Submanifolds and Minimal Surfaces

Book 8A ended with surfaces in $\mathbb{R}^3$ and the two ways they curve: intrinsically, through the metric, and extrinsically, through the way they sit in space. This chapter extends that picture to a submanifold of any Riemannian manifold, through the second fundamental form and the Gauss equation that links the two kinds of curvature. It then asks the variational question that soap films answer: which surfaces have the least area? The first variation of area says that minimal surfaces have zero mean curvature. The second variation brings in the ambient Ricci curvature, and that term ties minimal surfaces to the Ricci flow. It is how Schoen and Yau showed that a torus cannot carry positive scalar curvature, and minimal surfaces are one of the tools in the proof that the Ricci flow on certain 3-manifolds becomes extinct in finite time.

By the end of this chapter you will be able to:

• define the second fundamental form, shape operator and mean curvature of a submanifold, and recognise totally geodesic submanifolds;
• state and use the Gauss equation, including its traced form for hypersurfaces;
• derive the first variation of area and the minimal surface equation;
• state the second variation of area and decide the stability of simple examples;
• explain the Schoen–Yau argument that a stable minimal surface in a 3-manifold of positive scalar curvature is a sphere.

8.1Soap films

8.2The second fundamental form

Let $M \subset \bar M$ be a submanifold of a Riemannian manifold $(\bar M, \bar g)$, with the induced metric $g$ and Levi-Civita connections $\nabla$ and $\bar\nabla$. For vector fields $X, Y$ tangent to $M$, split the ambient derivative into tangential and normal parts:

(the Gauss formula; the tangential part is the induced connection, as in 9A.2 Connections). The normal part $\operatorname{II}(X, Y)$ is the second fundamental form, a symmetric tensor with values in the normal bundle. It measures how $M$ bends inside $\bar M$.

For a hypersurface with a unit normal field $\nu$, write $\operatorname{II}(X, Y) = h(X, Y)\nu$ with $h(X, Y) = \langle\bar\nabla_XY, \nu\rangle = \langle\mathcal SX, Y\rangle$, where $\mathcal SX = -\bar\nabla_X\nu$ is the shape operator: the conventions of 8A.9 The Curvature of Surfaces. The eigenvalues of $\mathcal S$ are the principal curvatures and the mean curvature is $H = \operatorname{tr}\mathcal S = g^{ij}h_{ij}$, their sum. A round sphere of radius $r$ in $\mathbb{R}^{n+1}$, with the inward normal, has $h = \frac1rg$ and $H = \frac nr$. $M$ is totally geodesic if $\operatorname{II} = 0$: then geodesics of $M$ are geodesics of $\bar M$. Great spheres in $S^n$, linear subspaces in $\mathbb{R}^n$, and hyperbolic subspaces of $\mathbb{H}^n$ are totally geodesic.

The proof (Lee, chapter 8) substitutes the Gauss formula into the definition of $\bar R$ and takes tangential parts. For a surface in $\mathbb{R}^3$, $\bar K = 0$ and the equation says $K = h_{11}h_{22} - h_{12}^2 = \det\mathcal S$: the Theorema Egregium. For the spheres $\{r = r_0\}$ in the warped product $dr^2 + \varphi^2g_{S^{n-1}}$, $h = \frac{\varphi'}{\varphi}g$ for the inward normal, the intrinsic curvature is $\frac{1}{\varphi^2}$, and the Gauss equation gives the tangential curvature $\frac{1 - \varphi'^2}{\varphi^2}$ used in 9A.5 Computing Curvature.

Tracing twice, for a hypersurface,

where $R$ is the scalar curvature of $M$ (Exercise 8.6). This identity is the heart of the Schoen–Yau argument below.

8.3The first variation of area

Let $M^k \subset \bar M$ be a compact submanifold, possibly with boundary, and deform it by $F_s$ with $F_0 = \operatorname{id}$ and variation field $V = \partial_sF_s|_{s = 0}$, vanishing on $\partial M$.

So $M$ is a critical point of area for all compactly supported variations if and only if $\vec H = 0$: such $M$ are called minimal (though they need not minimise). The mean curvature vector is minus the gradient of area, and the mean curvature flow $\partial_tx = \vec H$ of 6A.8 Curve Shortening and the First Geometric Flows is the steepest descent of area, the extrinsic cousin of the Ricci flow. For a graph $z = u(x, y)$, $H = 0$ is the minimal surface equation $\operatorname{div}\frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} = 0$ of 6A.9 Calculus of Variations and Gradient Flows.

Examples in $\mathbb{R}^3$. The plane; the catenoid, the surface of revolution of $r = c\cosh\frac zc$ (Euler, 1744); the helicoid $(s\cos t, s\sin t, ct)$ (Meusnier, 1776), traced by a line rotating as it rises (Figure 8.2, Exercise 8.7). The catenoid is the only minimal surface of revolution besides the plane, and the helicoid, by Catalan's theorem of 1842, the only ruled one besides the plane.

8.4The second variation and stability

A minimal hypersurface is stable if its area does not decrease to second order under any compactly supported variation.

This is the analogue for surfaces of the second variation of length in 9A.7 Jacobi Fields and Curvature versus Topology, and it is proved the same way, by differentiating the first variation once more (Colding and Minicozzi give the details). There the sectional curvature of planes containing the geodesic entered; here it is the Ricci curvature of the normal direction, plus the extra term $|h|^2$ from the shape of $M$. Both enter with a minus sign: positive ambient Ricci curvature destabilises minimal surfaces.

