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Comparison Geometry for Ricci Curvature

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Comparison Geometry for Ricci Curvature Comparison Geometry for Ricci Curvature Xianzhe Dai Guofang Wei 1 1Partially supported by NSF grant DMS-08 2 A Ricci curvature bound is weaker than a sectional curvature bound but stronger than a scalar curvature bound. Ricci curvature is also special that it occurs in the Einstein equation and in the Ricci flow. Comparison geometry plays a very important role in the study of manifolds with lower Ricci curva- ture bound, especially the Laplacian and the Bishop-Gromov volume compar- isons. Many important tools and results for manifolds with Ricci curvature lower bound follow from or use these comparisons, e.g. Meyers' theorem, Cheeger- Gromoll's splitting theorem, Abresch-Gromoll's excess estimate, Cheng-Yau's gradient estimate, Milnor's result on fundamental group. We will present the Laplacian and the Bishop-Gromov volume comparison theorems in the first lec- ture, then discuss their generalizations to integral Ricci curvature, Bakry-Emery Ricci tensor and Ricci flow in the rest of lectures. Contents 1 Basic Tools for Ricci Curvature 5 1.1 Bochner's formula . 5 1.2 Mean Curvature and Local Laplacian Comparison . 7 1.3 Global Laplacian Comparison . 10 1.4 Volume Comparison . 12 1.4.1 Volume of Riemannian Manifold . 12 1.4.2 Comparison of Volume Elements . 15 1.4.3 Volume Comparison . 16 1.5 The Jacobian Determinant of the Exponential Map . 20 1.6 Characterizations of Ricci Curvature Lower Bound . 22 1.7 Characterization of Warped Product . 24 2 Geometry of Manifolds with Ricci Curvature Lower Bound 27 2.1 Cheeger-Gromoll's Splitting Theorem . 27 2.2 Gradient Estimate . 31 2.2.1 Harmonic Functions . 31 2.2.2 Heat Kernel . 34 2.3 First Eigenvalue and Heat Kernel Comparison . 34 2.3.1 First Nonzero Eigenvalue of Closed Manifolds . 35 2.3.2 Dirichlet and Neumann Eigenvalue Comparison . 40 2.3.3 Heat Kernel Comparison . 41 2.4 Isoperimetric Inequality . 41 2.5 Abresch-Gromoll's Excess Estimate . 41 2.6 Almost Splitting Theorem . 44 3 Topology of Manifolds with Ricci Curvature Lower Bound 45 3.1 First Betti Number Estimate . 45 3.2 Fundamental Groups . 48 3.2.1 Growth of Groups . 49 3.2.2 Fundamental Group of Manifolds with Nonnegative Ricci Curvature . 50 3.2.3 Finiteness of Fundamental Groups . 54 3.3 Volume entropy and simplicial volume . 56 3.4 Examples and Questions . 57 3 4 CONTENTS 4 Gromov-Hausdorff convergence 61 5 Comparison for Integral Ricci Curvature 65 5.1 Integral Curvature: an Overview . 65 5.2 Mean Curvature Comparison Estimate . 67 5.3 Volume Comparison Estimate . 69 5.4 Geometric and Topological Results for Integral Curvature . 73 5.5 Smoothing . 79 6 Comparison Geometry for Bakry-Emery Ricci Tensor 81 6.1 N-Bakry-Emery Ricci Tensor . 81 6.2 Bochner formulas for the N-Bakry-Emery Ricci tensor . 83 6.3 Eigenvalue and Mean Curvature Comparison . 84 6.4 Volume Comparison and Myers' Theorems . 88 7 Comparison Geometry in Ricci Flow 93 7.1 Reduced Volume Monotonicity . 93 7.2 Heuristic Argument . 93 7.3 Laplacian Comparison for Ricci Flow . 98 8 Ricci Curvature for Metric Measure Spaces 99 8.1 Metric Space and Optimal Transportation . 99 8.1.1 Metric and Length Spaces . 99 8.1.2 Optimal Transportation . 100 8.1.3 The Monge transport . 102 8.1.4 Topology and Geometry of P (X) . 103 8.2 N-Ricci Lower Bound for Measured Length Spaces . 108 8.2.1 Via Localized Bishop-Gromov . 108 8.2.2 Entropy And Ricci Curvature . 113 8.2.3 The Case of Smooth Metric Measure Spaces . 116 8.2.4 Via Entropy Convexity . 119 8.3 Stability of N-Ricci Lower Bound under Convergence . 122 8.4 Geometric and Analytical Consequences . 122 8.5 Cheeger-Colding . 127 Chapter 1 Basic Tools and Characterizations of Ricci Curvature Lower Bound The most basic tool in studying manifolds with Ricci curvature bound is the Bochner formula, which measures the non-commutativity of the covariant deriva- tive and the connection Laplacian. Applying the Bochner formula to distance functions we get important tools like mean curvature and Laplacian comparison theorems, volume comparison theorem. Each of these tools can be used to give a characterization of the Ricci curvature lower bound. These tools have many applications, see next two chapters. 