Introduction to Ricci-Flow

Lecture 1 (2020-04-17)

• Basic Notation for Riemannian manifolds; 
• Course outline;  
• Symmetries and traces of the Riemann curvature tensor; 
• Decomposition of the Riemann curvature tensor; 

Lecture 2 (2020-04-17)

• Definition of Ricci-Flow;
• Flow of Einstein metrics as basic examples;
• Quasilinear parabolic system;
• Role of diffeomorphisms; Ricci solitons; 

Lecture 3 (2020-04-24)

• Short time existence: 
• Parabolic systems;
• Hamilton's approach for parabolic systems with an Integrability condition. 

Lecture 4 (2020-04-24)

• DeTurck's approach to short time existence with an 
• Explicit choice of diffeomorphism;
• Relation to Hamilton's approach;
• Analogy to mean curvature flow; 

Lecture 5 (2020-04-30)

• Evolution equations for the Riemann-, Ricci- and Scalar curvature;
• Parabolic comparison principles for the scalar curvature;
• Lower bounds on scalar curvature; blow-up of positive scalar curvature; 

Lecture 6 (2020-04-30)

• Evolution of curvature derivatives;
• Proof that Ricci-flow can be extended as long as the curvature remains bounded;

Lecture 7 (2020-05-08)

• The case of positive scalar curvature plus "Pinching": General theorem that once pinching is established,  convergence to spherical spaceform follows;
• Evolution of gradient of scalar curvature and traceless Ricci tensor;

Lecture 8 (2020-05-08)

• Continuation of proof of general pinching theorem; estimate on gradient of scalar curvature; use of Myer's theorem to compare min/max of scalar curvature.

Lecture 9 (2020-05-15)

• Different rescaling techniques for Ricci-Flow, in particular by fixing the volume;
• Upper bound on scalar curvature after rescaling; examples of deteriorating injectivity bound;

Lecture 10 (2020-05-15)

• Estimate on injectivity radius and lower bound on scalar curvature; exponential convergence of solution to the rescaled equation; conclusion of main pinching theorem for positive curvature;

Lecture 11 (2020-05-22)

• The tensor maximum principle; estimate on Ricci curvature in dimension 3;

Lecture 12 (2020-05-22)

• Proof of the pinching estimate in dimension 3 for positive Ricci-curvature;
• comparison to mean curvature flow;

Lecture 13 (2020-05-29)

• The estimate of Hamilton Ivey for dimension 3; rescalings of Ricci-flow singularities in dimension 3 have non-negative sectional curvature; comparison to convexity estimate in mean curvature flow;

Lecture 14 (2020-05-29)

• The pinching theorems in higher dimensions – a survey;

Lecture 15 (2020-06-12)

• Two dimensions: longtime existence via potential estimates; the negative curvature case and the zero-curvature case;

Lecture 16 (2020-06-19)

• The Li-Yau Harnack inequality for the linear heat-equation; the Harnack inequality for 2-d Ricci flow;

• An entropy estimate for the curvature in 2-d Ricciflow;

Lecture 17 (2020-06-19)

• Convergence of Ricci-Flow on the 2-sphere: The curvature estimate, convergence to a soliton; only soliton on S^2 has constant curvature; conclusion of the 2-d case.

Lecture 18 (2020-06-26)

• Outline of the approach to singularities of Ricci-flow in the 3-d case; description of major problems and type of singularities to be expected; role of the “cigar”-solution; concept of “collapsing”;

Lecture 19 (2020-06-26)

• Relation between Sobolev-inequality and isoperimetric problem and non-collapsing; the logarithmic Sobolev inequality in Euclidean space; equivalence to lower bound of Perelman’s W-functional;

Lecture 20 (2020-07-03)

• Entropy and conjugate heat equation; F-functional as derivative of entropy; monotonicity of the F-functional of Perelman;

Lecture 21 (2020-07-10)

• Ricci-flow as a gradient flow of the F-functional; eigenvalue interpretation of minimum of the F-functional; monotonicity of W-functional; existence of lower bound and existence of minimizer to the W-functional; analogies to Yamabe problem;

Lecture 22 (2020-07-17)

• Proof of Perelman’s non-collapsing estimate via monotonicity of the W-functional

Lecture 23 (2020-07-24)

• Non-collapsing implies lower bound on injectivity radius which implies compactness theorem;
• Rescaling of singularities and preliminary classification of singularities in dimension 3;
• Outlook to further steps in the proof of the Poincare conjecture.