PDE Problems in Geometry

建议的研究涉及几何流领域中的问题，特别是Ricci流和平均曲率流。具体地说，我们计划研究辛同构在平均曲率流下的演化。特别地，我们想要找到保证平均曲率流将给定的辛同构演化为双全纯等距的条件。PI还计划研究Ricci流的自相似解。PI最近在非折叠假设下得到了布莱恩特孤子的唯一性结果。试图构造折叠的、但不是旋转对称的自相似解似乎很有趣。最后，PI计划在Andrews和Clutterak最近工作的基础上研究薛定谔算子的第一和第二本征值之间的差距。这个项目的主要目标是使用分析中的工具来解决微分几何问题，特别是偏微分方程。这种方法在过去取得了成功，并导致了该领域中几个主要公开问题的解决，包括Min-Oo猜想、Yamabe问题的紧性猜想和可微球定理。由于拉格朗日平均曲率流与Strominger-Yau-Zaslow猜想的联系，它在数学物理中具有潜在的应用。

The proposed research is concerned with problems in the field of geometric flows, in particular the Ricci flow and the mean curvature flow. Specifically, we plan to study the evolution of symplectomorphisms under the mean curvature flow. In particular, we want to find condition that guarantee that the mean curvature flow evolves a given symplectomorphism to a biholomorphic isometry. The PI also plans to study self-similar solutions to the Ricci flow. The PI has recently obtained a uniqueness result for the Bryant soliton under a noncollapsing assumption. It seems interesting to try to construct self-similar solutions which are collapsed, but not rotationally symmetric. Finally, the PI plans to study the gap between the first and second eigenvalue of a Schroedinger operator, building on recent work of Andrews and Clutterbuck.The main goal of this project is to approach problems in differential geometry using tools from analysis, especially partial differential equations. This approach has been successful in the past, and has led to the solution of several major open in the problems in the field, including Min-Oo's Conjecture, the Compactness Conjecture for the Yamabe problem, and the Differentiable Sphere Theorem. The Lagrangian mean curvature flow offers potential applications to mathematical physics, due to its connection with the Strominger-Yau-Zaslow conjecture.