Variational Problems in Geometry

提出的研究重点是最小表面，即局部面积最小的表面。最小表面的经典实例可以通过将线框浸入肥皂溶液中并形成其边界为线框的肥皂膜表面而物理地获得。最小化的主题渗透到所有的自然科学中，因此最小表面是自然界中几种现象的重要模型。它们与分子工程、材料科学和广义相对论中的黑洞理论有关。提出的研究旨在了解这些表面如何能够局部最小化而不是全局最小化，此外，这如何揭示它们所居住空间的形状。最小子流形是几何中最基本的变分问题——面积最小化问题的关键点。自欧拉和拉格朗日的工作以来，它们一直是数学研究的重要对象，在他们的研究中发展起来的许多思想后来被证明是变分法、非线性偏微分方程和数学物理发展的关键。此外，最小子流形也是理解曲率的关键方法，在几何、低维拓扑和广义相对论中产生了许多引人注目的应用。最小超曲面存在理论的最新进展表明，这是该领域一个非常激动人心的时刻。拟议研究的一个主要组成部分是根据新的存在结果调查新的莫尔斯指数估计。一个目标是研究由该理论产生的新的最小超曲面的几何和拓扑性质，这反过来又可以导致几个应用。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 proposed research focuses on minimal surfaces, which are surfaces that locally minimize area. Classical examples of minimal surfaces can be obtained physically by immersing a wire frame into a soap solution and forming a soap film surface whose boundary is the wire frame. The theme of minimization permeates all of the natural sciences and minimal surfaces are thus an important model for several phenomena in nature. They have been linked to molecular engineering, materials science, and to the theory of black holes in general relativity. The proposed research aims to understand how such surfaces can be locally minimizing but not globally so and, moreover, how this can reveal the shape of the space they live in. Minimal submanifolds are critical points to the most fundamental variational problem in the geometry, that of minimizing area. They have been an essential object in mathematical research since the work of Euler and Lagrange, and many of the ideas developed in their study turned out to be key in the development of calculus of variations, nonlinear PDEs, and mathematical physics. In addition, minimal submanifolds have also been a crucial method in understanding curvature, yielding many striking applications to geometry, low-dimensional topology, and general relativity. Recent advances on the existence theory of minimal hypersurfaces suggest that this is a very exciting time for the field. A major component of the proposed research is to investigate new Morse index estimates in light of the new existence results. One goal is to study what are the geometrical and topological properties of the new minimal hypersurfaces produced by the theory, which in turn can lead to several applications.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.