Min-Max Problems for Families of Cycles in Riemannian Manifolds

奖项：DMS 1711053，首席研究员：Yevgeny liokumovich最小表面是横跨电线的肥皂膜的数学理想化，它最小化了跨越表面家族中的表面积。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有效路径的描述上的：在一条穿过山口的道路的附近选择中，有效选择将使所达到的最大高度最小。最小-最大方法是在20世纪60年代和70年代发展起来的，用于研究最小曲面的存在性和其他问题，近年来得到了更广泛的应用。这些项目涉及当前最小-最大理论的四个领域。对指数和多重界的研究有望应用于非haken 3流形中的Heegard曲面。第二个项目旨在为具有整数系数的最小-最大循环族开发最优界，并可能导致相关的推测参数共面积不等式。8维以上的最小-最大超曲面可能有奇点；第三个项目将旨在证明对于8流形上的一般度量集，可以构造光滑的最小超曲面。第四个项目涉及最小超曲面的k参数扫描构造的等分布性质。

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.