The Laplacian and the Bochner Formula

The Ricci flow is a heat equation, and a heat equation needs a Laplacian. This chapter builds calculus on a Riemannian manifold: gradient, divergence, Hessian and the Laplace–Beltrami operator, integration by parts on closed manifolds, and the eigenvalues of the Laplacian on spheres and tori. It then does the one computation every evolution equation of the Ricci flow depends on: commuting covariant derivatives, where curvature appears as the price of changing the order of differentiation. The first and most famous result of that kind is the Bochner formula, which turns a lower bound on Ricci curvature into control of harmonic and eigen-functions. Its first application, Lichnerowicz's eigenvalue estimate, says that positive Ricci curvature makes a manifold "ring" at a high pitch.

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

• compute gradients, divergences, Hessians and Laplacians in coordinates and in warped products;
• integrate by parts on a closed Riemannian manifold, and state Green's identities;
• describe the eigenvalues and eigenfunctions of the Laplacian on spheres and flat tori;
• commute covariant derivatives of vector fields, 1-forms and 2-tensors;
• prove the Bochner formula and Lichnerowicz's estimate, and define the Lichnerowicz Laplacian.

6.1The Earth's gravity in spherical harmonics

6.2Gradient, divergence, Hessian and Laplacian

On $(M, g)$:

• the gradient of a function is the vector field dual to $df$: $\langle\nabla f, X\rangle = Xf$, or $\nabla^if = g^{ij}\partial_jf$;
• the divergence of a vector field is the trace of its covariant derivative: $\operatorname{div}X = \nabla_iX^i$;
• the Hessian is $\nabla^2f = \nabla df$, with components $\nabla_i\nabla_jf = \partial_i\partial_jf - \Gamma_{ij}^k\partial_kf$, symmetric (9A.2 Connections);
• the Laplacian (Laplace–Beltrami operator) is $\Delta f = \operatorname{div}\nabla f = g^{ij}\nabla_i\nabla_jf$, the trace of the Hessian.

In coordinates, with $|g| = \det(g_{ij})$,

(Exercise 6.3). The sign convention is the analyst's: on $\mathbb{R}^n$, $\Delta = \sum\partial_i^2$, and $-\Delta$ has nonnegative eigenvalues (8A.7 Tensors and Index Notation). For a warped product $dr^2 + \varphi(r)^2g_{S^{n-1}}$,

where $(n - 1)\frac{\varphi'}{\varphi}$ is the mean curvature of the sphere $\{r = \text{const}\}$. On $\mathbb{R}^n$ it is $\frac{n - 1}{r}$, giving the familiar radial Laplacian $f'' + \frac{n - 1}{r}f'$ of 6A.2 Harmonic Functions.

Integration by parts. On a closed (compact, boundaryless) manifold the divergence theorem (8A.8 Differential Forms and Stokes’ Theorem) gives $\int_M\operatorname{div}X\,dV = 0$. Applied to $X = f\nabla h$:

So $\Delta$ is symmetric and $-\Delta$ is nonnegative: if $\Delta f = -\lambda f$, then $\lambda\int f^2 = \int|\nabla f|^2 \geq 0$, with $\lambda = 0$ only for constants (on a connected $M$). The maximum principle of 6A.4 Maximum Principles carries over word for word: at an interior maximum of $f$, $\nabla f = 0$ and $\nabla^2f \leq 0$, so $\Delta f \leq 0$. These are the two tools, integration and the maximum principle, of every estimate in the Ricci flow.

6.3Eigenvalues: spheres and tori

The Laplacian of a closed manifold has eigenvalues $0 = \lambda_0 < \lambda_1 \leq \lambda_2 \leq \dots \to \infty$ (of $-\Delta$), with smooth eigenfunctions forming an orthonormal basis of $L^2$ (the spectral theorem for compact self-adjoint operators, 4A.7 Compact Operators and Spectra). The first nonzero eigenvalue $\lambda_1$ is the decay rate of the heat equation towards its average, and the square of the lowest pitch of a drum shaped like the manifold. Two cases can be computed completely.

Flat tori. On $\mathbb{R}^n/\Lambda$, the eigenfunctions are $e^{2\pi i\langle\xi, x\rangle}$ for $\xi$ in the dual lattice $\Lambda^*$, with eigenvalues $4\pi^2|\xi|^2$: Fourier series (2B.7 Fourier Series and the First Heat Equation) again.

