Topics in Scalar Curvature Spring 2017 Richard M. Schoen Notes by Chao Li Contents Chapter 1. Introduction and motivation 5 1. Einstein equations of general relativity 5 2. Variation of total scalar curvature 6 3. Conformal geometry 7 4. Manifolds with negative scalar curvature 8 5. How about R > 0? 12 Chapter 2. The positive mass theorem 13 1. Manifolds admitting metrics with positive scalar curvature 13 2. Positive mass theorem: first reduction 14 3. Minimal slicing 21 4. Homogeneous minimal slicings 38 5. Top dimensional singularities 43 6. Compactness of minimal slicings 47 7. Dimension reduction 50 8. Existence and proof of the main theorem 52 Bibliography 57 3 CHAPTER 1 Introduction and motivation Let (M; g) be a smooth Riemannian manifold with Riemannian curvature tensor Rijkl. In the study of differential geometry, there are three types of curvatures that has received a lot of attention. The sectional curvature measures the curvature of two dimensional submanifolds of (M; g). In local coordinate, the sectional curvature of a two dimensional tangent plan spanned by ei; ej is given by Rijji. The Ricci curvature is the trace of the sectional curvature. In local coordinate, the P Ricci tensor is defined as Ricij = Rikkj. The scalar curvature is the trace of the Ricci curvature: P k R = i;j Rijji. Note that in our convention the scalar curvature of a two dimensional surface is twice its Gauss curvature. The study of curvature dates back to the time of Gauss and Riemann, where curvature was first observed as the quantity that distinguish locally a general Riemannian manifold and the Euclidean space. Since then lots of studies have been carried out to understand the connection between the curvature with local and global geometric and topological aspects of the manifold. Here we just name a few of well known results of this flavor: the Gauss-Bonnet theorem, Toponogov compara- sion theorem, the Bochner-Weizenb¨ock formula, Bonnet-Myers theorem, Cheeger-Gromoll's volume comparasion theorem, Gromov's finiteness theorem, etc. From our very crude list one readily sees that results concerning scalar curvature are much fewer compared to those of the sectional or Ricci curvature. Indeed, scalar curvature measures the average the sectional curvature among all dis- tinct two planes, and is believed to carry relatively few information about the manifold. This adds considerable delicacy into the study of the scalar curvature. In this chapter we introduce three motivations to study the scalar curvature, and various phenamina unique to it. 1. Einstein equations of general relativity The first senario where scalar curvature naturally occurs is the theory of general relativity. Suppose (Sn+1; g) is a Minkowski manifold representing the space time. Then it satisfies the Einstein equation, namely 1 RicS − RSg = T; 2 where T is called the stress-energy tensor representing matter in the space time. Assume that M n is a submanifold of Sn+1. At a point in M, take a normal frame of S, feigi=0;:::;n such that e0 is the unit normal vector of M. Restrict the Einstein equation on M with respect to e0; e0, we get 1 Ric00 − RS = T00: 2 P Expanding in local coordinates, Ric00 = j R0jj0, and 1 X X X RS = Rijji = R0jj0 + Rijji: 2 0≤i≤j≤n j 1≤i≤j≤n 5 6 1. INTRODUCTION AND MOTIVATION Denote RM the Riemannian tensor on M and h the second fundamental form of the embedding M ⊂ S. By the Gauss equation, M 2 Rijji = Rijji + hiihjj − hij: Therefore 1 1 1 2 1 2 RS = Ric00 + RM + (tr h) − khk : 2 2 2 2 By the Einstein equation, one sees that 1 2 2 (1.1) (RM ) + (tr h) − khk = T00 := µ. 2 In the theory of general relativity, T00 = µ is called the mass density. One may also derive other relations from the Einstein equation by taking the 0j component. By a similar calculation, we have (1.2) div(h − (tr h)g) = T0j: Together, equation 1.1 and 1.2 are called the constraint equations. They provide a necessary condition for a Riemannian manifold to be realized as a submanifold of a space time. A particularly important case is when h is identically zero on M. In other words, M is a totally geodecis submanifold of S. The physical reason is that, when h ≡ 0, the reflection of the time variable t ! −t is an isometry. It is called time symmetric space time. We see from the previous equation that in this case RM = 2µ. Note that since µ is the mass density, we always assume that µ is nonnegative. Then RM is nonnegative. We then conclude that Any Riemannian manifold satisfying the constraint equations must have nonnegative scalar curvature. From this prospective, the study of Riemannian manifold with nonnegative scalar curvature can be embedded into the study of the constraint equations of general relativity. 