Topics in Geometric Analysis, Fall 2009

Lectures: Monday / Wednesday / Friday, 2:15 -- 3:05 PM in Building 380, Room 381T
Instructor: Simon Brendle, Room 382Z, brendle "at" math.stanford.edu
Office Hours: Tuesday / Thursday, 2:15 -- 3:00 PM or by appointment

This course will present an introduction to Hamilton's Ricci flow, as well as a survey of recent developments, such as the Differentiable Sphere Theorem.

  • Class 1 (Monday, September 21): Riemannian geometry background; a survey of sphere theorems in geometry
  • Class 2 (Wednesday, September 23): Hamilton's Ricci flow; special solutions; Einstein metrics; Ricci solitons; the cigar soliton and the Rosenau solution; DeTurck's trick; strict parabolicity of Ricci-DeTurck flow
  • Class 3 (Friday, September 25): Equivalence of Ricci flow and Ricci-DeTurck flow; shorttime existence and uniqueness for Ricci flow; the evolution of the Levi-Civita connection; evolution of the Riemann curvature tensor
  • Class 4 (Monday, September 28): The evolution equation for the Riemann curvature tensor (continued); the evolution equations for Ricci and scalar curvature
  • Class 5 (Wednesday, September 30): Estimates for the scalar curvature via the maximum principle; formation of finite-time singularities for initial metrics with positive scalar curvature; Shi's estimates for the covariant derivatives of the curvature tensor; interior estimates for tensors satisfying linear parabolic equations
  • Class 6 (Friday, October 2): Curvature blow-up at finite-time singularities; estimates for evolving metrics
  • Class 7 (Monday, October 5): Any gradient Ricci soliton on S^2 has constant curvature; Hamilton's entropy formula for the Ricci flow on S^2
  • Class 8 (Wednesday, October 7): Ricci flow on S^2; curvature bounds; convergence to a constant curvature metric
  • Class 9 (Friday, October 9): The tangent and normal cone to a convex set; a necessary and sufficient condition for a closed set to be invariant under an ODE
  • Class 10 (Monday, October 12): Pointwise curvature estimates in higher dimensions; Hamilton's maximum principle for systems
  • Class 11 (Wednesday, October 14): Pinching sets; Hamilton's convergence criterion for the Ricci flow; asymptotic roundness
  • Class 12 (Friday, October 16): Hamilton's convergence criterion for the Ricci flow (continued); C^\infty convergence of the rescaled metrics
  • Class 13 (Monday, October 19): Curvature pinching in dimension 3; Hamilton's convergence theorem for three-manifolds with positive Ricci curvature
  • Class 14 (Wednesday, October 21): The curvature estimate of Hamilton and Ivey; any compact Ricci soliton in dimension 3 has constant sectional curvature
  • Class 15 (Friday, October 23): Preserved curvature conditions in higher dimensions; nonnegative curvature operator and 2-nonnegative curvature operator
  • Class 16 (Monday, October 26): Nonnegative isotropic curvature is preserved by the Ricci flow: the four-dimensional case
  • Class 17 (Wednesday, October 28): Nonnegative isotropic curvature is preserved by the Ricci flow: the general case
  • Class 18 (Friday, October 30): Nonnegative isotropic curvature is preserved by the Ricci flow: the general case (continued)
  • Class 19 (Monday, November 2): Results from complex linear algebra; curvature conditions related to nonnegative isotropic curvature; the condition that M x R has nonnegative isotropic curvature
  • Class 20 (Wednesday, November 4): The condition that M x R^2 has nonnegative isotropic curvature; the condition that M x S^2 has nonnegative isotropic curvature; the logical implications among various curvature conditions
  • Class 21 (Friday, November 6): The condition that M x S^2 has nonnegative isotropic curvature is preserved by the Ricci flow
  • Class 22 (Monday, November 9): The pinching set construction; if M x R has positive isotropic curvature, then the Ricci flow evolves M to a manifold of constant curvature
  • Class 23 (Wednesday, November 11): Berger's classification of holonomy groups; locally symmetric and locally irreducible manifolds; Hamilton's theorem on three-manifolds with nonnegative Ricci curvature
  • Class 24 (Friday, November 13): A variant of Bony's maximum principle for degenerate elliptic equations
  • Class 25 (Monday, November 16): A variant of Bony's maximum principle for degenerate elliptic equations (continued); structure of manifolds with nonnegative isotropic curvature
  • Class 26 (Wednesday, November 18): TBA
  • Class 27 (Friday, November 20): TBA
  • Class 28 (Monday, November 30): TBA
  • Class 29 (Wednesday, December 2): TBA
  • Class 30 (Friday, December 4): TBA