Spheres. On the unit $S^n$, the eigenvalues of $-\Delta$ are

and the eigenfunctions with eigenvalue $\lambda_k$ are the restrictions to $S^n$ of the homogeneous harmonic polynomials of degree $k$ on $\mathbb{R}^{n + 1}$, the spherical harmonics of degree $k$ (Exercise 6.5). On $S^2$, $\lambda_k = k(k + 1)$ with multiplicity $2k + 1$: the functions $Y_k^m$, $|m| \leq k$, of quantum mechanics, chemistry and geodesy (Figure 6.1). The first nonzero eigenvalue of the unit $S^n$ is $\lambda_1 = n$, with eigenfunctions the coordinate functions $x_1, \dots, x_{n + 1}$.

6.4Commuting derivatives

Second covariant derivatives of functions commute ($\nabla_i\nabla_jf = \nabla_j\nabla_if$, because the connection is torsion-free). Second covariant derivatives of tensors do not, and the commutator is curvature. For a vector field, a 1-form and a 2-tensor, the Ricci identities are

one curvature term for each index, with $+$ for upper and $-$ for lower indices, like the Christoffel terms of 9A.2 Connections (Lee, chapter 7). The first is the definition of $R$ applied to $X$; the second follows by applying the first to $\omega(X)$, a function (Exercise 6.6); the third follows from the second by the product rule.

The most used consequence, the one in the rehearsal exercise of this chapter and in every soliton identity of 11B.1 Ricci Solitons, is the commutation of the Laplacian with the gradient:

Taking the Laplacian of a gradient produces a Ricci term. That term is the bridge between Ricci curvature and analysis.

6.5The Bochner formula

The formula says that for a harmonic function ($\Delta f = 0$), $\frac12\Delta|\nabla f|^2 = |\nabla^2f|^2 + \operatorname{Ric}(\nabla f, \nabla f)$: if $\operatorname{Ric} \geq 0$, then $|\nabla f|^2$ is subharmonic. That is the start of the Bochner technique: on a closed manifold with $\operatorname{Ric} \geq 0$, integrating shows every harmonic 1-form is parallel, so the first Betti number is at most $n$ (Bochner, 1946). In the Ricci flow the same pattern, "a square plus a curvature term", appears in Hamilton's Harnack inequality (11B.2 Ancient Solutions and the Harnack Inequality), the Li–Yau estimate (9B.7 The Heat Equation on a Manifold) and Perelman's entropy (12A.3 The 𝓦-Entropy).

The unit sphere has $\operatorname{Ric} = (n - 1)g$ and $\lambda_1 = n$, so the estimate is sharp, and Morio Obata (1962) proved the sphere is the only case of equality. Ricci curvature bounded below by a positive constant forces a spectral gap, and hence exponentially fast decay of heat: positive curvature makes diffusion efficient.

6.6Laplacians on tensors

The rough Laplacian of a tensor field is $\Delta T = g^{ij}\nabla_i\nabla_jT$, the trace of its second covariant derivative. For a symmetric 2-tensor $h$, the Lichnerowicz Laplacian adds curvature terms:

(in this guide's conventions; it is checked on the sphere in Exercise 6.8). It arises from linearising the Ricci tensor, and under the Ricci flow the Ricci tensor itself evolves by

derived in 11A.2 How Curvature Evolves. You verify it on the shrinking sphere in Exercise 6.8.

6.7History

Laplace and Legendre introduced spherical harmonics in the 1780s, studying the gravitational attraction of planets. Eugenio Beltrami defined the Laplacian of a Riemannian metric in the 1860s. Salomon Bochner published his formula and its consequences for Betti numbers in 1946. André Lichnerowicz proved the eigenvalue estimate in 1958 and introduced his Laplacian in his work on Einstein metrics; Obata's rigidity theorem followed in 1962.

6.8Exercises

Solution

$\Gamma_{ik}^i = \frac12g^{il}(\partial_ig_{kl} + \partial_kg_{il} - \partial_lg_{ik}) = \frac12g^{il}\partial_kg_{il}$, the other two terms cancelling by symmetry of $g^{il}$; this is $\frac12\partial_k\log|g|$. So $\nabla_iX^i = \partial_iX^i + \Gamma_{ik}^iX^k = \partial_iX^i + X^k\partial_k\log\sqrt{|g|} = \frac{1}{\sqrt{|g|}}\partial_i(\sqrt{|g|}X^i)$. With $X = \nabla f$ this gives $\Delta$. For $g = dr^2 + \varphi^2g_S$, $\sqrt{|g|} = \varphi^{n - 1}\sqrt{|g_S|}$, and for $f = f(r)$, $\Delta f = \varphi^{1 - n}(\varphi^{n - 1}f')' = f'' + (n - 1)\frac{\varphi'}{\varphi}f'$.

