Min-Max Problems for Families of Cycles in Riemannian Manifolds

黎曼流形中循环族的最小-最大问题

• 批准号: 1711053
• 负责人: Yevgeny Liokumovich
• 金额: $ 15.17万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1711053&HistoricalAwards=false
• 关键词: Min; Max; Problems; Families; Cycles

Award: DMS 1711053, Principal Investigator: Yevgeny LiokumovichA minimal surface is the mathematical idealization of a soap film spanning a wire, which minimizes surface area within the family of spanning surfaces. The min-max theory for minimal surfaces and other variational problems is modeled on a description of an efficient path over a mountain range that goes through a mountain pass: among nearby choices for a road over a mountain, the efficient choice will minimize the maximum altitude attained. A min-max method was developed in the 1960s and 1970s to study existence and other questions for minimal surfaces and has been made more useable in recent years.These projects address four areas of current min-max theory. An investigation of index and multiplicity bounds is expected to have applications to Heegard surfaces in non-Haken 3-manifolds A second project is intended to develop optimal bounds for min-max families of cycles with integer coefficients and may lead to a related, conjectural parametric coarea inequality. Min-max minimal hypersurfaces in dimensions eight or more may have singularities; a third project will aim to show that for a generic set of metrics on an 8-manifold, smooth minimal hypersurfaces may be constructed. A fourth project concerns equidistributional properties of k-parameter sweepout constructions of minimal hypersurfaces.

奖项：DMS 1711053，首席研究员：Yevgeny Liokumovicha最小表面是跨越电线的肥皂膜的数学理想化，可最大程度地减少跨度表面家族中的表面积。最小表面和其他变分问题的最小表面理论的模型是在对山脉穿过山脉的有效路径上的描述上建立的：在山上的附近选择附近的选择中，有效的选择将最大程度地减少达到的最大高度。 在1960年代和1970年代开发了一种最小的方法，以研究生存和其他最小表面的问题，并在近年来更有可用。这些项目涉及当前最小值理论的四个领域。 对指数和多样性界限的研究预计将在非生存的3个manifolds中对Heegard表面进行应用。第二个项目旨在为具有整数系数的最小循环中的最佳范围开发最佳界限，并可能导致相关的，猜想的参数coarea coarea coarea不平等。 最小最小的高度曲面八个或更多，可能具有奇异性。第三个项目的目的是表明，对于在8个manifold，平滑，最小的超曲面上的一套通用指标中。 第四个项目涉及K-Parameter扫描最小曲面的构造的等分分配特性。