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Analysis on Manifolds 

SS 2024    08.04.2024 - 19.07.2024

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

Lectures

Lecture notes

Exercises

Deadline for submission of homeworks is always Friday. Joint submission of homework by groups of students is not allowed.
At least 50% of points for homework is required for admission to the exam. 
This percentage is calculated as A/M, where A is the number the earned points and M is the maximum possible number of points. 
The exercises marked with *  are taken into account in A, but not in M.

Blatt 0+1
Blatt 2

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.