Partial Differential Equations in Riemannian Geometry

拟议的研究涉及几何学和偏微分方程式的十字路口问题。近年来，涉及偏微分方程的方法在几何学中变得越来越重要：也许最引人注目的例子是理查德·汉密尔顿首创的Ricci流方法和Grigoriy Perelman对Poincare猜想的解；在理解极小曲面方面也取得了其他重要进展。这些表面的面积最小，可以作为肥皂膜的模型。这个项目的目标是加深我们对这些方程和类似方程的理解。具体地说，PI和Gerhard Huisken正计划研究几何流动的奇点形成。除了Ricci流，一个特别的焦点是2-凸超曲面的流。PI还计划研究极小曲面理论中的问题。在这个方向上，他计划研究极小曲面上拉普拉斯算子的特征值；另一个方向是具有自由边界的极小环的分类。最后，PI计划研究用胶合方法构造几何偏微分方程组的解。

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.