Laplacian Comparison

Book 9A turned curvature into local statements: at a point, along a geodesic, for small balls. Book 9B turns curvature bounds into global control. If the Ricci curvature is at least some value everywhere, how fast can geodesics spread, how large can balls be, how small can a manifold get before it collapses, and what can limits of such manifolds look like? These are the facts that the Ricci flow uses in every compactness argument, and the Ricci flow books quote them in a few lines. This book proves the core ones and states the rest in the exact form used later.

The first tool is the distance function $r(x) = d(p, x)$ from a point. Its Hessian measures how the geodesic spheres around $p$ bend, and it obeys a first-order matrix ODE along each geodesic, a Riccati equation, driven by curvature. Comparing that ODE with the one for a model space gives Laplacian comparison: if $\operatorname{Ric} \geq 0$, then $\Delta r \leq \frac{n - 1}{r}$, as in Euclidean space. The same equation, read in spacetime, is Raychaudhuri's focusing equation, the engine of the singularity theorems of general relativity.

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

• describe where the distance function is smooth, and compute its gradient and Hessian;
• derive the Riccati equation for the Hessian of distance and its traced form;
• prove Laplacian comparison under a lower Ricci bound, and state Hessian comparison under sectional bounds;
• explain in what sense the comparison holds across the cut locus;
• relate the traced Riccati inequality to the focusing of light in general relativity.

1.1Focusing of light

1.2The distance function

Fix $p \in M$, with $M$ complete, and let $r(x) = d(p, x)$. Then $r$ is smooth on $M\setminus(\{p\}\cup\operatorname{Cut}(p))$, where $\operatorname{Cut}(p)$ is the cut locus of 9A.3 Geodesics and the Exponential Map, a closed set of measure zero. On that set:

• $|\nabla r| = 1$ (the Gauss lemma), and $\nabla r = \partial_r$ is the velocity of the unit-speed radial geodesics from $p$;
• the integral curves of $\nabla r$ are those geodesics, so $\nabla_{\partial_r}\partial_r = 0$;
• the level sets $\{r = s\}$ are the geodesic spheres, and $\partial_r$ is their unit normal.

The Hessian $\nabla^2r$, viewed as the endomorphism $\mathcal S(X) = \nabla_X\partial_r$, has $\partial_r$ in its kernel ($\nabla_{\partial_r}\partial_r = 0$) and is symmetric. On vectors tangent to a geodesic sphere it is the sphere's shape operator with respect to the inward normal $-\partial_r$, in the convention of 9A.8 Submanifolds and Minimal Surfaces up to that sign. Its trace is the mean curvature of the sphere, and also the Laplacian: $\Delta r = \operatorname{tr}\mathcal S$. In the model spaces of curvature $k$ (9A.1 Riemannian Metrics and Model Spaces),

so $\Delta r$ is $(n - 1)\cot r$ on the unit sphere, $\frac{n - 1}{r}$ in Euclidean space and $(n - 1)\coth r$ in hyperbolic space (Figure 1.1). The Euclidean value is the familiar radial term of the Laplacian, $f'' + \frac{n - 1}{r}f'$, of 9A.6 The Laplacian and the Bochner Formula.

1.3The Riccati equation

On the unit sphere, $\mathcal S = \cot r$ on the tangential directions, and indeed $-\csc^2r + \cot^2r + 1 = 0$ (Exercise 1.4). The equation is the infinitesimal form of the Jacobi equation of 9A.7 Jacobi Fields and Curvature versus Topology: if $J$ is a Jacobi field along a radial geodesic with $J(0) = 0$, then $\nabla_{\partial_r}J = \mathcal S(J)$, and differentiating once more gives back $J'' + R(J, \partial_r)\partial_r = 0$.

1.4Comparison

Sectional curvature bounds give the stronger, matrix version (Exercise 1.6).

Across the cut locus. At a cut point, $r$ is not smooth, but it can only have a concave kink: near a point $q$ that is reached by two minimising geodesics, $r$ is the minimum of two smooth functions, the distances measured along nearby families of geodesics. A minimum of smooth functions bends down, never up. Eugenio Calabi (1958) made this precise: $\Delta r \leq (n - 1)\frac{\operatorname{sn}_k'}{\operatorname{sn}_k}$ holds on all of $M\setminus\{p\}$ in the barrier sense. At every $q$ there is a smooth function $\phi \geq r$ near $q$, with $\phi(q) = r(q)$, whose Laplacian at $q$ is at most the bound plus any $\varepsilon > 0$. (Take $\phi(x) = \delta + d(\gamma(\delta), x)$ for a point $\gamma(\delta)$ a little way along a minimising geodesic from $p$ to $q$; $q$ is not a cut point of $\gamma(\delta)$.) Upper barriers are what the maximum principle needs: at a maximum of $u - \phi$, where $\phi$ touches from above, the usual argument goes through. The comparison is valid in the sense of distributions too, and the cut locus can only help.

