<![CDATA[Second variation formula for minimal surfaces]]>

<![CDATA[Second variation formula for minimal surfaces]]>If $f$ is a smooth function on a manifold $M$ , and $p$ is a critical point of $f$ , recall that the Hessian $H_pf$ is the quadratic form $\nabla df$ on $T_pM$ (in local co-ordinates, the coefficients of the Hessian are the second partial derivatives of $f$ at $p$ ). Since $H_pf$ is symmetric, it has a well-defined index, which is the dimension of the subspace of maximal dimension on which $H_pf$ is negative definite. The Hessian does not depend on a choice of metric. One way to see this is to give an alternate definition $H_pf(X(p),Y(p)) = X(Yf)(p)$ where $X$ and $Y$ are any two vector fields with given values $X(p)$ and $Y(p)$ in $T_pM$ . To see that this does not depend on the choice of $X,Y$ , observe

$X(Yf)(p) - Y(Xf)(p) = [X,Y]f(p) = df([X,Y])_p = 0$

because of the hypothesis that $df$ vanishes at $p$ . This calculation shows that the formula is symmetric in $X$ and $Y$ . Furthermore, since $X(Yf)(p)$ only depends on the value of $X$ at $p$ , the symmetry shows that the result only depends on $X(p)$ and $Y(p)$ as claimed. A critical point is nondegenerate if $H_pf$ is nondegenerate as a quadratic form.

In Morse theory, one uses a nondegenerate smooth function $f$ (i.e. one with isolated nondegenerate critical points), also called a Morse function, to understand the topology of $M$ : the manifold $M$ has a (smooth) handle decomposition with one $i$ -handle for each critical point of $f$ of index $i$ . In particular, nontrivial homology of $M$ forces any such function $f$ to have critical points (and one can estimate their number of each index from the homology of $M$ ). Morse in fact applied his construction not to finite dimensional manifolds, but to the infinite dimensional manifold of smooth loops in some finite dimensional manifold, with arc length as a “Morse” function. Critical “points” of this function are closed geodesics. Any closed manifold has a nontrivial homotopy group in some dimension; this gives rise to nontrivial homology in the loop space. Consequently one obtains the theorem of Lyusternik and Fet:

Theorem: Let $M$ be a closed Riemannian manifold. Then $M$ admits at least one closed geodesic.

In higher dimensions, one can study the space of smooth maps from a fixed manifold $S$ to a Riemannian manifold $M$ equipped with various functionals (which might depend on extra data, such as a metric or conformal structure on $S$ ). One context with many known applications is when $M$ is a Riemannian $3$ -manifold, $S$ is a surface, and one studies the area function on the space of smooth maps from $S$ to $M$ (usually in a fixed homotopy class). Critical points of the area function are called minimal surfaces; the name is in some ways misleading: they are not necessarily even local minima of the area function. That depends on the index of the Hessian of the area function at such a point.

Let $\rho(t)$ be a (compactly supported) $1$ -parameter family of surfaces in a Riemannian $3$ -manifold $M$ , for which $\rho(0)$ is smoothly immersed. For small $t$ the surfaces $\rho(t)$ are transverse to the exponentiated normal bundle of $\rho(0)$ ; hence locally we can assume that $\rho$ takes the form $\rho(t,u,v)$ where $u,v$ are local co-ordinates on $\rho(0)$ , and $\rho(\cdot,u,v)$ is contained in the normal geodesic to $\rho(0)$ through the point $\rho(0,u,v)$ ; we call such a family of surfaces a normal variation of surfaces. For such a variation, one has the following:

Theorem (first variation formula): Let $\rho(t)$ be a normal variation of surfaces, so that $\rho'(0) = f\nu$ where $\nu$ is the unit normal vector field to $\rho(0)$ . Then there is a formula:

$\frac d {dt} \text{area}(\rho(t))|_{t=0} = \int_{\rho(0)} -\langle f\nu,\mu\rangle d\text{area}$

where $\mu$ is the mean curvature vector field along $\rho(0)$ .

Proof: let $T,U,V$ denote the image under $d\rho$ of the vector fields $\partial_t,\partial_u,\partial_v$ . Choose co-ordinates so that $u,v$ are conformal parameters on $\rho(0)$ ; this means that $\langle U,V\rangle = 0$ and $\|U\|=\|V\|$ at $t=0$ .

