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I am currently teaching a class at the University of Chicago on hyperbolic groups, and I have just introduced the concept of $\delta$-hyperbolic (geodesic) metric spaces. A geodesic metrix space $(X,d_X)$ is $\delta$-hyperbolic if for any geodesic triangle $abc$, and any $p \in ab$ there is some $q \in ac \cup bc$ with $d_X(p,q)\le \delta$. The quintessential $\delta$-hyperbolic space is the hyperbolic plane, the unique (up to isometry) simply-connected complete Riemannian 2-manifold of constant curvature $-1$. It follows that any simply-connected complete Riemannian manifold of constant curvature $K<0$ is $\delta$-hyperbolic for some $\delta$ depending on $K$; roughly one can take $\delta \sim (-K)^{-1/2}$.

What gives this condition some power is the rich class of examples of spaces which are $\delta$-hyperbolic for some $\delta$. One very important class of examples are simply-connected complete Riemannian manifolds with upper curvature bounds. Such spaces enjoy a very strong comparison property with simply-connected spaces of constant curvature, and are therefore the prime examples of what are known as CAT(K) spaces.

Definition: A geodesic metric space $(X,d_X)$ is said to be $CAT(K)$, if the following holds. If $abc$ is a geodesic triangle in $X$, let $\bar{a}\bar{b}\bar{c}$ be a comparison triangle in a simply connected complete Riemannian manifold $Y$ of constant curvature $K$. Being a comparison triangle means just that the length of $\bar{a}\bar{b}$ is equal to the length of $ab$ and so on. For any $p \in bc$ there is a corresponding point $\bar{p}$ in the comparison edge $\bar{b}\bar{c}$ which is the same distance from $\bar{b}$ and $\bar{c}$ as $p$ is from $b$ and $c$ respectively. The $CAT(K)$ condition says, for all $abc$ as above, and all $p \in bc$, there is an inequality $d_X(a,p) \le d_Y(\bar{a},\bar{p})$.

The term CAT here (coined by Gromov) is an acronym for Cartan-Alexandrov-Toponogov, who all proved significant theorems in Riemannian comparison geometry. From the definition it follows immediately that any $CAT(K)$ space with $K<0$ is $\delta$-hyperbolic for some $\delta$ depending only on $K$. The point of this post is to give a short proof of the following fundamental fact:

CAT(K) Theorem: Let $M$ be a complete simply-connected Riemannian manifold with sectional curvature $\le K_0$ everywhere. Then $M$ with its induced Riemannian (path) metric is $CAT(K_0)$.

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.

1. 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).
2. 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 $\text{Ric}(\nu) + |A|^2$.
3. 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.