Differential Geometry II Lecture 10: Einstein metrics and Ricci flow Robert Haslhofer

In this lecture we further discuss some aspects of the analysis and geometry of Ricci curvature, namely Einstein metrics and Hamilton’s Ricci flow. The goal is to give an overview of the field, and thus many parts are in a “story telling format”.

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Einstein manifolds

An Einstein manifold is a Riemannian manifold (M, g) such that Rc(g) = λg

(1.1)

for some λ ∈ R. In other words, the Ricci curvature is a constant multiple of the metric. Depending on the sign of λ, we talk about positive Einstein metrics, negative Einstein metrics, and Ricci flat metrics, respectively. Einstein metrics are of great interest and importance in geometry, topology, and physics. We recall from Lecture 9 that the Ricci curvature can be thought of as nonlinear Laplacian of the metric; it is helpful to compare (1.1) with similar canonical equations from related fields, namely the harmonic map equation, the minimal surface or constant mean curvature equation, the Yang-Mills equation, etc. Lemma 1.1 (Schur) If (M n , g) is a connected Riemannian manifold of dimension n ≥ 3, that satisfies Rc(g) = f g for some differentiable function f : M → R, then f is constant. Proof. Recall the second Bianchi identity (from Lecture 6), ∇i Rjklm + ∇j Rkilm + ∇k Rijlm = 0.

(1.2)

Contracting with g km and g il we obtain the so-called contracted Bianchi identity, 2∇i Rij − ∇j R = 0. Thus, if Rij = f gij , then (n − 2)∇j f = 0, which implies the claim. Proposition 1.2 (n = 3) Every three dimensional Einstein metric has constant sectional curvature. 1

(1.3)

Proof. We recall from Lecture 7 that in three dimensions Rm = g ∧ Rc◦ +

R g ∧ g. 12

(1.4)

The Einstein equation is equivalent to Rc◦ = 0, and the claim follows. Exercise 1. Show that every two dimensional Riemannian manifold is Einstein. Remark 1.3 (Positive Einstein in dimension four) As we discussed in Lecture 7, Einstein metrics on compact oriented 4-manifolds are constrained by the Hitchin-Thorpe inequality, and the known examples of positive Einstein metrics (up to quotients and products of lower dimensional manifolds) are S 4 with the round metric, CP 2 with the Fubini-Study metric (c.f. Lecture 1), 2 2 CP 2 ]CP with the Page-metric, CP 2 ]2CP with the Chen-LeBrun-Weber metric, 2 and CP 2 ]kCP (3 ≤ k ≤ 8) with its K¨ ahler-Einstein metrics. Remark 1.4 (Ricci-flat metrics) Up to quotients, the known compact Ricci flat metrics in dimension four are the flat metrics on the torus T 4 , and K3 with its Ricci flat K¨ ahler-metrics. In higher dimensions, there are many examples of compact Ricci-flat manifolds due to Yau’s solution of the Calabi-problem, the gluing construction by Joyce, etc. However, all of them have special holonomy (namely SUn/2 , Spn/4 , G2 , or Spin7 ). The most fundamental open question about Einstein metrics is the following. Open problem 1.5 (Compact Ricci flat metrics) Is there any compact Ricci flat manifold with holonomy equal to SOn ? Remark 1.6 (Twisted Kummer construction) Page suggested a twisted Kummer construction, where one flips the orientation of some Eguchi-Hanson metrics, but as of today no one succeeded in making this work.

A glimps at general relativity Einstein metrics (with Lorentzian signature) are the vacuum solutions in general relativity. The simplest vacuum solution is flat Minkowski space time R3+1 , g = −dt2 + dx2 + dy 2 + dz 2 .

(1.5)

The most important example of a nonflat space time is the Schwarzschild solution g = −(1 −

2m 2 2m −1 2 )dt + (1 − ) dr + r2 (dθ2 + sin2 θdϕ2 ), r r

(1.6)

which is the rotationally symmetric solution in the exterior of a star or a black hole. It is straightforward to check that Rc(g) = 0. The planets in our solar system 2

move along geodesics in the gravitational field generated by the sun; in particular, equation (1.6) implies that the orbits of planets differ in a small quantitative way from the Newtonian prediction. This perihelion precession is particularly visible for the planet Mercury, and was one of the first experimental tests of general relativity. Einstein’s theory is described by a single beautiful equation. It has the form E + Λg = 8πT.