2. Variation of total scalar curvature On a Riemannian manifold (M; g), consider Einstein-Hilbert functional R(g) = Rgdµg ˆM on all Riemannian metrics with unit volume. It is shown by Hilbert that for n ≥ 3 the critical points of this functional are Einstein metrics, i.e. metrics satisfying Ric(g) = c · R(g)g. This can be seen from the following calculation. Let g be a Riemannian metric with unit volume which is a critical point of R. Set gt = g + th, t 2 (−, ) be a smooth family of Riemannian metrics, h a smooth compactly supported (0; 2) −2=n tensor. Letg ¯t = Vol(t) gt be normalized with unit volume. Theng ¯t is a valid variation. To calculate the derivative of R(¯gt), we first observe that in local coordinates, 1 X Γk = gkl(g + g − g ); ij 2 il;j jl;i ij;l l ! X k k X k l k l Ricij = Γij;k − Γki;j + (ΓklΓji − ΓjlΓki) ; k l where the comma denotes a partial derivative in the given direction. We then see the derivatives of the Christoffel symbol and the Ricci curvature under this variation are k 1 X kl X k k (2.1) Γ_ = g (h + h − h ); Ric_ ij = (Γ_ − Γ_ ): ij 2 il;j jl;i ij;l ij;k ki;j k 3. CONFORMAL GEOMETRY 7 Therefore we find that R_ can be written as X ij R_ = − h Ricij +divergence terms; i;j ij P ik jl here h = k;l g g hkl. As for the volume form, one easily checks that d 1 Volg = trg (h)d Volg : dt t 2 t t Putting these together and use the divergence theorem, we find d 1 R(gt) = − h; Ric(gt) − Rtgt d Volgt : dt ˆM 2 (2−n)=n Since R(¯gt) = Vol(gt) R(gt) we find that d (2−n)=n 1 n − 2 jt=0R(¯gt) = − Vol(g) h; Ric(g) − R(g)g + R(g)g d Volgt : dt ˆM 2 2n Therefore g is a critical point for R functional if and only if Ric(g) = cg for some constant c. The existence and geometric properties of Einstein metrics have been long standing questions that have stimulated the development of many area. In dimension 2 and 3 Einstein metrics are metrics with constant sectional curvature, and can be handled by Ricci flow. In dimension 4 or above, relatively few is known. One major difficulty, among others, is that we do not know whether the extremal of the function R is local maximum or minimum. 3. Conformal geometry Let's continue looking at the Einstein-Hilbert functional. As discussed previously, we do not know whether R has an infimum or supreme in general. However, we are going to see that R does attain a minimum when we restrict our consideration within a special family of metrics, namely, the conformal class of metrics. n Let g0 be a background metric on a compact manifold M , n ≥ 3. Define the conformal class of g0 by 2u 1 [g0] = fg = e g0 : u 2 C (M)g: And the Yamabe invariant of this conformal class Y ([g0]) = inffR(g): g 2 [g0]; Vol(g) = 1g: We are going to see shortly that Y ([g0]) is finite for any smooth background metric g0. For the 4 4 convenience of calculation, let's use u n−2 as the conformal factor, namely let g = u n−2 g0. We have the following important formula that will be used throughout our discussion. 4 Lemma 3.1 (Conformal formula). The scalar curvature of g = u n−2 g0 and the scalar curvature of g0 are related by −1 − n+2 (3.1) R(g) = −c(n) u n−2 Lu; n−2 the constant c(n) = 4(n−1) , L is an elliptic operator Lu = ∆g0 u − c(n)R(g0)u, called the conformal Laplacian. 2 2f Proof. Denote f = n−2 log u, then g = e g0. Let Γ;Rijkl; Ric, etc. denote the geometric quantities of the metric g and let Γ0; (R0)ijkl; Ric0, etc. denote those of g0. Take a local coordinate normal at one point. Then we first see the new Christoffel symbols are given by k k k k Γij = (Γ0)ij + δi @jf + δj @if − gij@kf: 8 1. INTRODUCTION AND MOTIVATION By the previous formula relating the Christoffel symbol and the Ricci tensor, 2 Ricij = (Ric0)ij − (n − 2)[@i@jf − (@if)(@jf)] + (∆0f − (n − 2)krfk )(g0)ij: Taking trace we get −2f 2 R = e (R0 + 2(n − 1)∆f − (n − 2)(n − 1)krfk ): 2 Now replace f = n−2 log u, we obtain the desired formula for the scalar curvature.