Partial Differential Equations in Riemannian Geometry

点击翻译按钮获取中文摘要

The proposed research is concerned with questions at the crossroads of geometry and partial differential equations. In recent years, methods involving partial differential equations have become increasingly important in geometry: Perhaps the most spectacular example is the Ricci flow method pioneered by Richard Hamilton and the solution of the Poincare conjecture by Grigoriy Perelman; other important advances were made in the understanding of minimal surfaces. These are surfaces which minimize area and serve as models for soap films. The goal of this project is to further our understanding of these and similar equations.Specifically, the PI and Gerhard Huisken are planning to study singularity formation for geometric flows. Besides the Ricci flow, a particular focus are flows for 2-convex hypersurfaces. The PI is also planning to work on problems in minimal surface theory. In this direction, he is planning to study the eigenvalues of the Laplace operator on a minimal surface; another direction is the classification of minimal annuli with free boundary. Finally, the PI is planning to study the gluing approach to the construction of solutions to geometric PDEs.