Solution

$f' = -\sin r$, $f'' = -\cos r$, and $\Delta f = -\cos r + (n - 1)\cot r(-\sin r) = -n\cos r$. With $\nabla^2f = -fg$, $|\nabla^2f|^2 = nf^2$ and $\frac1n(\Delta f)^2 = \frac1n n^2f^2 = nf^2$.

Solution

$0 = \Delta P = (k(k - 1) + nk)r^{k - 2}Y + r^{k - 2}\Delta_SY$, so $\Delta_SY = -k(k + n - 1)Y$. $\Delta(x_1x_2) = 0$ in $\mathbb{R}^3$; with $k = 2$, $n = 2$: $2\cdot3 = 6$.

Solution

By the product rule, $\nabla_i\nabla_j(\omega_kX^k) = (\nabla_i\nabla_j\omega_k)X^k + \nabla_j\omega_k\nabla_iX^k + \nabla_i\omega_k\nabla_jX^k + \omega_k\nabla_i\nabla_jX^k$. Antisymmetrise in $i, j$: the left side and the two middle terms are symmetric, so $0 = (\nabla_i\nabla_j\omega_k - \nabla_j\nabla_i\omega_k)X^k + \omega_kR_{ijl}{}^kX^l$. Renaming, $(\nabla_i\nabla_j\omega_k - \nabla_j\nabla_i\omega_k)X^k = -R_{ijk}{}^l\omega_lX^k$ for all $X$.

Solution

$\Lambda^* = \frac1a\mathbb{Z}\times\frac1b\mathbb{Z}$, so $\lambda_1 = \frac{4\pi^2}{a^2}$, from $\xi = (\frac1a, 0)$. The estimate needs $\operatorname{Ric} \geq (n - 1)k$ with $k > 0$, and here $\operatorname{Ric} = 0$. With $ab = 1$ and $a \to \infty$, $\lambda_1 = 4\pi^2/a^2 \to 0$: a long thin torus is a slow drum.

Solution

$\operatorname{Ric}$ is parallel, so $\Delta R_{jk} = 0$. With $R^{pq} = (n - 1)c^{-1}g^{pq}$ (raising with $g$): $2R_{pjkq}R^{pq} = 2c^{-2}(n - 1)(ng_{jk} - g_{jk}) = 2(n - 1)^2c^{-2}g_{jk}$, and $2R_{jp}R^p{}_k = 2(n - 1)^2c^{-2}g_{jk}$; they cancel. With the opposite sign the right-hand side would be $-4(n - 1)^2c^{-2}g_{jk} \neq 0$, contradicting the exact solution. The sphere test fixes the sign, as in 9A.4 Curvature and What It Means.

Solution

(a) The Ricci identity with indices $(j, i, k)$: $\nabla_j\nabla_i\omega_k - \nabla_i\nabla_j\omega_k = -R_{jik}{}^l\omega_l$. Contract with $g^{ik}$: $\nabla_j\Delta f - \nabla^i\nabla_i\nabla_jf = -g^{ik}R_{jikm}\nabla^mf$, using $\nabla_j\omega_k = \nabla_k\omega_j$ to rewrite $g^{ik}\nabla_i\nabla_j\omega_k = \nabla^i\nabla_i\nabla_jf$. And $g^{ik}R_{jikm} = g^{ik}R_{kmji} = R_{mj}$ (contract the first and last slots). So $\nabla^i\nabla_i\nabla_jf = \nabla_j\Delta f + R_{jm}\nabla^mf$. (b) Trace: $R + \Delta f = n\lambda$, so $\nabla_j\Delta f = -\nabla_jR$. Divergence: $\frac12\nabla_jR + \nabla_j\Delta f + R_{jk}\nabla^kf = 0$, so $R_{jk}\nabla^kf = \frac12\nabla_jR$. (c) $\nabla_j(R + |\nabla f|^2 - 2\lambda f) = \nabla_jR + 2\nabla_j\nabla_kf\nabla^kf - 2\lambda\nabla_jf = \nabla_jR + 2(\lambda g_{jk} - R_{jk})\nabla^kf - 2\lambda\nabla_jf = \nabla_jR - 2R_{jk}\nabla^kf = 0$.