(1.7)

The tensor T = Tij on the right hand side is the energy momentum tensor. It is the source of curvature, which appears on the left hand side in form of the Einstein tensor 1 E = Rc − Rg. (1.8) 2 We have written down the equation including the cosmological constant Λ. The energy momentum tensor satisfies the conservation law ∇i Tij = 0.

(1.9)

Using (1.3) we see that the Einstein tensor indeed also satisfies ∇i Eij = 0,

(1.10)

and in fact this divergence free condition determines the equation (1.7) in an essentially unique way (once we agree that the equation should be quasilinear second order for the metric, and coordinate independent and thus tensorial). For vacuum solutions the energy momentum tensor vanishes, and using Lemma 1.1 we see that the equation (1.7) with T = 0 is equivalent to Rc = λg

(1.11)

with the constant λ = 21 R − Λ = n1 R. Exercise 2 (Mass of the Schwarzschild metric). Consider the t = 0 slice of the Schwarzschild metric, i.e. the Riemannian three-manifold R3 \ B2m with the metric g = (1 −

2m −1 2 ) dr + r2 (dθ2 + sin2 θdϕ2 ). r

(1.12)

Show that R(g) ≥ 0, and that the mass is given by the ADM-formula m=

1 lim 16π r→∞

Z

3 X

(∂i gij − ∂j gii )ν j dA.

|x|=r i,j=1

3

(1.13)

The Einstein-Hilbert functional Einstein metrics arise as critical points of the Einstein-Hilbert functional Z E(g) = Rg dµg . (1.14) M

In the following we assume that either the underlying manifold is compact, or that the scalar curvature is integrable and that the variations are compactly supported. Theorem 1.7 (First variation) The first variation of the Einstein-Hilbert functional is given by the formula Z d |0 E(g + εh) = − hE, hidµ, (1.15) dε M where E = Rc − 21 Rg is the Einstein tensor. In particular, if M is compact then the critical points under fixed volume are precisely the Einstein metrics, E + Λg = 0. For the proof (and also for later use), we need the following lemma. Lemma 1.8 (Variation of volume and scalar curvature) If we vary a Riemannian metric g in direction of a symmetric two-tensor h, then d |0 dµg+εh = dε

1 2

trg h dµg ,

d |0 Rg+εh = −4g trg h + ∇i ∇j hij − hRc, hig . (1.16) dε

p Proof of Lemma 1.8. Recall from Lecture 2 that locally dµg = det(gij )dx1 . . . dxn , d |0 det(A + εB) = det(A) tr(A−1 B), since and recall from linear algebra that dε det(A + εB) = det(A) det(I + εA−1 B) = det(A)(1 + ε tr(A−1 B) + O(ε2 )). (1.17) Thus

d 1 |0 dµg+εh = g ij hij dµg = 12 trg h dµg . dε 2 To prove formula for the variation of R, we recall from Lecture 6 that

where

(1.18)

R = g jk (∂i Γijk − ∂j Γiik ) + Γ ∗ Γ,

(1.19)

1 Γijk = g il (∂j gkl + ∂k gjl − ∂l gjk ). 2

(1.20)

d Writing δΓ = dε |0 Γ(g + εh), we compute (using normal coordinates and the fact that the difference of two connections is a tensor)

1 δΓijk = g il (∇j hkl + ∇k hjl − ∇l hjk ), 2 4

(1.21)

and (using also δA−1 = −A−1 δAA−1 which follows from differentiating AA−1 = I) 1 1 δR = ∇i (∇j hji + ∇j hji − ∇i hjj ) − ∇j ∇j hii − Rij hij . 2 2

(1.22)

This proves the lemma. Proof of Theorem 1.7. Using Lemma 1.8 and integrating by parts, we compute Z d (−4 tr h + ∇i ∇j hij − hRc, hi + R 12 tr h)dµ |0 E(g + εh) = (1.23) dε MZ hRc − 12 Rg, hidµ. (1.24) =− M

If the volume is kept fixed, then

2

R M

tr h dµ = 0, and the claim follows.

