PDE Problems in Geometry

本文主要研究几何流动问题，特别是里奇流和平均曲率流。具体来说，我们计划研究平均曲率流下的辛形态的演化。特别地，我们想要找到保证平均曲率流从给定的辛形态演化为生物全纯等距的条件。PI还计划研究里奇流的自相似解。PI最近得到了Bryant孤子在非坍缩假设下的唯一性结果。尝试构造自相似的解似乎很有趣它们是坍缩的，但不是旋转对称的。最后，PI计划以Andrews和Clutterbuck最近的工作为基础，研究薛定谔算子的第一和第二特征值之间的差距。这个项目的主要目标是利用分析工具，特别是偏微分方程，来解决微分几何中的问题。这种方法在过去取得了成功，并导致了该领域几个主要开放问题的解决，包括Min-Oo猜想、Yamabe问题的紧性猜想和可微球定理。拉格朗日平均曲率流由于与斯特罗明格-尤尔-扎斯洛猜想的联系，为数学物理提供了潜在的应用。

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.