• A totally geodesic $S^{n-1}$ in the unit $S^n$ (a great sphere) has $h = 0$ and $\overline{\operatorname{Ric}}(\nu, \nu) = n - 1$. With $f = 1$ the second variation is $-(n - 1)\operatorname{Area} < 0$: pushing the equator towards a pole shrinks it. Great spheres are minimal but unstable.
• The catenoid spanning two coaxial rings of radius $R$ exists only while the separation is at most about $1.33R$; for smaller separations there are two catenoids, and only the fatter one is stable (Figure 8.3, Exercise 8.9). Pull the rings further apart and the film collapses, the experiment of 6A.9 Calculus of Variations and Gradient Flows.

Schoen and Yau (1979) combined this with the existence of area-minimising surfaces in homology classes to show that the 3-torus carries no metric of positive scalar curvature: an area-minimising torus in $T^3$ would be stable, contradicting the theorem. That is the start of the theory of 9B.6 Scalar Curvature and Topology. In the Ricci flow, area-minimising and min-max minimal surfaces measure how a 3-manifold shrinks: Perelman, and Colding and Minicozzi, showed that a quantity defined by sweeping out the manifold with minimal discs or spheres must reach zero in finite time when the manifold's topology allows it (10A.8 Min–Max and Width, 12C.2 Finite Extinction).

8.5History

Lagrange (1760) derived the minimal surface equation, Euler had found the catenoid (1744) and Meusnier the helicoid (1776), and Meusnier identified the condition as zero mean curvature. Gauss proved his equation for surfaces in 1827. Plateau's Statique expérimentale et théorique des liquides appeared in 1873. Jesse Douglas and Tibor Radó solved the Plateau problem of spanning a curve by a minimal disc in 1930–31, and Douglas received one of the first two Fields Medals, in 1936, for it. Jean Taylor's proof of Plateau's laws appeared in 1976, and Schoen and Yau's work on positive scalar curvature in 1979.

8.6Exercises

Solution

$\bar\nabla_X\nu = -\frac1rX$ for tangent $X$, so $\mathcal SX = \frac1rX$ and $h = \frac1rg$. With $\bar K = 0$: $K(X, Y) = h(X, X)h(Y, Y) - h(X, Y)^2 = \frac{1}{r^2}$ for orthonormal $X, Y$.

Solution

$R = \sum_{i \neq j}K(e_i, e_j) = \sum_{i \neq j}\bar K(e_i, e_j) + \sum_{i \neq j}(h_{ii}h_{jj} - h_{ij}^2)$. The second sum is $(\sum h_{ii})^2 - \sum_ih_{ii}^2 - \sum_{i \neq j}h_{ij}^2 = H^2 - |h|^2$. The first is the sum over all pairs of distinct vectors of the frame $e_1, \dots, e_{n-1}, \nu$ of $\bar M$, minus the pairs involving $\nu$: $\bar R - 2\sum_i\bar K(e_i, \nu) = \bar R - 2\overline{\operatorname{Ric}}(\nu, \nu)$.

Solution

$x_s = (\cos t, \sin t, 0)$, $x_t = (-s\sin t, s\cos t, c)$: $E = 1$, $F = 0$, $G = s^2 + c^2$. The unit normal is $\nu = \frac{(c\sin t, -c\cos t, s)}{\sqrt{s^2 + c^2}}$. $x_{ss} = 0$, so $L = 0$; $x_{tt} = (-s\cos t, -s\sin t, 0)$, so $N = x_{tt}\cdot\nu = 0$; $x_{st} = (-\sin t, \cos t, 0)$, so $M = \frac{-c}{\sqrt{s^2 + c^2}}$. Then $H = \frac{EN - 2FM + GL}{EG - F^2} = 0$, and $K = \frac{LN - M^2}{EG - F^2} = -\frac{c^2}{(s^2 + c^2)^2}$. For each fixed $t$, the curve $s \mapsto x(s, t)$ is a straight line.

Solution

The equator is the fixed set of the reflection $x_{n+1} \mapsto -x_{n+1}$, an isometry, so it is totally geodesic. Then $\int(0 - (n - 1)\cdot1)\,dA = -(n - 1)\operatorname{Area}$. Directly: $\frac{d^2}{ds^2}\cos^{n-1}s\,\big|_{s = 0} = -(n - 1)$.

Solution

$c = \frac{h}{2u}$ and $R = \frac{h}{2u}\cosh u$, so $\frac hR = \frac{2u}{\cosh u}$. Its derivative vanishes where $\cosh u - u\sinh u = 0$, that is $u\tanh u = 1$, at $u \approx 1.1997$, where $\frac{2u}{\cosh u} \approx \frac{2.3994}{1.8102} \approx 1.3255$. The function rises from $0$ to this maximum and decreases to $0$, so each smaller value is taken twice: two values of $u$, two catenoids.

Solution

(a) $H = \frac nr$, so $r' = -\frac nr$ and $(r^2)' = -2n$. (b) The Ricci flow has $(r^2)' = -2(n - 1)$; both shrink spheres to points in finite time, at rates set by their curvatures. (c) $\frac{d}{dt}\operatorname{Area} = -\int fH = -\int H^2$. For the sphere: $\frac{d}{dt}(\omega r^n) = n\omega r^{n-1}r' = -n^2\omega r^{n-2}$, where $\omega = \operatorname{Area}(S^n(1))$, and $-\int H^2 = -\frac{n^2}{r^2}\omega r^n$, the same.