Geometric Variational Theory and Application

该研究计划的主要课题是跨越封闭电线和肥皂泡的肥皂膜的数学模型。根据最小作用量原理，在所有横跨金属丝的膜或包括固定体积的所有气泡中，肥皂膜或肥皂泡的表面积将最小化。这种表面的物理特性表明，肥皂膜和肥皂泡分别是面积或面积减去封闭体积的临界点。在数学上，一个肥皂膜横跨一条线被称为最小曲面，肥皂泡被称为常数平均曲率曲面（缩写为CMC）。这两类曲面早在18世纪60年代就引起了物理学家和数学家的兴趣，并得到了广泛的研究，这也激发了数学和科学中许多其他学科的进步。一个一般的理论证明存在这样的几何对象，所谓的“最小最大理论”，最近取得了惊人的成功，并将在本研究计划的主要研究对象。 更具体地说，PI将研究最小-最大理论及其在CMC曲面和具有自由边界的极小曲面上的应用。在第一个主题中，PI打算证明在给定的三个流形中的Heegaard亏格的min-max CMC曲面的拓扑界，以及证明对于给定的平均曲率的多个CMC曲面的存在性。PI还计划建立最小-最大理论，用于构造具有由任意光滑函数规定的平均曲率的曲面，推广曲率函数为常数的CMC曲面。在第二个主题中，PI将研究自由边界min-max极小曲面的紧性，适当性和莫尔斯指数上界;作为应用，PI计划通过使用自由边界解的近似来研究奇异或非紧空间中的极小曲面。PI将完成一个利用调和映射理论构建具有自由边界的最小-最大最小圆盘的项目。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The main topics of this research program are mathematical models for soap films spanning a closed wire and soap bubbles. By the least action principle, the surface area of a soap film or a soap bubble will minimize among all films spanning the wire or among all bubbles including a fixed volume. The physical characterization of such surfaces says that the soap film and the soap bubble are respectively critical points of area or area subtracting enclosed volume. Mathematically, a soap film spanning a wire is called a minimal surface, and a soap bubble is called a surface of constant mean curvature (abbreviated as CMC). These two types of surfaces already caught interests by physicists and mathematicians in 1760s, and have been extensively-studied topics which also inspired the advances of many other subjects in mathematics and science. One general theory for proving the existence of such geometric objects, called the "min-max theory", has had striking recent successes, and will be the major object of study in this research program. More specifically, the PI will conduct research on the min-max theory and its applications for CMC surfaces and minimal surfaces with free boundary. In the first subject, the PI intends to prove topological bounds for the min-max CMC surfaces by Heegaard genus in a given three manifold, as well as to prove the existence of multiple CMC surfaces for a given mean curvature. The PI also plans to establish the min-max theory for constructing surfaces with mean curvature prescribed by an arbitrary smooth function, generalizing that of the CMC surfaces where the curvature functions are constants. In the second subject, the PI will investigate the compactness property, properness property, and Morse index upper bounds of the free boundary min-max minimal surfaces obtained by the PI with collaborators before; as applications, the PI plans to study minimal surfaces in singular or non-compact spaces by approximations using the free boundary solutions. The PI will finish a program on constructing min-max minimal disks with free boundary using the theory of harmonic maps.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.