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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

Mo 12-14 U2-232 Fr 10-12 U2-113
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

Blatt 3

Blatt 4

Blatt 5

Blatt 6

Blatt 7

Blatt 8

Blatt 9

## 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
**R**^{n}^{
}, **H**^{n
}, **S**^{n
}.

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 **H**^{3}.
The heat kernel in **S**^{1}.

**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

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