Minimal surfaces and quantitative topology of the space of cycles

所提出的研究方案涉及黎曼流形中极小曲率曲面和常曲率曲面的存在性和性质。肥皂泡和带电粒子在磁场中的轨迹就是这些现象的一些真实例子。更广泛地说，极小曲面理论中的问题与偏微分方程组、广义相对论、工程等许多领域的许多问题密切相关。20世纪60年代，Almgren提出了一种构造闭流形中极小曲面的新方法，现在称为Almgren-Pitts Min-Max理论。在过去的几年里，最小-最大理论经历了复兴。马奎斯和内维斯对威尔莫尔猜想的证明是一项了不起的成就，随后又有许多其他重要的结果，回答了一些长期存在的猜想。目前的研究方案集中在关于循环族的一些基本问题上，涉及到极小极大理论中的问题以及在几何和拓扑学中的应用。它包含了一些与黎曼流形中平坦圈空间的几何性质以及满足某些特殊条件的圈族的存在性有关的问题，目的是得到关于极小-极大极小超曲面的正则性、几何和拓扑的信息。该项目依赖于格罗莫夫、古斯、马奎斯和内维斯开发的技术和思想，以及来自拓扑学、几何测量理论和分析的其他思想。

The proposed research program concerns existence and properties of minimal and constant curvature surfaces in Riemannian manifolds. Soap bubbles and trajectories of charged particles in a magnetic field are some of the real world examples of these phenomena. More generally, problems in minimal surface theory are closely related to many problems in partial differential equations, general relativity, engineering and many other areas. ******In 1960s Almgren suggested a new technique for the construction of minimal surfaces in closed manifolds, which is now known as the Almgren-Pitts Min-Max Theory. In the last several years the Min-Max Theory has experienced a renaissance. The proof of the Willmore conjecture by Marques and Neves was one remarkable achievement with many other important results that followed, answering a number of long-standing conjectures. Current research proposal is focused on some fundamental questions about families of cycles in connection with problems in Min-Max Theory and with applications in geometry and topology.******This research project lies at the interface of Geometric calculus of variations and quantitative topology. It contains a number of problems related to the geometric properties of the space of flat cycles in a Riemannian manifold and existence of families of cycles satisfying certain special conditions with the goal of obtaining information about regularity, geometry and topology of min-max minimal hypersurfaces. The project relies on techniques and ideas developed by Gromov, Guth, Marques and Neves, as well as other ideas from topology, geometric measure theory and analysis.**