Geometric Variational Theory and Application

本研究计划的主要课题是横跨封闭电线和肥皂泡的肥皂膜的数学模型。根据最小作用原理，肥皂膜或肥皂泡的表面积在横跨电线的所有薄膜中或在包括固定体积的所有气泡中最小。这种表面的物理表征表明，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。数学上，横跨一根电线的肥皂膜称为最小表面，而肥皂泡称为恒定平均曲率表面（简称CMC）。这两种曲面早在18世纪60年代就引起了物理学家和数学家的兴趣，并成为广泛研究的主题，也激发了数学和科学中许多其他学科的进步。一个证明这种几何物体存在的一般理论，被称为“最小-最大理论”，最近取得了惊人的成功，并将成为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最小曲面上的应用。在第一个主题中，PI打算用Heegaard格证明给定三流形中最小-最大CMC曲面的拓扑界，以及证明给定平均曲率下多个CMC曲面的存在性。PI还计划建立用于构造由任意光滑函数规定的平均曲率曲面的最小-最大理论，推广曲率函数为常数的CMC曲面的最小-最大理论。在第二主题中，PI将研究PI与合作者之前获得的自由边界最小-最大最小曲面的紧性，性质和莫尔斯指数上界；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最小曲面。PI将完成一个利用调和映射理论构造具有自由边界的最小-最大最小磁盘的程序。该奖项反映了美国国家科学基金会的法定使命，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.