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Course description: 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. The course is oriented on students in Mathematics and Physics, and is one of the modules in the Mathematics Master Programme as well as the Master Programme in Theoretical and Mathematical Physics (TMP); it is worth 9 ECTS points. Pre-requisites: The core module "Differenzierbare Mannigfaltigkeiten/Differential geometry". The knowledge of the module "Riemannian geometry" is beneficial, but not necessary. Lectures schedule: Lectures will be given in English twice a week; 10.00-12.00 Mon and 10.00-12.00 Wed, Room B 040. Course outline: The course covers the materail on basic principles (maximum principles, mean-value inequalities, Harnack inequalities, unique continuation), properties of harmonic functions on manifolds (existence, gradient estimates, Liouville principle), geometric eigenvalue problems, and harmonic maps. Reading list: 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 harmonic functions in geometry: 3. Li, P. Harmonic functions and applications to complete manifolds. Lecture notes, 2004. 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. For eigenvalue problems in geometry: 5. Chavel, I. Eigenvalues in Riemannian geometry. Including a chapter by Burton Randol. With an appendix by Jozef Dodziuk. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984. xiv+362 pp. 6. 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. For harmonic maps: 7. Hélein, F. Harmonic maps, conservation laws and moving frames. Translated from the 1996 French original. With a foreword by James Eells. Second edition. Cambridge Tracts in Mathematics, 150. Cambridge University Press, Cambridge, 2002. xxvi+264 pp. 8. Jost, J. Two-dimensional geometric variational problems. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1991. x+236 pp. Exercise classes (Übungen): The exercise classes will hold at 10.00-12.00 Fri, Room B040. The problem sets for the exercise classes are posted on this web-page on Wednesdays and are due a week later by Wednesday 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.
Final Exam: There will be an oral exam during the period 11-15 Feb. If you would like to take the exam please register by sending an email. The registration deadline is 28 Jan. |
Course Programme: 1. Introduction 1.1. Euler-Lagrange equations as examples of PDEs: notion of a variational problem, Euler-Lagrange equations, examples, natural boundary conditions. 1.2. 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. 1.3. Sub-harmonic functions and Perron method: existence theorems based on maximum principle. 1.4. Background on Sobolev and Holder spaces, Poincare inequality, Sobolev embedding and Sobolev inequality, Campanato's theorem and its consequences. 2. Laplace-Beltrami operator on Riemannian manifolds: basic facts and geometric perspectives 2.1. Preliminaries on Laplace-Beltrami operator; Sobolev spaces on manifolds. 2.2. Background on geodesic spherical coordinates, constant curvature spaces. Harmonic functions on constant curvature spaces: Green's representation, mean-value properties, Harnack inequalities. 2.3. Comparison theorems for sectional and Ricci curvature: volume comparison, Laplacian and mean-value comparison theorems. Application to the Cheng-Toponogov theorem. 2.4. Splitting theorem and the structure of manifolds with non-negative Ricci curvature; Bochner formula and the properties of Busemann functions. 3. Further analytic methods 3.1. 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. 3.2. Moser estimates and Harnack inequality; Moser iteration method. Applications, including Holder continuity of solutions. 3.3. Existence of minimisers of certain quadratic like functionals. The Dirichlet problem for harmonic functions on compact Riemannian manifolds. 3.4. Elements of the regularity theory: Campanato's approach to de Giorgi's theorem. 4. Harmonic functions on Riemannian manifolds 4.1. Spaces with lower Ricci curvature bound I: Cheng-Yau gradient estimate and its applications. 4.2. Spaces with lower Ricci curvature bound II: Poincare inequality, mean-value inequality for subharmonic functions. 4.3. Harmonic functions on manifolds of negative curvature: survey of some results. 5. Eigenvalue problems on Riemannian manifolds 5.1. Eigenvalues problems and basic principles: Rayleigh theorem, Courant's nodal domain theorem, and their consequences. 5.2. Comparison theorems for eigenvalues: Cheng's theorems, Lichnerowicz-Obata and Li-Yau theorems. 5.3. Eigenvalue problems on Riemannian surfaces: Hersch'es isoperimetric inequality, Yang-Yau inequality, notion of conformal volume. |