Min-Max Problems for Families of Cycles in Riemannian Manifolds

奖项：DMS 1711053，主要研究者：Yevgeny Liokumovich最小表面是跨越导丝的肥皂膜的数学理想化，其最小化跨越表面家族内的表面积。最小曲面和其他变分问题的最小-最大理论是建立在一条穿过山口的山脉上的有效路径的描述上的：在附近的山区道路选择中，有效的选择将使所达到的最大高度最小化。 极小极大方法是在20世纪60年代和70年代发展起来的，用于研究极小曲面的存在性和其他问题，近年来变得更加实用。 指数和重数界的调查，预计有应用到Heegard曲面在非哈肯3-流形第二个项目是打算开发最佳的界限，最小最大家庭的周期与整数系数，并可能导致相关的，几何参数coarea不等式。 8维或更高维的最小-最大极小超曲面可能有奇点;第三个项目旨在证明，对于8维流形上的一组通用度量，可以构造光滑的极小超曲面。 第四个项目关注的是极小超曲面的k-参数sweepout构造的等分布性质。

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.