Time: Fridays 10-12, 20 January-23 March 2012 (10 weeks)
Lecturer: Hans-Joachim Hein
Description: Our main goal is to prove some of the structure theorems for Riemannian manifolds with uniform lower bounds on their Ricci curvature due to Cheeger and Colding in the mid-1990's. This involves an interesting mix of Riemannian geometry and geometric analysis. In particular, we will use our main goal as an excuse to talk about many different ideas and techniques in the field. The only prerequisites are basic courses on manifolds and Riemannian geometry. Some previous experience with linear elliptic equations is helpful but not necessary.
Lecture notes: 7, 8, 9 are missing. For what it's worth I'm posting the Starboard files until these are typed up, which may still take a while (12.04.).
Lecture 1 (20.01.): review of Riemannian geometry and tensor calculus
Lecture 2 (27.01.): Weitzenböck formula, Hodge theorem, sketch of L^2 theory for the Dirichlet problem
Lecture 3 (03.02.): maximum principles, harmonic functions on R^n, monotonicity of frequency
Lecture 4 (10.02.): warped product version of volume comparison, applications, eigenvalue estimates
Lecture 5 (17.02.): some recap, 1/2-Hölder bound for Jacobi fields, Ricci and \pi_1, splitting theorem
Lecture 6 (24.02.): proof of the splitting theorem, Gromov-Hausdorff convergence, Gromov compactness
Lecture 7 (02.03.): the CC theorems, Poincare-Sobolev and segment inequalities, two proofs of the Poincare
Lecture 8 (09.03.): clarifications, proof of the segment, proof of the Sobolev via isoperimetric estimates
Lecture 9 (16.03.): completion of proof, Moser iteration, Cheng-Yau gradient estimate, Li-Yau Harnack
Lecture 10 (23.03.): clarifications on Cheng-Yau, good cutoff functions, excess inequality, volume convergence
Problem sheets:
Outline: We covered Chapters 1-3 of this outline almost completely, and a basic selection of topics from Chapters 4-7. Unfortunately I would have needed another hour or two at the end to finish the proof of the volume convergence theorem. I may post my compiled notes for Chapter 3 at some point.
(1) Review of Riemannian geometry and Hodge theory
arclength, geodesics, normal coordinates, Jacobi fields, cut locus, statement of the Rauch comparison theorem, differential calculus on Riemannian manifolds, divergence theorem, Killing fields, Weitzenböck formula, statement of the Hodge decomposition theorem
(2) The Laplace equation, mostly on R^n
Dirichlet problem, overview of L^2 theory and Sobolev type inequalities, maximum principles, gradient estimates, Liouville, Harnack, eigenfunctions on the sphere and harmonic polynomials, Green's function, harmonic vs holomorphic on C^n, monotonicity of frequency
(3) Basic geometry of lower Ricci bounds
volume monotonicity and rigidity of warped products, monotonicity "applied in reverse" (e.g. Myers), 1/2-Hölder bound on Jacobi fields, eigenvalue estimates on 1-forms, the fundamental group (Milnor conjecture, polynomial growth, Margulis lemma), Cheeger-Gromoll splitting theorem
(4) Convergence of metric spaces and manifolds
Gromov-Hausdorff distance, Gromov compactness, sketch of applications (groups of polynomial growth, existence of Einstein metrics, statements of the Cheeger-Colding theorems), Anderson's counterexample, examples of limit spaces with Ricci bounds, tangent cones
(5) Functional inequalities and applications
effective Poincaré/Sobolev/isoperimetric/Morrey inequalities under lower Ricci bounds, global versions on open manifolds, Weyl type bounds on the spectral counting function, harmonic functions of polynomial growth, Kleiner's proof of Gromov's theorem on groups of polynomial growth
(6) Linear analysis with lower Ricci bounds
Cheng-Yau gradient estimate, Li-Yau Harnack inequality, good cutoff functions, upper and lower heat kernel bounds, Moser iteration and generalized mean value inequalities, global behavior of the Green's function, monotonicity of entropy, analogies with Ricci flow
(7) Cheeger-Colding theory
Abresch-Gromoll excess inequality, volume convergence, almost splitting, almost metric cone theorem, Hölder continuity of tangent cones
References:
Background
- Burago & Burago & Ivanov, A course in metric geometry
- Cheeger & Ebin, Comparison theorems in Riemannian geometry
- Evans, Partial differential equations (2nd edition)
- Gromov, Sign and geometric meaning of curvature
- Viaclovsky, Topics in Riemannian geometry
Geometry
- Cheeger, Degeneration of Riemannian metrics under Ricci curvature bounds
- Colding & Naber, Sharp Hölder continuity of tangent cones
- Gromov, Metric structures for Riemannian and non-Riemannian spaces
- Karcher, Riemannian comparison constructions
- Wei, Manifolds with a lower Ricci curvature bound
- Wei, Lecture notes on Ricci curvature
Analysis
- Colding & Minicozzi, An excursion into geometric analysis
- Donaldson, Geometric analysis
- Li, Lecture notes on geometric analysis (also try this)