1.1 Bochner's formula For a smooth function u on a Riemannian manifold (M n; g), the gradient of u is the vector field ru such that hru; Xi = X(u) for all vector fields X on M. The Hessian of u is the symmetric bilinear form Hess (u)(X; Y ) = XY (u) − rX Y (u) = hrX ru; Y i; and the Laplacian is the trace ∆u = tr(Hess u). For a bilinear form A, we denote jAj2 = tr(AAt). The Bochner formula for functions is Theorem 1.1.1 (Bochner's Formula) For a smooth function u on a Rie- mannian manifold (M n; g), 1 ∆jruj2 = jHess uj2 + hru; r(∆u)i + Ric(ru; ru): (1.1.1) 2 5 6 CHAPTER 1. BASIC TOOLS FOR RICCI CURVATURE Proof: We can derive the formula by using local geodesic frame and commuting the derivatives. Fix x 2 M, let feig be an orthonormal frame in a neighborhood of x such that, at x, rei ej(x) = 0 for all i; j. At x, 1 1 X ∆jruj2 = e e hru; rui 2 2 i i i X X = eihrei ru; rui = eiHess u(ei; ru) i i X X = eiHess u(ru; ei) = eihrruru; eii i i X = hrei rruru; eii i X = hrrurei ru; eii + hr[ei;ru]ru; eii + hR(ei; ru)ru;(1.1.2) eii : i Now at x, X X hrrurei ru; eii = [ruhrei ru; eii − hrei ru; rrueii] i i = ru(∆u) = hru; r(∆u)i; (1.1.3) and X X hr[ei;ru]ru; eii = Hess u([ei; ru]; ei) i i X = Hess u(ei; rei ru) i X 2 = hrei ru; rei rui = jHess uj : (1.1.4) i Combining (1.1.2), (1.1.3) and (1.1.4) gives (1.1.1). 2 2 (∆u) Applying the Cauchy-Schwarz inequality jHess uj ≥ n to (1.1.1) we obtain the following inequality 1 (∆u)2 ∆jruj2 ≥ + hru; r(∆u)i + Ric(ru; ru); (1.1.5) 2 n 1 with equality if and only if Hess u = hIn for some h 2 C (M). If Ric ≥ (n − 1)H, then 1 (∆u)2 ∆jruj2 ≥ + hru; r(∆u)i + (n − 1)Hjruj2: (1.1.6) 2 n The Bochner formula simplifies whenever jruj or ∆u are simply. Hence it is natural to apply it to the distance functions, harmonic functions, and the eigenfunctions among others, getting many applications. The formula has a more general version (Weitzenb¨ock type) for vector fields (1-forms). 1.2. MEAN CURVATURE AND LOCAL LAPLACIAN COMPARISON 7 1.2 Mean Curvature and Local Laplacian Com- parison Here we apply the Bochner formula to distance functions. We call ρ : U ! R, where U ⊂ M n is open, is a distance function if jrρj ≡ 1 on U. Example 1.2.1 Let A ⊂ M be a submanifold, then ρ(x) = d(x; A) = inffd(x; y)jy 2 Ag is a distance function on some open set U ⊂ M. When A = q is a point, the distance function r(x) = d(q; x) is smooth on M n fq; Cqg, where Cq is the cut locus of q. When A is a hypersurface, ρ(x) is smooth outside the focal points of A. For a smooth distance function ρ(x), Hess ρ is the covariant derivative of the normal direction @r = rρ. Hence Hess ρ = II, the second fundamental form of the level sets ρ−1(r), and ∆ρ = m, the mean curvature. For r(x) = n−1 d(q; x), m(r; θ) ∼ r as r ! 0; for ρ(x) = d(x; A), where A is a hypersurface, m(y; 0) = mA, the mean curvature of A, for y 2 A. Putting u(x) = ρ(x) in (1.1.1), we obtain the Riccati equation along a radial geodesic, 2 0 0 = jIIj + m + Ric(@r;@r): (1.2.1) By the Cauchy-Schwarz inequality, m2 jIIj2 ≥ : n − 1 Thus we have the Riccati inequality m2 m0 ≤ − − Ric(@ ;@ ): (1.2.2) n − 1 r r If RicM n ≥ (n − 1)H, then m2 m0 ≤ − − (n − 1)H: (1.2.3) n − 1 From now on, unless specified otherwise, we assume m = ∆r, the mean n curvature of geodesic spheres. Let MH denote the complete simply connected n space of constant curvature H and mH (or mH when dimension is needed) the mean curvature of its geodesics sphere, then m2 m0 = − H − (n − 1)H: (1.2.4) H n − 1 Let snH (r) be the solution to 00 snH + HsnH = 0 8 CHAPTER 1. BASIC TOOLS FOR RICCI CURVATURE 0 such that snH (0) = 0 and snH (0) = 1, i.e. snH are the coefficients of the Jacobi n fields of the model spaces MH : p 8 p1 sin Hr H > 0 <> H snH (r) = r H = 0 : (1.2.5) > p1 sinh pjHjr H < 0 : jHj Then 0 snH mH = (n − 1) : (1.2.6) snH n−1 As r ! 0, mH ∼ r . The mean curvature comparison is Theorem 1.2.2 (Mean Curvature Comparison) If RicM n ≥ (n−1)H, then along any minimal geodesic segment from q, m(r) ≤ mH (r): (1.2.7) Moreover, equality holds if and only if all radial sectional curvatures are equal to H.
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