Variational Problems in Geometry

所提出的研究集中在极小曲面，即局部最小化面积的曲面。最小曲面的经典例子可以通过将线框浸入肥皂溶液并形成以线框为边界的肥皂膜表面来获得。极小化的主题渗透到所有自然科学中，极小曲面因此是自然界中几个现象的重要模型。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。这项拟议的研究旨在了解这些表面如何局部最小化而不是全局最小化，以及这如何揭示它们所居住的空间的形状。极小子流形是几何中最基本的变分问题--面积最小化问题的临界点。自从欧拉和拉格朗日的工作以来，它们一直是数学研究的基本对象，他们研究中形成的许多想法被证明是变分法、非线性偏微分方程组和数学物理发展的关键。此外，极小子流形也是理解曲率的重要方法，在几何学、低维拓扑学和广义相对论中产生了许多引人注目的应用。极小超曲面存在理论的最新进展表明，这是该领域一个非常令人兴奋的时期。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极小超曲面的几何和拓扑性质，这反过来可以导致几个应用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.