Universität Wien

250124 VO Topics in Geometric Analysis (2015S)

Topics in minimal surface theory

5.00 ECTS (3.00 SWS), SPL 25 - Mathematik

Details

Sprache: Deutsch

Prüfungstermine

Lehrende

Termine (iCal) - nächster Termin ist mit N markiert

Dienstag Termine 14:15-15:00 finden im Sprechungszimmer im 2. Stock statt!

  • Montag 09.03. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 09.03. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 16.03. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 16.03. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 23.03. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 23.03. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 13.04. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 13.04. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 20.04. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 20.04. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 27.04. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 27.04. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 04.05. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 04.05. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 11.05. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 11.05. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 18.05. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 18.05. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 01.06. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 01.06. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 08.06. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 08.06. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 15.06. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 15.06. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 22.06. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 22.06. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 29.06. 13:15 - 14:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Montag 29.06. 14:15 - 15:00 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock

Information

Ziele, Inhalte und Methode der Lehrveranstaltung

Syllabus:

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 possibly including a discussion of pseudoholomorphic curves and the theory of horizons in general relativity.

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. Our next goal is a a slick derivation of the curvature estimates for stable minimal hypersurfaces of E. Heinz, R. Schoen, L. Simon, and S.-T. Yau. This will be a crucial ingredient for an unusual proof of existence of area minimizing hypersurfaces spanning a given boundary in $\mathbb{R}^{n+1}$ where $2 \leq n \leq 4$. We will completely dodge geometric measure theory in this, but clearly recognize its advent as a necessary and logical development. The class will conclude with selected topics in minimal surface theory that suit the gusto of the audience.

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. You should be prepared to work very hard to take advantage of this class.

Art der Leistungskontrolle und erlaubte Hilfsmittel

Jointly for VO+PS: There will be a thorough 30-minute oral exam based in part on exercises that I suggest in the course of the semester.

Mindestanforderungen und Beurteilungsmaßstab

Prüfungsstoff

Literatur

Zuordnung im Vorlesungsverzeichnis

MGEV, MANV

Letzte Änderung: Mo 07.09.2020 15:40