Lecture Series: Introduction to Ricci-Flow, Course at Tübingen University, Summer 2020

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; 
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Lecture 2 (2020-04-17)

  • Definition of Ricci-Flow;
  • Flow of Einstein metrics as basic examples;
  • Quasilinear parabolic system;
  • Role of diffeomorphisms; Ricci solitons; 
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Lecture 3 (2020-04-24)

  • Short time existence: 
  • Parabolic systems;
  • Hamilton's approach for parabolic systems with an Integrability condition. 
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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; 
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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; 
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Lecture 6 (2020-04-30)

  • Evolution of curvature derivatives;
  • Proof that Ricci-flow can be extended as long as the curvature remains bounded;
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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;
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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.
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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;
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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;
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Lecture 11 (2020-05-22)

  • The tensor maximum principle; estimate on Ricci curvature in dimension 3;
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Lecture 12 (2020-05-22)

  • Proof of the pinching estimate in dimension 3 for positive Ricci-curvature;
  • comparison to mean curvature flow;
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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;
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Lecture 14 (2020-05-29)

  • The pinching theorems in higher dimensions – a survey;
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Lecture 15 (2020-06-12)

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

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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.
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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”;
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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;
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Lecture 20 (2020-07-03)

  • Entropy and conjugate heat equation; F-functional as derivative of entropy; monotonicity of the F-functional of Perelman;
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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;
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Lecture 22 (2020-07-17)

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

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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.
Watch online (81 min) Download (1,31 GB)

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