Renormalized Energy Between Vortices in Some Ginzburg–Landau Models on 2-Dimensional Riemannian Manifolds

Abstract

We study a variational Ginzburg–Landau type model depending on a small parameter $\varepsilon >0$ for (tangent) vector fields on a 2-dimensional Riemannian manifold S. As $\varepsilon \rightarrow 0$, these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus ${\mathfrak {g}}$ of S is different than 1. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorticity-dependent lattice in the $2{\mathfrak {g}}$-dimensional space of harmonic 1-forms on S if ${\mathfrak {g}}\geqq 1$. Our second main result determines the interaction energy (called renormalized energy) between vortex points as a $\Gamma$-limit (at the second order) as $\varepsilon \rightarrow 0$. The renormalized energy governing the optimal location of vortices depends on the Gauss curvature of S as well as on the quantized flux. The coupling between flux quantization constraints and vorticity, and its impact on the renormalized energy, are new phenomena in the theory of Ginzburg–Landau type models. We also extend this study to two other (extrinsic) models for embedded hypersurfaces $S\subset {{\mathbb {R}}}^3$, in particular, to a physical model for non-tangent maps to S coming from micromagnetics.

Appendix A. Ball Construction. Proof of Proposition 8.2

In this section we present the vortex ball construction leading to Proposition 8.2. We start with several lemmas in which we verify, largely by adapting classical proofs to our setting, that basic ingredients needed for the vortex ball argument on the Euclidean plane remain available in the present setting. Once these ingredients are available, we follow classical arguments. Since the arguments are rather standard, our exposition is terse in places.

We first fix positive constants $c_1(S), r_0(S)$ such that $\partial B_r(x)$ is a homeomorphic to a circle for every $x\in S$, and $0<r<r_0$ (thus, $r_0$ is at most the injectivity radius of S) and we recall (121). In several places later in our argument, we will impose additional smallness conditions on $r_0$.

In view of Lemma 5.1, in proving lower energy bounds we may restrict our attention to smooth vector fields.

Lemma A.1

Given $u\in {{\mathcal {X}}}(S)$, let $\rho := |u|_g$. Assume that $\varepsilon<r<r_0(S)$. Then there exist positive constants $c_2,c_3, c_4$ such that the following hold. First,

$$\begin{aligned} \frac{1}{2} \int _{\partial B_r(x)} | d \rho |_g^2 + \frac{1}{2\varepsilon ^2}F(\rho ^2) \, d{{\mathcal {H}}}^1 \geqq \frac{c_2}{\varepsilon } \Vert 1-\rho \Vert _{L^\infty }^2 \end{aligned}$$

(122)

where $|{d}\rho |_g^2(x) := ({d}\rho (\tau _1))^2 + ({d}\rho (\tau _2))^2$ for any orthonormal basis $\{ \tau _1,\tau _2 \}$ of $T_xS$. Second,

$$\begin{aligned} \int _{\partial B_r(x)} e_\varepsilon ^{in}(u) d{{\mathcal {H}}}^1 \geqq \lambda _\varepsilon \left( \frac{r}{|d|} \right) \quad \hbox { for } d = {\left\{ \begin{array}{ll} \deg (u; \partial B_r(x))&{}\hbox { if }\rho \geqq \frac{1}{2} \hbox { on }\partial B_r(x) \\ \hbox {any positive integer} &{}\hbox { if not } \end{array}\right. } \nonumber \\ \end{aligned}$$

(123)

where

$$\begin{aligned} \lambda _\varepsilon (r) := \min _{0<s\leqq 1} \left[ \frac{c_2}{4 \varepsilon }(1-s)^2 + s^2 \frac{\pi }{r}(1- c_3 r^2)\,\right] \geqq \frac{\pi (1-c_3 r^2)}{r+c_4 \varepsilon } \end{aligned}$$

(124)

is a nonincreasing function, and we use the convention that for $r>0$, $\lambda _\varepsilon (r/0) = \lambda _\varepsilon (+\infty ) = 0$.

Proof

At $y\in \partial B_r(x)$, let $\tau \in T_yS$ denote the unit tangent to $\partial B_r(x)$, oriented in the standard way, and let ${}'$ denote differentiation with respect to $\tau$. Further define $\zeta := (1 - \rho )^2$. Then by (1)

$$\begin{aligned} |\zeta '| + \frac{1}{\varepsilon }|\zeta | = 2|\rho -1| |\rho '| + \frac{1}{\varepsilon }\zeta \leqq \varepsilon |\rho '|^2 + \frac{2}{\varepsilon }(1-\rho )^2 \leqq C \varepsilon \left( |d \rho |_g^2 + \frac{1}{2\varepsilon ^2}F(\rho ^2)\right) . \end{aligned}$$

Then (123) follows from a (suitably scaled) Sobolev embedding $W^{1,1}\hookrightarrow L^\infty$ on $\partial B_r(x)$, taking into account the fact that ${{\mathcal {H}}}^1(\partial B_r(x)) \geqq \varepsilon$.

