Minimal surfaces and quantitative topology of the space of cycles

循环空间的最小曲面和定量拓扑

• 批准号: RGPIN-2019-06912
• 负责人: Liokumovich, Yevgeniy
• 金额: $ 2.26万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=739181
• 关键词: Minimal; surfaces; quantitative; topology; space

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.

拟议的研究计划涉及黎曼流形中最小曲率曲面和恒定曲率曲面的存在性和性质。肥皂泡和磁场中带电粒子的轨迹是这些现象的一些现实例子。更一般地说，极小曲面理论中的问题与偏微分方程、广义相对论、工程学和许多其他领域中的许多问题密切相关。  20 世纪 60 年代，阿尔姆格伦 (Almgren) 提出了一种在封闭流形中构建最小曲面的新技术，该技术现在被称为阿尔姆格伦-皮茨最小-最大理论。在过去的几年里，最小-最大理论经历了复兴。马克斯和内维斯对威尔莫尔猜想的证明是一项了不起的成就，随后出现了许多其他重要结果，回答了许多长期存在的猜想。目前的研究计划集中于与最小-最大理论中的问题以及几何和拓扑中的应用相关的循环族的一些基本问题。该研究项目位于几何变分法和定量拓扑学的交汇处。它包含许多与黎曼流形中平循环空间的几何性质以及满足某些特殊条件的循环族的存在性有关的问题，其目的是获得有关最小-最大最小超曲面的正则性、几何和拓扑的信息。该项目依赖于 Gromov、Guth、Marques 和 Neves 开发的技术和想法，以及来自拓扑、几何测量理论和分析的其他想法。