Partial Differential Equations in Riemannian Geometry

提出的研究是关于在几何和偏微分方程的十字路口的问题。近年来，涉及偏微分方程的方法在几何中变得越来越重要：也许最引人注目的例子是理查德·汉密尔顿（Richard Hamilton）开创的里奇流方法和格里高利·佩雷尔曼（Grigoriy Perelman）对庞加莱猜想的求解；其他重要的进展是对最小表面的理解。这些表面将面积最小化，并作为肥皂膜的模型。这个项目的目标是进一步加深我们对这些方程和类似方程的理解。具体来说，PI和Gerhard Huisken正计划研究几何流的奇点形成。除了Ricci流，特别关注的是2凸超曲面的流。PI还计划研究最小曲面理论中的问题。在这个方向上，他计划研究最小曲面上拉普拉斯算子的特征值；另一个方向是自由边界最小环空的分类。最后，PI计划研究粘接方法来构造几何偏微分方程的解。

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.