Geometric Analysis; Minimal Surfaces, Geometric Flows, and Function Theory

近年来，极小曲面理论在许多长期存在的问题上取得了突破，许多数学家作出了重要贡献。最小曲面是测地线的自然推广。它是一个局部面积最小的曲面。层压定理、单侧曲率估计、嵌入曲面的Calabi-Yau猜想的解决以及固定格曲面的结构结果（都是PI和Bill Minicozzi的共同工作）在最近的突破中发挥了关键作用，并被许多人使用。PI提议与Bill Minicozzi一起，继续我们对最小曲面和几何分析相关领域的研究，包括几何演化方程，如平均曲率和里奇流，以及函数理论。我们还提出了与Bruce Kleiner关于平均凸集的平均曲率流的联合方案和与Nancy Hingston关于测地线莫尔斯指数界的联合方案。最后，我们提出了一个与Camillo De Lellis关于Min-Max结构和应用的项目，以及与Camillo De Lellis和Bill Minicozzi关于通用度量的联合项目。

Recent years have seen breakthroughs on many long-standing problems in the theory of minimal surfaces, with important contributions from many mathematicians. A minimal surface is a natural generalization of geodesics. It is a surface that locally minimize area. The lamination theorem, the one-sided curvature estimate, the resolution of the Calabi-Yau conjectures for embedded surfaces, and the structure results for fixed genus surfaces (all joint work of the PI and Bill Minicozzi) have played a key role in recent breakthroughs and have been used by many people.The PI proposes, jointly with Bill Minicozzi, to continue our investigations on minimal surfaces and related areas of geometric analysis, including geometric evolution equations such as the mean curvature and Ricci flow, and on function theory.We also propose a joint project with Bruce Kleiner about mean curvature flow of mean convex sets and a joint project with Nancy Hingston about Morse index bounds of geodesics. Finally, we propose a project with Camillo De Lellis about Min-Max constructions and applications and a joint project with Camillo De Lellis and Bill Minicozzi on generic metrics.