The infinitesimal area form on $\rho(t)$ is $\sqrt{\|U\|^2\|V\|^2 - \langle U,V \rangle^2} dUdV$ which we abbreviate by $E^{1/2}$ , and write

$\frac d {dt} \text{area}(\rho(t)) = \int_{\rho(t)} \frac {dUdV} {2E^{1/2}} (\|U\|^2\langle V,V\rangle' + \|V\|\langle U,U\rangle' - 2\langle U,V\rangle\langle U,V\rangle')$

Since $V,T$ are the pushforward of coordinate vector fields, they commute; hence $[V,T]=0$ , so $\nabla_T V = \nabla_V T$ and therefore

$\langle V,V\rangle' = 2\langle \nabla_T V,V\rangle = 2\langle \nabla_V T,V\rangle = 2(V\langle T,V\rangle - \langle T,\nabla_V V\rangle)$

and similarly for $\langle U,U\rangle'$ . At $t = 0$ we have $\langle T,V\rangle = 0$ , $\langle U,V\rangle = 0$ and $\|U\|^2 = \|V\|^2 = E^{1/2}$ so the calculation reduces to

$\frac d {dt} \text{area}(\rho(t))|_{t=0} = \int_{\rho(0)} -\langle T,\nabla_U U + \nabla_V V\rangle dUdV$

Now, $T|_{t=0} = f\nu$ , and $\nabla_U U + \nabla_V V = \mu E^{1/2}$ so the conclusion follows. qed.

As a corollary, one deduces that a surface is a critical point for area under all smooth compactly supported variations if and only if the mean curvature $\mu$ vanishes identically; such a surface is called minimal.

The second variation formula follows by a similar (though more involved) calculation. The statement is:

Theorem (second variation formula): Let $\rho(t)$ be a normal variation of surfaces, so that $\rho'(0)=f\nu$ . Suppose $\rho(0)$ is minimal. Then there is a formula:

$\frac {d^2} {dt^2} \text{area}(\rho(t))|_{t=0} = \int_{\rho(0)} -\langle f\nu,L(f)\nu\rangle d\text{area}$

where $L$ is the Jacobi operator (also called the stability operator), given by the formula

$L = \text{Ric}(\nu) + |A|^2 + \Delta_\rho$

where $A$ is the second fundamental form, and $\Delta_\rho = -\nabla^*\nabla$ is the metric Laplacian on $\rho(0)$ .

This formula is frankly a bit fiddly to derive (one derivation, with only a few typos, can be found in my Foliations book; a better derivation can be found in the book of Colding-Minicozzi) but it is easy to deduce some significant consequences directly from this formula. The metric Laplacian on a compact surface is negative self-adjoint (being of the form $-X^*X$ for some operator $X$ ), and $L$ is obtained from it by adding a $0$ th order perturbation, the scalar field $|A|^2 + \text{Ric}(\nu)$ . Consequently the biggest eigenspace for $L$ is $1$ -dimensional, and the eigenvector of largest eigenvalue cannot change sign. Moreover, the spectrum of $L$ is discrete (counted with multiplicity), and therefore the index of $-L$ (thought of as the “Hessian” of the area functional at the critical point $\rho(0)$ ) is finite.

A surface is said to be stable if the index vanishes. Integrating by parts, one obtains the so-called stability inequality for a stable minimal surface $S$ :

$\int_S (\text{Ric}(\nu) + |A|^2)f^2d\text{area} \le \int_S |\nabla f|^2 d\text{area}$

for any reasonable compactly supported function $f$ . If $S$ is closed, we can take $f=1$ . Consequently if the Ricci curvature of $M$ is positive, $M$ admits no stable minimal surfaces at all. In fact, in the case of a surface in a $3$ -manifold, the expression $\text{Ric}(\nu) + |A|^2$ is equal to $R - K + |A|^2/2$ where $K$ is the intrinsic curvature of $S$ , and $R$ is the scalar curvature on $M$ . If $S$ has positive genus, the integral of $-K$ is non-negative, by Gauss-Bonnet. Consequently, one obtains the following theorem of Schoen-Yau:

Corollary (Schoen-Yau): Let $M$ be a Riemannian $3$ -manifold with positive scalar curvature. Then $M$ admits no immersed stable minimal surfaces at all.

On the other hand, one knows that every $\pi_1$ -injective map $S \to M$ to a $3$ -manifold is homotopic to a stable minimal surface. Consequently one deduces that when $M$ is a $3$ -manifold with positive scalar curvature, then $\pi_1(M)$ does not contain a surface subgroup. In fact, the hypothesis that $S \to M$ be $\pi_1$ -injective is excessive: if $S \to M$ is merely incompressible, meaning that no essential simple loop in $S$ has a null-homotopic image in $M$ , then the map is homotopic to a stable minimal surface. The simple loop conjecture says that a map $S \to M$ from a $2$ -sided surface to a $3$ -manifold is incompressible in this sense if and only if it is $\pi_1$ -injective; but this conjecture is not yet known.

