}_{g_{0}},$ (5.7)

where $v_{\varepsilon}(p)=(\det G_{\varepsilon})^{1/2}$. Now, by (4.44), every
eigenvalue $\mu$ of $S_{\varepsilon}$ satisfies $|\mu|\leq C\varepsilon$, so
the eigenvalues $1+\mu$ of $G_{\varepsilon}={\mathbb{I}}_{2n}+S_{\varepsilon}$
lie in $[1-C\varepsilon,1+C\varepsilon]$. Hence, for all $\varepsilon<1/C$,

$(1-C\varepsilon)^{n}\leq v_{\varepsilon}(p)\leq(1+C\varepsilon)^{n}.$ (5.8)

Furthermore, the eigenvalues $\nu=(1+\mu)^{-1}$ of $H_{\varepsilon}$ lie in
$[(1+C\varepsilon)^{-1},(1-C\varepsilon)^{-1}]$. Now defining, for any given
smooth function $f$, $v$ to be the column matrix with entries
$\mathrm{d}f_{p}(E_{i})$,

$|\mathrm{d}f_{p}|_{g_{\varepsilon}}^{2}=v^{T}H_{\varepsilon}v,\qquad|\mathrm{d}f_{p}|_{g_{0}}^{2}=v^{T}v.$
(5.9)

Hence

$\frac{|\mathrm{d}f_{p}|_{g_{0}}^{2}}{(1+C\varepsilon)}\leq\nu_{min}|\mathrm{d}f_{p}|_{g_{0}}^{2}\leq|\mathrm{d}f_{p}|_{g_{\varepsilon}}^{2}\leq\nu_{max}|\mathrm{d}f_{p}|_{g_{0}}^{2}\leq\frac{|\mathrm{d}f_{p}|_{g_{0}}^{2}}{(1-C\varepsilon)}.$
(5.10)

The bounds (5.8) and (5.10) are independent of $p\in M_{n}$, so hold globally.
Hence, for all $f\in C^{\infty}(M_{n})$,

$\displaystyle(1-C\varepsilon)^{n}\|f\|_{L^{2},g_{0}}^{2}\leq$
$\displaystyle\|f\|_{L^{2},g_{\varepsilon}}^{2}$
$\displaystyle\leq(1+C\varepsilon)^{n}\|f\|_{L^{2},g_{0}}^{2},$
$\displaystyle\frac{(1-C\varepsilon)^{n}}{1+C\varepsilon}\|\mathrm{d}f\|_{L^{2},g_{0}}^{2}\leq$
$\displaystyle\|\mathrm{d}f\|_{L^{2},g_{\varepsilon}}^{2}$
$\displaystyle\leq\frac{(1+C\varepsilon)^{n}}{1-C\varepsilon}\|\mathrm{d}f\|_{L^{2},g_{0}}^{2},$
(5.11)

and so, for all $f\neq 0$,

$\displaystyle\frac{\|\mathrm{d}f\|_{L^{2},g_{\varepsilon}}^{2}}{\|f\|_{L^{2},g_{\varepsilon}}^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{(1+C\varepsilon)^{n}}{(1-C\varepsilon)^{n+1}}\frac{\|\mathrm{d}f\|_{L^{2},g_{0}}^{2}}{\|f\|_{L^{2},g_{0}}^{2}}$
$\displaystyle\frac{\|\mathrm{d}f\|_{L^{2},g_{0}}^{2}}{\|f\|_{L^{2},g_{0}}^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{(1+C\varepsilon)^{n+1}}{(1-C\varepsilon)^{n}}\frac{\|\mathrm{d}f\|_{L^{2},g_{\varepsilon}}^{2}}{\|f\|_{L^{2},g_{\varepsilon}}^{2}}.$
(5.12)

Hence, for any $(k+1)$ dimensional subspace $V$ of $C^{\infty}(M_{n})$,

$\displaystyle\Lambda_{g_{\varepsilon}}(V)$ $\displaystyle\leq$
$\displaystyle\frac{(1+C\varepsilon)^{n}}{(1-C\varepsilon)^{n+1}}\Lambda_{g_{0}}(V),$
$\displaystyle\Lambda_{g_{0}}(V)$ $\displaystyle\leq$
$\displaystyle\frac{(1+C\varepsilon)^{n+1}}{(1-C\varepsilon)^{n}}\Lambda_{g_{\varepsilon}}(V),$
(5.13)

and so

$\displaystyle\lambda_{k}(g_{0})$ $\displaystyle\geq$
$\displaystyle\frac{(1-C\varepsilon)^{n+1}}{(1+C\varepsilon)^{n}}\lambda_{k}(g_{\varepsilon}),$
$\displaystyle\lambda_{k}(g_{\varepsilon})$ $\displaystyle\geq$
$\displaystyle\frac{(1-C\varepsilon)^{n}}{(1+C\varepsilon)^{n+1}}\lambda_{k}(g_{0}),$
(5.14)

which completes the proof. ∎

The spectrum of $g_{0}$ is known explicitly [5],

$\lambda_{k}(g_{0})=4k(n+k),\qquad
d_{k}(g_{0})=n(n+2k)\left(\begin{array}[]{c}n-1+k\\\ k\end{array}\right)^{2},$
(5.15)

where we have changed our labelling convention so that
$0=\lambda_{0}<\lambda_{1}<\lambda_{2}<\cdots$ and $d_{k}$ is the degeneracy
of $\lambda_{k}$. In order to extract explicit bounds on
$\lambda_{k}(g_{\varepsilon})$ from Corollary 10, we need an explicit upper
bound on the constant $C$. Such a bound should be obtainable when $\Sigma$ is
given the round metric, since explicit formulae for $\widehat{\varphi}$ are
available in this case [4]. A particularly important issue, if one is to apply
Corollary 10 to the study of the quantum statistical mechanics of vortices
[13], is the $n$ dependence of $C$.

## Acknowledgements

JMS wishes to thank Gerasim Kokarev for bringing reference [2] to his
attention.

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