1.5History

The Riccati equation, named after Jacopo Riccati's study of first-order quadratic ODEs in the 1720s, entered Riemannian geometry through the second variation and Jacobi fields. Comparison theorems for sectional curvature go back to Rauch (1951); the Laplacian comparison under Ricci bounds is implicit in Bishop's volume estimate (1963). Calabi introduced the barrier trick in 1958, in his work on a maximum principle on manifolds. Amal Kumar Raychaudhuri published his equation in 1955; Penrose's singularity theorem appeared in 1965.

1.6Exercises

Solution

$\operatorname{sn}_k'' = -k\operatorname{sn}_k$, so $\big(\frac{\operatorname{sn}_k'}{\operatorname{sn}_k}\big)' = \frac{\operatorname{sn}_k''\operatorname{sn}_k - \operatorname{sn}_k'^2}{\operatorname{sn}_k^2} = -k - \big(\frac{\operatorname{sn}_k'}{\operatorname{sn}_k}\big)^2$. Multiplying by $n - 1$ gives the second identity.

Solution

$\partial_ir = \frac{x_i}{r}$, $\partial_i\partial_jr = \frac{\delta_{ij}}{r} - \frac{x_ix_j}{r^3} = \frac1r(\delta_{ij} - \partial_ir\,\partial_jr)$, so $\Delta r = \frac{n - 1}{r}$. And $\partial_i\partial_j(r^2) = 2\delta_{ij}$.

Solution

$\lambda' = \langle(\nabla_{\partial_r}\mathcal S)E, E\rangle = -|\mathcal SE|^2 - \operatorname{Rm}(E, \partial_r, \partial_r, E) \leq -\lambda^2 - k$, since $|\mathcal SE| \geq |\langle\mathcal SE, E\rangle|$ and $K(E, \partial_r) \geq k$. With $\mu = \frac{\operatorname{sn}_k'}{\operatorname{sn}_k}$, which satisfies $\mu' = -\mu^2 - k$, the function $\operatorname{sn}_k^2(\lambda - \mu)$ has derivative $\leq -\operatorname{sn}_k^2(\lambda - \mu)^2 \leq 0$ and tends to $0$ at $r = 0$, so $\lambda \leq \mu$.

Solution

Geodesics from $p$ are straight lines in the universal cover, so the distance to $x$ is the shortest distance to a lift. On $x_1 = \frac12$, the two closest lifts of $p$ are $0$ and $(1, 0)$, at equal distance. Across the edge, $r$ switches from one to the other, with a corner pointing up: the gradient jumps from $\frac{x}{|x|}$ (with positive first component) to $\frac{x - (1, 0)}{|x - (1, 0)|}$ (negative first component). A jump of $\partial_1r$ from positive to negative gives a negative multiple of a delta measure on the edge in $\partial_1^2r$, so $\Delta r$ as a distribution is $\frac1r$ plus a negative measure, below the bound.

Solution

While the geodesic minimises, $r$ is smooth along it (just before a cut point) and $m \leq (n - 1)\sqrt k\cot(\sqrt kr)$, which tends to $-\infty$ as $r \to \frac{\pi}{\sqrt k}$. But $m$ is finite at every smooth point, so the geodesic must reach a cut point at or before $\frac{\pi}{\sqrt k}$. Every point of $M$ is therefore within distance $\frac{\pi}{\sqrt k}$ of $p$.

Solution

$\nabla\phi = \frac{\eta'}{R}\nabla r$, so $|\nabla\phi| \leq \frac{\sup|\eta'|}{R}$. And $\Delta\phi = \frac{\eta''}{R^2}|\nabla r|^2 + \frac{\eta'}{R}\Delta r$. Since $\eta' \leq 0$ and $\Delta r \leq \frac{n - 1}{r}$, the second term is $\geq \frac{\eta'}{R}\cdot\frac{n - 1}{r}$, and $\eta' \neq 0$ only where $R \leq r \leq 2R$, so it is $\geq -\frac{(n - 1)\sup|\eta'|}{R^2}$. Hence $\Delta\phi \geq -\frac{\sup|\eta''| + (n - 1)\sup|\eta'|}{R^2}$. (Across the cut locus the same holds in the barrier sense, because $\eta' \leq 0$ turns Calabi's upper barrier for $r$ into a lower barrier for $\phi$.)