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Analysis on Manifolds 

WS 2016/17    17.10.2016 - 10.02.2017

Link to the previous course "Analysis of Elliptic Differential Equations" 

Exams

1. Analysis of Elliptic Differential Equations:  Fr 03.02.2017   12:00-14:00   V3-201
2. Analysis on Manifolds:  Di   07.02.2017  16:00-18:00  T2-227

Lectures

Di  16-18   T2-227
Do 12-14   T2-204

Lecture notes

Tutorials

Fr 16-18  V4-112   Tutor: Eryan Hu

Exercises

The problems marked by * are additional. They do bring points to those who solve them but they do not contribute to the maximal possible number of points. 

Contents of the course

1. Riemannian manifolds
Smooth manifolds and tangent vectors. Submanifolds. Riemannian metric. Riemannian measure. Divergence theorem. Laplace-Beltrami operator. Weighted manifolds. Product manifolds. Polar coordinates in Rn , Hn , Sn . Model manifolds. Length of paths and the geodesic distance. Smooth mappings and isometries.

2. Weak Laplace operator and spectrum
Weak gradient and Sobolev spaces on Riemannian manifolds. Weak Laplacian. Regularity theory. Compact Embedding Theorem. The Dirichlet problem in precompact domains and its resolvent. The eigenvalue problem. Positivity of the bottom eigenvalue.

3. The heat semigroup
Caloric functions. The initial-boundary problem for the heat equation in precompact domains: uniqueness and existence of solution. The heat semigroup. Smoothness of solutions. Weak maximum principle for caloric functions. Markovian properties of the heat semigroup. The heat kernel in precompact domains.  
4. The heat kernel on a manifold
Construction of the global heat semigroup by exhaustion. Existence and smoothness of the heat kernel. Heat kernel as a fundamental solution of the heat equation. Heat kernels on model manifolds. Heat kernels and change of measure. The heat kernel in H3. The heat kernel in S1.

5*. Stochastic completeness
Bounded Cauchy problem and stochastic completeness. Geodesically complete manifolds. Geodesic balls. Volume test for stochastic completeness. Stochastic completeness of model manifolds.

6*. Integrated estimates of the heat kernel
The integrated maximum principle with the Gaussian weight. The Davies-Gaffney inequality. Integrated estimate of the heat kernel.  Application to eigenvalue estimates.

Literature

  1. Chavel I., Riemannian geometry : a modern introduction, Cambridge University Press, 1993.
  2. Chavel I., Eigenvalues in Riemannian geometry, Academic Press, 1984.
  3. Grigor'yan A., Heat kernel and analysis on manifolds, AMS/IP, 2009.
  4. Rosenberg S., The Laplacian on a Riemannian manifold, Cambridge University Press, 1997.