Next, if $\rho \geqq \frac{1}{2}$ on $\partial B_r(x)$, then we can define $v = u/\rho$ and $d:= \deg (u; \partial B_r(x))$ . Since $|v|_g=1$, we have $|Dv|_g = |j(v)|_g$, and thus

$$\begin{aligned} \int _{\partial B_r(x)}|Dv|_g^2 \, d{\mathcal {H}}^1 = \int _{\partial B_r(x)}|j(v)|_g^2 \, d{\mathcal {H}}^1&\ \geqq \ \frac{1}{{{\mathcal {H}}}^1(\partial B_r(x))} \big |\int _{\partial B_r(x)}j(v) \big |^2_g\\&\overset{(5)}{=} \frac{1}{{{\mathcal {H}}}^1(\partial B_r(x))} \left( 2\pi d - \int _{B_r(x)}\kappa \ \,\hbox {vol}_g \right) ^2. \end{aligned}$$

Since S is compact and smooth, it follows from this and (121) that

$$\begin{aligned} \int _{\partial B_r(x)} \frac{1}{2} |Dv|_g^2 d{{\mathcal {H}}}^1 \geqq \frac{\pi d^2}{r}(1- c_3 r^2)\, . \end{aligned}$$

(125)

Finally, if we write $s :=\min _{\partial B_r(x)} (\rho \wedge 1) >0$, then $|Du|_g^2 = |{d}\rho |{_g}^2 + \rho ^2|Dv|{_g}^2 \geqq |{d}\rho |_g^2 + s^2 |Dv|_g^2$. Then one may deduce (124) from (123) and (126), after first taking $r_0$ small enough so that $c_3r_0^2 \leqq 1/2$, which yields $|d|(1-c_3r^2)\geqq 1-c_3\frac{r^2}{d^2}$ for every $|d|\geqq 1$. Then (124) follows directly in the case $\rho \geqq \frac{1}{2}$ on $\partial B_r(x)$, if $d\ne 0$, whereas if $d=0$ it is immediate. If $\min _{\partial B_r(x)}\rho < \frac{1}{2}$, then $\Vert 1-\rho \Vert _{L^\infty } > \frac{1}{2}$, and thus

$$\begin{aligned} \int _{\partial B_r(x)} e_\varepsilon ^{in}(u) d{{\mathcal {H}}}^1&\geqq \frac{1}{2} \int _{\partial B_r(x)} | d \rho |_g^2 + \frac{1}{2\varepsilon ^2}F(\rho ^2) \, d{{\mathcal {H}}}^1 \overset{(123)}{\geqq }\frac{c_2}{4\varepsilon } \\&\geqq \min _{0<s\leqq 1} \left[ \frac{c_2}{4 \varepsilon }(1-s)^2 + s^2 \frac{\pi }{r}(1- c_3 r^2)\,\right] = \lambda _\varepsilon (r) \geqq \lambda _\varepsilon \left( \frac{r}{|d|}\right) \end{aligned}$$

for any d. (If $\rho =0$ somewhere on $\partial B_r(x)$, then the definition $v = u/\rho$ may not make sense, but the proof of (124) relies only on (123) and makes no mention of v.) $\square$

We also need:

Lemma A.2

Assume that u is a smooth vector field on S and that for some $0<r<r_0(S)$ and $x\in S$,

$$\begin{aligned} \rho := |u|_g \geqq \frac{1}{2} \hbox { on }\partial B_r(x), \qquad \deg (u;\partial B_r(x)) = d\ne 0. \end{aligned}$$

Then if $r_0$ is sufficiently small,

$$\begin{aligned} \int _{B_r(x)} |Du|_g^2 \,\hbox {vol}_g \geqq \frac{\pi }{4} |d| \, . \end{aligned}$$