Ricci flow

In 1982, Richard Hamilton introduced the evolution equation ∂t g(t) = −2 Rc(g(t)),

g(0) = g0 .

(2.1)

Recalling Proposition 1.1 from Lecture 9, the Ricci flow can be thought of as heat equation for Riemannian manifolds. Motivated by this, one hopes that the flow improves the metric and converges to an optimal geometry, and indeed this idea has been extremely sucessful. For starters, given any initial metric g0 on a compact manifold M , there exists a unique solution of (2.1) on a maximal time interval [0, T ) for some T ≤ ∞. This follows from standard parabolic theory (after fixing the gauge). Moreover, if T < ∞ then lim sup|Rm(g(t))| = ∞. (2.2) t→T M

Example 2.1 (Einstein metrics only move by scaling) If the initial metric is Einstein, Rc(g0 ) = λg0 , then the solution of (2.1) is given by g(t) = (1 − 2λt)g0 .

(2.3)

We have T = ∞ for λ ≤ 0, and T = 1/(2λ) for λ > 0. Equation (2.1) implies evolution equations for all geometric quantitites, e.g. Proposition 2.2 (Evolution of scalar curvature) If g(t) evolves by Ricci flow, then the scalar curvature R = R(g(t)) evolves by ∂t R = 4R + 2|Rc|2 .

(2.4)

In particular, the minimum of the scalar curvature is nondecreasing along the flow, and if min R(g0 ) > 0 then T < ∞. 5

Proof. Using Lemma 1.8 with h = −2 Rc we obtain ∂t R = 24R − 2∇i ∇j Rij + 2|Rc|2 .

(2.5)

Using the contracted Bianchi identity (1.3), this implies (2.4). By the maximum principle, the minimum of the scalar curvature is nondecreasing. Moreover, using n . |Rc|2 ≥ n1 R2 , we see that in the case R0 = min R(g0 ) > 0 we must have T ≤ 2R 0 In his first paper about Ricci flow, Hamilton proved the following landmark result. Theorem 2.3 (Hamilton, 1982) If (M, g0 ) is a compact manifold with positive Ricci curvature, then the Ricci flow starting at g0 converges modulo scaling to a metric of constant positive sectional curvature. In particular, if M is simply connected then it is diffeomophic to S 3 . Note that by Proposition 2.2 we have T < ∞, and that by Proposition 1.2 it suffices to prove convergence to an Einstein metric for t → T (modulo rescaling). In essence, Hamilton’s proof is based on a very careful analysis of the evolution equation for the Ricci curvature, which turns out to be ∂t Rij = 4Rij + 2Ripjq Rpq − 2Rip Rpj

(2.6)

Using the tensor maximum principle, Hamilton proved that positive Ricci curvature is preserved in dimension three, and in fact that the metric becomes rounder and rounder as t → T . Remark 2.4 (Differentiable sphere theorem) The Brendle-Schoen proof of the differentiable sphere theorem is based on a similar strategy (see also B¨ ohm-Wilking). The evolution equation for the curvature tensor in higher dimensions is quite complicated, but can be written neatly as Dt Rm = 4 Rm +2(Rm ◦ Rm + Rm ] Rm),

(2.7)

where Rm ◦ Rm is the composition of Rm with itself viewed as operator Λ2 T M → Λ2 T M , and Rm ] Rm is the so-called Lie-algebra square which is defined using the adjoint representation of Λ2 Tp M = so(Tp M ). The analysis of (2.7) relies in a crucial way on the structure of the space of algebraic curvature tensors (Lecture 7). Without positive assumptions on the curvature, the main part of the story is studying the formation of singularities (e.g. the neckpinch), and trying to continue the flow beyond them. The most spectacular result is Perelman’s construction of Ricci flow with surgery in dimension three in his proof of the geometrization conjecture.

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