Michael Eichmair: Katalogdaten im Frühjahrssemester 2012 |  |
| Name | Herr Dr. Michael Eichmair |
| Departement | Mathematik |
| Beziehung | Assistenzprofessor |
| Nummer | Titel | ECTS | Umfang | Dozierende | |
|---|---|---|---|---|---|
| 401-3492-12L | Topics in Geometric Analysis | 6 KP | 3G | M. Eichmair | |
| Kurzbeschreibung | Classical theory for minimal and constant mean curvature surfaces from a modern perspective, including the Bernstein theorem for minimal graphs, curvature estimates for stable minimal surfaces, the Bernstein theorem for complete stable minimal immersions, interior gradient estimates, and the Alexandrov-Hopf theorem. Sketch of one of the deeper existence and regularity theorems for such surfaces. | ||||
| Lernziel | A synthetic presentation of techniques in minimal surface theory with an eye towards "modern applications" (blow up techniques, "tangent object" analysis, monotonicity formulae). The goal of the lecturer is to add to the toolkit of students hoping to do research work in geometric analysis. | ||||
| Inhalt | (Tentative) - Review of the theory of isometric immersions into Euclidean space. - Minimal immersions. First and second variation. - The monotonicity formula. Proof of a "smooth" version of Allard's theorem. - The Choi-Schoen curvature estimates (curvature bounds from energy bounds). Sketch that the space of minimal embeddings in closed three-manifolds with positive Ricci curvature is compact. This is very similar to regularity results for harmonic maps and pseudoholomorphi curves. - Review and brief discussion of the De Giorgi-Nash-Moser theory for elliptic equations in divergence form. - Discussion of Jacobi fields and stability of minimal hyper surfaces. The Bernstein theorem, due to Fischer-Colbrie and Schoen, for stable minimal immersions into $R^3$. - Reilly's formula and Reilly's proof of the Alexandrov theorem. - Simons' identity. The Simons' cone. - Curvature estimates of Schoen-Simon-Yau for stable minimal hypersurfaces in low dimensions. - Interior gradient estimates for minimal graphs. - Overview of the proof of one of the following theorems: Douglas/Rado solution of the Plateau problem, De Giorgi's proof that the reduced boundary of a least perimeter set is smooth, Allard's theorem, Schoen-Simon regularity theory for stable minimal surfaces, Leon Simon's proof of the existence of minimizers for the Willmore functional. | ||||
| Literatur | Lecture notes and references to the literature will be provided. | ||||
| 401-5350-00L | Analysis Seminar  | 0 KP | 1K | M. Struwe, F. Da Lio, M. Eichmair, N. Hungerbühler, T. Kappeler, T. Rivière, D. A. Salamon | |
| Kurzbeschreibung | Forschungskolloquium | ||||
| Lernziel | |||||
| Inhalt | Research seminar in Analysis | ||||