Proof

First, let $O := \{y\in B_r(x) : \rho (y)<t \}$, where t is a regular value of $\rho (\cdot )$ such that $\frac{1}{8}<t<\frac{1}{4}$. Then O is an open set with smooth boundary, compactly contained in $B_r(x)$, and

$$\begin{aligned} d = \deg (u;\partial B_r(x))&= \frac{1}{2\pi } \left( \int _{\partial B_r(x)} j\left( \frac{u}{|u|_g}\right) + \int _{B_r(x)} \kappa \,\hbox {vol}_g \right) \\&= \frac{1}{2\pi } \left( \int _{\partial O} \frac{ j(u)}{t^2} + \int _O \kappa \,\hbox {vol}_g \right) \end{aligned}$$

where the final equality follows from Lemma 5.4 and Stokes’ Theorem, as well as the fact that $|u|_g=t$ on $\partial O$. Thus if $r_0$ is sufficiently small (depending on $\Vert \kappa \Vert _\infty$) then

$$\begin{aligned} 2\pi t^2 \left( |d| - \frac{1}{2} \right) \leqq \big |\int _{\partial O} j(u) \big |_g = \big |\int _{O} dj(u) \big |_g \leqq \int _O|d j(u)|_g \, \leqq \int _O |Du|_g^2, \end{aligned}$$

where the last inequality follows from Lemma 5.3, see (40). $\square$

Finally we recall

Lemma A.3

Assume that $u_\varepsilon$ is a smooth vector field satisfying (51). Then there exists $\varepsilon _0>0$ such that whenever $0<\varepsilon <\varepsilon _0$, there exists a collection $\tilde{{\mathcal {B}}}^0 = \{ {\tilde{B}}^0_j \}$ of closed pairwise disjoint balls that cover the set where $|u_\varepsilon |_g\leqq \frac{1}{2}$, and such that

$$\begin{aligned} \sum _j {\tilde{r}}^0_j \leqq C \varepsilon \int _S e_\varepsilon ^{in}(|u_\varepsilon |_g)\, \,\hbox {vol}_g, \qquad \hbox { where }{\tilde{r}}^0_j\hbox { denotes the radius of } {\tilde{B}}^0_j. \end{aligned}$$

Proof

Let $\rho := |u_\varepsilon |_g$. Then $\frac{1}{2} |{d}\rho |_g^2 + \frac{1}{4\varepsilon ^2}F(\rho ^2) \geqq \frac{1}{\varepsilon \sqrt{2}} |{d}\rho |_g \sqrt{F(\rho ^2)} \geqq \frac{c}{\varepsilon }|1-\rho | \, |{d}\rho |_g$, by (1). Thus the coarea formula, which remains valid on a smooth manifold, implies that

$$\begin{aligned} \varepsilon \int _S e_\varepsilon ^{in}(\rho ) \,\hbox {vol}_g \geqq c \int _0^\infty |1-s| {{\mathcal {H}}}^1(\rho ^{-1}(s))\, ds \geqq c \int _{1/2}^{3/4}{{\mathcal {H}}}^1(\rho ^{-1}(s))\, ds. \end{aligned}$$

In particular we may find some $\alpha \in [\frac{1}{2}, \frac{3}{4}]$, a regular value of $\rho$, such that ${{\mathcal {H}}}^1(\rho ^{-1}(\alpha )) \leqq C \varepsilon \int _S e_\varepsilon ^{in}(\rho ) \,\hbox {vol}_g$. Following standard arguments, we may start with an efficient finite cover of $\rho ^{-1}(\alpha )$ and then merge balls to find a collection of closed pairwise disjoint balls that cover $\rho ^{-1}(\alpha )$ and whose radii sum to at most $2{{\mathcal {H}}}^1(\rho ^{-1}(\alpha ))$. This is $\tilde{{\mathcal {B}}}^0$. The complement of the union of these balls is connected as long as $\varepsilon$ is small enough, so on the complement, either $\rho > \alpha$ or $\rho <\alpha$ everywhere. The latter case is impossible by (51) and (123), if $\varepsilon$ is small enough, and this proves the lemma. $\square$

A few more definitions are needed before we prove Proposition 8.2. W.l.o.g., we may assume that $\frac{1}{2}$ is a regular value of $\rho =|u|_g$. First, we set

$$\begin{aligned} Z&:= \{ x\in S : |u(x)|_g \leqq \frac{1}{2}\}, \\ Z_E&:= \cup \{ \hbox { connected components }Z_l\hbox { of }Z \ : \ \deg (u; \partial Z_l) \ne 0 \}. \end{aligned}$$

Next, for any set $V\subset S$ such that $\partial V \cap Z_E = \emptyset$ we define the generalized degree

$$\begin{aligned} \hbox {dg}(u;\partial V) := \sum \{ \deg (u; \partial Z_l) \ : \ \hbox { components }Z_l\hbox { of }Z_E\hbox { such that }Z_l\subset \subset V\}. \end{aligned}$$

