This course is an introduction to analytical methods used widely in modern differential geometry and physics. It focuses on the study of solutions to PDEs on manifolds and its relationship to the underlying geometry. One of the purposes of the course is to highlight purely analytical phenomena behind many results in Riemannian geometry.
The course is oriented on students in Mathematics and Physics, and covers the modules:
WP34 (Fortgeschrittene Themen aus der Differentialgeometrie) and
WP30 (Fortgeschrittene Themen aus der Analysis und der Mathematischen Physik)
in the Mathematics Master Programme as well as the Master Programme in Theoretical and Mathematical Physics (TMP). It is worth 9 ECTS points.
The core module "Differenzierbare Mannigfaltigkeiten/Differential geometry". The knowledge of the more advanced modules, such as "Riemannian geometry" or "Partial Differential Equations", is beneficial, but not necessary.
Lectures will be given in English twice a week; 12.00-14.00 Tue and 14.00-16.00 Wed, Room B041.
The course covers the materail on basic principles (maximum principles, mean-value inequalities, Harnack inequalities, gradient estimates) for solutions of classical elliptic and parabolic equations on Riemannian manifolds. More advanced material includes Laplacian comparison theorems, the Cheeger-Gromol splitting theorem, Sobolev inequalities, and their applications. At the end of the course we plan to discuss the classical Yamabe problem on the existence of constant scalar curvature metrics in conformal classes on Riemannian manifolds.
For background on analysis:
1. Gilbarg, D., Trudinger, N. S. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp.
2. Jost, J. Partial differential equations. Second edition. Graduate Texts in Mathematics, 214. Springer, New York, 2007. xiv+356 pp.
For main material:
3. Li, P. Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press, Cambridge, 2012. x+406 pp.
4. Schoen, R., Yau, S.-T. Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. v+235 pp.
5. Chavel, I. Eigenvalues in Riemannian geometry. Pure and Applied Mathematics, 115. Academic Press, 1984. xiv+362 pp. ISBN: 0-12-170640-0
Exercise classes (Übungen):
The exercise classes will hold at 14.00-16.00 Fri, Room B041.
The problem sets for the exercise classes are posted on this web-page on Fridays and are due a week later by Friday 4pm. Students are encouraged to do as many problems in these sets as possible, since they comprise an important part of the course and similar problems are likely to appear on the exam.
There will be an oral exam in February. If you would like to take the exam please send an email to register and arrange the date and time.
1. Introduction: basic notions and concepts
1.1. Harmonic functions in the Euclidean space and their properties: Green's representation, Poisson formula, mean-value properties, Weyl's lemma, maximum principles, removal of isolated singularities, gradient estimates, Liouville principle, Harnack inequality.
1.2. Background on Sobolev spaces, Sobolev inequalities (with sharp constants), Trudinger inequality, Poincare inequalities.
1.3. General maximum principles and their applications: Hopf's strong maximum principle, apriori bounds, Alexandrov-Bakelman-Pucci, small volume maximum principle, maximum principle for Monge-Ampere, maximum principles for other non-linear equations.
1.4. Background on heat equation: gradient estimates, maximum principles, Harnack inequalities.
2. Laplace-Beltrami operator on Riemannian manifolds
2.1. Preliminaries on Laplace-Beltrami operator; Sobolev spaces on Riemannian manifolds.
2.2. Background on geodesic spherical coordinates, constant curvature spaces. Harmonic functions on constant curvature spaces.
2.3. Local comparison theorems under upper sectional curvature bound (volume comparison, local Laplacian comparison, mean-value comparison).
2.4. Global comparison theorems under lower Ricci curvature bound (volume comparison, distributional Laplacian comparison, mean-value comparison). Toponogov-Cheng theorem.
2.5. Cheeger-Gromoll-Toponogov splitting theorem and the structure of complete manifolds with non-negative Ricci curvature; Bochner formula and the properties of Busemann functions.
2.6. Analysis on spaces with a lower Ricci curvature bound I: Cheng-Yau gradient estimate and its applications, Harnack inequalities, Poincare inequalities (Li-Schoen and Buser theorems), mean-value inequality for subharmonic functions.
2.7. Analysis on spaces with a lower Ricci curvature bound II: Sobolev inequalities and their applications (Varopoulos-Aubin theorem, Ledoux rigidity, Bonnet-Meyers via Sobolev inequalities).
2.8. Green functions on Riemannian manifolds: existence theorems.
3. Heat equation on Riemannian manifolds
3.1. Heat equation and heat kernel on subdomains in complete manifolds: existence and relationship with Laplace eigenvalues.
3.2. Basic properties, asymptotics and Pleijel's recursion formulas.
3.3. Li-Yau gradient estimate and its applications; existence of heat kernels on non-compact manifolds.
3.4. Cheeger-Yau comparison theorems for heat kernels.