Update 8/26: It is probably worth making a few more remarks about the stability operator.

The first remark is that the three terms $\text{Ric}(\nu)$ , $|A|^2$ and $\Delta$ in $L$ have natural geometric interpretations, which give a “heuristic” justification for the second variation formula, which if nothing else, gives a handy way to remember the terms. We describe the meaning of these terms, one by one.

Suppose $f \equiv 1$ , i.e. consider a variation by flowing points at unit speed in the direction of the normals. In directions in which the surface curves “up”, the normal flow is focussing; in directions in which it curves “down”, the normal flow is expanding. The net first order effect is given by $\langle \nu,\mu\rangle$ , the mean curvature in the direction of the flow. For a minimal surface, $\mu = 0$ , and only the second order effect remains, which is $|A|^2$ (remember that $A$ is the second fundamental form, which measures the infinitesimal deviation of $S$ from flatness in $M$ ; the mean curvature is the trace of $A$ , which is first order. The norm $|A|^2$ is second order).
There is also an effect coming from the ambient geometry of $M$ . The second order rate at which a parallel family of normals $\nu$ along a geodesic $\gamma$ diverge is $\langle R(\gamma',\nu)\gamma',\nu\rangle$ where $R$ is the curvature operator. Taking the average over all geodesics $\gamma$ tangent to $S$ at a point gives the Ricci curvature in the direction of $\nu$ , i.e. $\text{Ric}(\nu)$ . This is the infinitesimal expansion of area of a geodesic plane under the normal flow, and has second order. The interactions between these terms have higher order, so the net contribution when $f \equiv 1$ is $\text{Ric}(\nu) + |A|^2$ .
Finally, there is the contribution coming from $f$ itself. Imagine that $S$ is a flat plane in Euclidean space, and let $S_\epsilon$ be the graph of $\epsilon f$ . The infinitesimal area element on $S_\epsilon$ is $\sqrt{1+\epsilon^2 |\nabla f|^2} \sim 1+\epsilon^2/2 |\nabla f|^2$ . If $f$ has compact support, then differentiating twice by $\epsilon$ , and integrating by parts, one sees that the (leading) second order term is $\Delta f$ . When $S$ is not totally geodesic, and the ambient manifold is not Euclidean space, there is an interaction which has higher order; the leading terms add, and one is left with $L = \text{Ric}(\nu) + |A|^2 + \Delta$ .

The second remark to make is that if the support of a variation $f$ is sufficiently small, then necessarily $|\nabla f|$ will be large compared to $f$ , and therefore $-L$ will be positive definite. In other words all variations of a (fixed) minimal surface with sufficiently small support are area increasing — i.e. a minimal surface is locally area minimizing (this is local in the surface itself, not in the “space of all surfaces”). This is a generalization of the important fact that a geodesic in a Riemannian manifold is locally length minimizing (though typically not globally length minimizing).

One final remark is that when $|A|^2$ is big enough at some point $p \in S$ , and when the injectivity radius of $S$ at $p$ is big enough (depending on bounds on $\text{Ric}(\nu)$ in some neighborhood of $p$ ), one can find a variation with support concentrated near $p$ that violates the stability inequality. Contrapositively, as observed by Schoen, knowing that a minimal surface in a $3$ -manifold $M$ is stable gives one a priori control on the size of $|A|^2$ , depending only on the Ricci curvature of $M$ , and the injectivity radius of the surface at the point. Since stability is preserved under passing to covers (for $2$ -sided surfaces, by the fact that the largest eigenvalue of $L$ can’t change sign!) one only needs a lower bound on the distance from $p$ to $\partial S$ . In particular, if $S$ is a closed stable minimal surface, there is an a priori pointwise bound on $|A|^2$ . This fact has many important topological applications in $3$ -manifold topology. On the other hand, when $S$ has boundary, the curvature can be arbitrarily large. The following example is due to Thurston (also see here for a discussion):

Example (Thurston): Let $\Delta$ be an ideal simplex in $\mathbb{H}^3$ with ideal simplex parameter imaginary and very large. The four vertices of $\Delta$ come in two pairs which are very close together (as seen from the center of gravity of the simplex); let $P$ be an ideal quadrilateral whose edges join a point in one pair to a point in the other. The simplex $\Delta$ is bisected by a “square” of arbitrarily small area; together with four “cusps” (again, of arbitrarily small area) one makes a (topological) disk spanning $P$ with area as small as desired. Isotoping this disk rel. boundary to a least area (and therefore stable) representative can only decrease the area further. By the Gauss-Bonnet formula, the curvature of such a disk must get arbitrarily large (and negative) at some point in the interior.

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