The course is an introduction to the eigenvalue problems on Riemannian manifolds, an important part of the modern geometric analysis with deep links to many classical problems. The programme covers a number of key results and develops basic techniques, currently used in the subject. It is designed for students specialising in geometry and/or analysis, and is particularly suitable for students taking the course on 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 6 ECTS points.
The core module "Differenzierbare Mannigfaltigkeiten/Differential geometry".
Lectures will be given in English once a week;
New time and room:12.00-14.00 Thu, Room B134.
The course starts with an exposition of basic facts on eigenvalue problems, covering such topics as Weyl's asymptotics formula. min-max methods, comparison theorems in geometry and analysis. Later we plan to study isoperimetric inequalities for eigenvalues and heat kernel on Riemannian manifolds.
1. Chavel, I. Eigenvalues in Riemannian geometry. Pure and Applied Mathematics, 115. Academic Press, 1984. xiv+362 pp. ISBN: 0-12-170640-0
2. Chavel, I. Riemannian geometry. A modern introduction. Second edition. Cambridge Studies in Advanced Mathematics, 98. Cambridge University Press, Cambridge, 2006. xvi+471 pp. ISBN: 978-0-521-61954-7; 0-521-61954-8
Exercise classes (Übungen):
The exercise classes will hold at 14.00-16.00 Fri, Room B045.
The problem sets for the exercise classes are posted on this web-page on Fridays and are due a week later by Friday 12am. 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 at the end of July. If you would like to take the exam please send an email to register and arrange the date and time. The registration deadline is 4 July 2014.
1. Introduction: background material and examples
1.0. Laplace operator on Riemannian manifolds.
1.1. Basic facts on eigenvalue problems. Weyl's asymptotic formula.
1.2. Variational principles and their consequences.
1.3. Courant's nodal domain and Pleijel's theorems.
1.4. Eigenvalue problems on disks in constant curvature spaces I: separation of variables
1.5. Eigenvalue problems on disks in constant curvature spaces II: bounds and qualitative behaviour
2. Comparison theorems and isoperimetric inequalities
2.1. Comparison theorems I (upper sectional curvature bound): Gunter-Bishop's volume comparison, Cheng's eigenvalue comparison, McKean's theorem.
2.2. Comparison theorems II (lower Ricci curvature bound): background on cut-locus and volume of metric balls, Bishop's volume comparison, Gromov's relative volume comparison, Cheng's eigenvalue comparison and their applications.
2.3. Euclidean isoperimetric inequality and Faber-Krahn theorem.
2.4. Cheeger isoperimetric constants and eigenvalue bounds; applications.