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250047 VO Topics in Geometric Analysis (2022S)

3.00 ECTS (2.00 SWS), SPL 25 - Mathematik

Moodle

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Note: The time of your registration within the registration period has no effect on the allocation of places (no first come, first served).

Details

Language: English

Examination dates

Lecturers

Michael Eichmair

Classes (iCal) - next class is marked with N

Wednesday 09.03. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 16.03. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 23.03. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 30.03. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 06.04. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 27.04. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 04.05. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 11.05. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 18.05. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 25.05. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 01.06. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 08.06. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 15.06. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 22.06. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 29.06. 13:15 - 14:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

Soap films have every intention to span their boundary with the least amount of area possible. The development of their mathematical theory and its generalization is a cornerstone of twentieth century mathematics. My goal in this class is to give a reasonably self-contained introduction to this theory of "minimal surfaces" and to illustrate its applications in several areas of mathematics and physics.
To get us started, I will review aspects of the theory of submanifolds of Euclidean space and Riemannian geometry. We then turn to the classical mathematical theory of soap films spanning a given boundary as developed by J. Douglas (who received one of the first two Fields medals for this contribution) and T. Rado. This discussion will contain a proof of the uniformization theorem in complex analysis as a special case. We then turn to properties of general minimal hypersurfaces including the crucial monotonicity formula and a discussion of minimal graphs. Time permitting, we will discuss a slick derivation of the curvature estimates for stable minimal hypersurfaces of E. Heinz, R. Schoen, L. Simon, and S.-T. Yau.

Prerequisites:
It will be very useful to have familiarity with the basics of elliptic partial differential equations and differential geometry. The existence of solutions for the Dirichlet problem for the minimal surface equation will be used as a black box.

Assessment and permitted materials

There will be a thorough 30-minute oral exam based in part on exercises that I suggest in the course of the semester.

Minimum requirements and assessment criteria

ability to state, prove, discuss, and expand upon results covered in class

Examination topics

all topics discussed in class

Reading list

Association in the course directory

MGEV; MANV;

Last modified: We 14.06.2023 07:47