Min-Max Problems for Families of Cycles in Riemannian Manifolds

奖：DMS 1711053，首席研究员：Yevgeny Liokumovich极小曲面是跨越金属丝的肥皂膜的数学理想化，它使跨越曲面族中的表面积最小化。极小曲面和其他变分问题的最小-最大理论是建立在描述穿过山口的山脉上的有效路径的基础上的：在附近的翻山道路选择中，有效的选择将使所达到的最大海拔最小化。20世纪60年代和70年代发展了一种极小极大方法，用于研究极小曲面的存在性和其他问题，并在最近几年变得更加实用。这些项目涉及当前极小极大理论的四个方面。指数和多重界的研究可望应用于非Haken三维流形中的Heegard曲面。第二个项目旨在发展具有整数系数的极小-极大圈族的最优界，并可能导致一个相关的、猜想的参数余面积不等式。8维或更多维的极小极小超曲面可能有奇点；第三个项目的目标是证明对于8维流形上的一组一般度量，可以构造光滑的极小超曲面。第四个项目涉及极小超曲面的k参数扫掠结构的等分布性质。

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.