Note that $\hbox {dg}(u;\partial V) = \deg (u,\partial V)$ if $\partial V$ is $C^1$, say, and $|u|_g > \frac{1}{2}$ on $\partial V$. Finally we define

$$\begin{aligned} \Lambda _\varepsilon (\sigma ) := \int _0^\sigma \lambda _\varepsilon (r)\, dr. \end{aligned}$$

It is straightforward to check that

$$\begin{aligned} \Lambda _\varepsilon (\sigma ) \geqq \pi \log (1+ \frac{\sigma }{c_4\varepsilon }) - C \sigma ^2 \geqq \pi (\log \frac{\sigma }{\varepsilon }- C)\qquad \hbox { for }0\leqq \sigma \leqq r_0(S).\nonumber \\ \end{aligned}$$

(126)

We record several other properties. First, since $\Lambda _\varepsilon (\cdot )$ is the integral over $[0,\sigma ]$ of a positive nonincreasing function, it is easy to see that

$$\begin{aligned} \Lambda _\varepsilon (\sigma _1+\sigma _2) \leqq \Lambda _\varepsilon (\sigma _1) + \Lambda _\varepsilon (\sigma _2), \qquad \sigma \mapsto \frac{1}{\sigma }\Lambda _\varepsilon (\sigma )\hbox { is nonincreasing}. \end{aligned}$$

Finally, consider two radii $r_1<r_2$ such that $\varepsilon \leqq r_j \leqq r_0$ for $j=1,2$, and assume that $x\in S$ is a point such that $Z_E$ does not intersect the annulus $B_{r_2}{\setminus } B_{r_1}(x)$. Then $\hbox {dg}(u;\partial B_r(x)) = \hbox {dg}(u; \partial B_{r_1}(x))$ for all $r\in (r_1,r_2)$, so one may use the coarea formula and integrate (124) from $r_1$ to $r_2$ to find that

$$\begin{aligned} \int _{B_{r_2}{\setminus } B_{r_1}(x)}e_\varepsilon ^{in}(u) \, \,\hbox {vol}_g \geqq {|d|}\left[ \Lambda _\varepsilon \left( \frac{r_2}{{|d|}}\right) - \Lambda _\varepsilon \left( \frac{r_1}{{|d|}}\right) \right] ,\qquad d:= \hbox {dg}(u; \partial B_{r_1}(x)). \qquad \end{aligned}$$

(127)

Proof of Proposition 8.2

We divide the proof in several steps:

Step 1. An initial covering of $Z_E$. We claim first that there exists a collection ${\mathcal {B}}^0 = \{ B_{l,0}\}_{k=1}^K$ of closed, pairwise disjoint balls with centers $a_{l,0}$ and radii $r_{l,0}\geqq \varepsilon$ for all l, such that $Z_E\subset \cup B^0_k$, and (after possibly decreasing the constant $c_2$ in the definition (125) of $\lambda _\varepsilon$, in a way that depends only on the geometry of S)

$$\begin{aligned} \int _{B_{l,0}} e^{in}_\varepsilon (u) \,\hbox {vol}_g \ \geqq \frac{c_2}{4\varepsilon } r_{l,0} \geqq \ \Lambda _\varepsilon (r_{l,0}) \qquad \hbox { for every }l. \end{aligned}$$

(128)

We first cover $Z_E$ with balls that satisfy (129). Indeed, for every $x\in Z_E$, this estimate holds for $B_r(x)$, for the smallest $r\geqq \varepsilon$ such that $\min _{\partial B_r(x)}\rho \geqq 1/2$. This is a result of Lemma A.2, if its $r\leqq 2\varepsilon$, and otherwise it follows from (123) and the coarea formula. One can then choose a finite subcover. The balls obtained in this fashion may overlap. If so, they may be combined into pairwise disjoint balls that still satisfy (129), by exactly the arguments in [24], proof of Proposition 3.3, where the same result is proved in the Euclidean setting. This argument involves a slightly more careful choice of balls (so that no center is contained in any other ball) and use of the Besicovitch covering lemma. For our present purposes, we may appeal to Federer [16, Sections 2.8.9–2.8.14], for a difficult but doubtless correct version of the covering lemma that is valid on a smooth compact Riemannian manifold, and indeed in much greater generality. Adjustments to the constant $c_2$ depend on constants appearing in this covering lemma, which are explicitly described in the above reference.

Step 2. Growing and merging balls. Now let $d_{l,0} := \hbox {dg}(u;\partial B_{l,0})$. We will assume for this discussion that $d_{l,0}\ne 0$ for some l, as the other case is both easier and less relevant for our main results. Using Lemma A.1 and associated properties of $\Lambda _\varepsilon$, such as those in (129), we may now follow the algorithm from [24], proof of Proposition 4.1, to which one may refer for the details omitted here. We describe it briefly. First, define

$$\begin{aligned} \sigma _0 := \min _{{\mathcal {B}}^0} r_{l,0}/|d_{l,0}| \overset{(51), (129)}{\leqq }C \ \varepsilon {|\!\log \varepsilon |}\ . \end{aligned}$$

We claim that for $\sigma \in (\sigma _0, r_0(S))$, there exists a finite collection of pairwise disjoint closed balls ${\mathcal {B}}^\sigma = \{ B_{l,\sigma }\}_{l=1}^{K_\sigma }$ with centers $a_{l,\sigma }$ and radii $r_{l,\sigma }$, such that

$$\begin{aligned} Z_E \subset \cup B_{l,\sigma },\int _{B_{l,\sigma }} e_\varepsilon ^{in}(u)\,\hbox {vol}_g \geqq \frac{r_{l,\sigma }}{\sigma }\Lambda _\varepsilon (\sigma ) \,,\ \hbox { and } \ r_{l,\sigma }\geqq \sigma |d_{l,\sigma }|\quad \hbox { for all }l, \nonumber \\ \end{aligned}$$

(129)

where $d_{l,\sigma } = \hbox {dg}(u;B_{l,\sigma })$. We take ${\mathcal {B}}^{\sigma _0}$ to be the collection found in Step 1 above. Given any $\sigma _1 \geqq \sigma _0$ for which such a collection exists, we say that the minimizing balls are those for which $r_{l,\sigma _1} = \sigma _1 |d_{l,\sigma _1}|$. Since the balls are closed and pairwise disjoint, there is some $\delta >0$ such that for $\sigma _1 \leqq \sigma \leqq \sigma +\delta$, we can expand the minimizing balls, while leaving the centers fixed, by enclosing them in pairwise disjoint annuli chosen so that, for every $\sigma$, the equality $r_{l,\sigma } = {\sigma } |d_{l,\sigma }|$ holds for all minimizing balls. We add balls to the collection of minimizing balls as $\sigma$ increases, when necessary. This preserves (130) due to properties of $\Lambda _\varepsilon$ summarized above, such as (128). At certain values of $\sigma$, for example $\sigma = \sigma _1+\delta$, the expansion process will lead to two or more balls colliding. When this occurs, one can regroup them into larger, pairwise disjoint balls in a way that preserves the properties (130). (Details of all these assertions can be found in [24].) This process can be continued as long as every minimizing ball has radius at most $r_0(S)$, which happens as long as $\sigma < r_0(S)$.

Step 3. Stopping the process, and covering all of Z. Recalling that $n>T-1$ by hypothesis, we fix $q \in (0, 1-\frac{T}{n+1})$, which implies that $\frac{T}{1-q}< n+1$. It then follows from (130), (127), and (51) that if $\varepsilon ^q\leqq \sigma < r_0(S)$, then

$$\begin{aligned} \sigma \sum |d_{l,\sigma }| \leqq \sum r_{l,\sigma } \leqq \sigma \frac{T\pi {|\!\log \varepsilon |}+ C}{\pi |\log (\sigma /\varepsilon )|-C} \leqq \sigma \Big (\frac{T}{1-q} + \frac{C}{{|\!\log \varepsilon |}}\Big ). \end{aligned}$$

(130)

Thus there exists $\varepsilon _0>0$ (depending on S, q, and the constant in (51)) such that if $0<\varepsilon <\varepsilon _0$, then $\sum |d_{l,\sigma }| < n+1$, and thus $\sum |d_{l,\sigma }|\leqq n$.

These balls have all the desired properties (the bound (57) on the sum of the radii follows from (131)) except that they cover $Z_E$ rather than all of Z. To rectify this, recall from Lemma A.3 that $Z{\setminus } Z_E$ can be covered by a finite collection of balls whose radii sum to at most $C \varepsilon {|\!\log \varepsilon |}$. We can add these balls to ${\mathcal {B}}^\sigma$, merging as necessary to obtain a pairwise disjoint collection (still denoted ${\mathcal {B}}^\sigma$) that covers all of Z, and with the sum of the radii increased by at most $C\varepsilon {|\!\log \varepsilon |}$. Since $\frac{T}{1-q}<n+1$, it follows from (131) that these still satisfy $\sum r_{l,\sigma } < \sigma (n+1)$ for $0<\varepsilon <\varepsilon _0$. The bound on the total degrees (56) and the energy lower bound (58) for this modified collection of balls are directly inherited from the previous collection. $\square$