Stability in Geometric Variational Problems

在科学、工程和数学中自然产生的许多现象都可以通过寻求最小化某些能量的配置来描述。例如，选择最佳驾驶路线可能涉及使车辆行驶的总距离或消耗的总能量最小化。对这些问题的数学研究被称为变分法。数学家试图提高我们对诸如“最佳配置是什么样的？”或者“如果我稍微偏离最佳配置，我会多使用多少能量？““主要研究人员的工作集中在几何上出现的问题上。例如，就像最佳驾驶路线可能会最小化起点和终点之间的长度一样，跨越金属线环的肥皂膜可以通过说它倾向于形成使跨越环的所有可能形状中的表面积最小化的配置来建模。与之密切相关的思想包括材料科学中的泛函，它模拟了物质不同相之间的接触。尽管这些都是自然的和充分研究的设置，许多基本问题的形状和性质的最佳配置仍然没有解决。 这些项目将集中在“稳定性”的概念，这是有关的问题，如何配置比较接近少的最佳配置，特别是将研究稳定性的后果，这些问题的最佳配置。这些活动的一个关键组成部分将涉及培训下一代研究人员来解决这些问题。这将通过指导和教学以及创建公开访问的教育材料来完成，这些材料描述了前沿的研究课题。该研究计划将专注于稳定的极小超曲面和相关问题。与Chao Li一起，首席研究员最近解决了四维中的稳定伯恩斯坦问题：四维欧氏空间中完全稳定的极小超曲面是平坦的。本文将研究与稳定极小超曲面相关的一系列问题以及相关的标量曲率问题，最终目标是理解高维稳定伯恩斯坦问题。类似的问题将被调查的相关领域，如艾伦-卡恩方程。这些项目还将考虑稳定性和标量曲率比较几何的关系，以及调查较弱形式的稳定性（有限莫尔斯指数），因为它涉及到极小（和其他）曲面的最小-最大构造。例如，这些项目将研究其他表面的面积谱（p宽度），然后与Christine Mantoulidis一起计算两个球体的p宽度。PI将继续指导研究生和博士后，并继续提供与这些研究领域相关的课程和小型课程。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Many phenomena arising naturally in science, engineering, and mathematics can be described by configurations seeking to minimize some energy. For example, choosing an optimal driving route could involve minimizing the total distance traveled or the total energy consumed by the vehicle. The mathematical study of such questions is known as the Calculus of Variations. Mathematicians seek to improve our big-picture understanding of questions like "what does the optimal configuration look like?" or "if I deviate slightly from the optimal configuration, how much more energy do I use?" The principal researcher's work is focused on such problems that arise geometrically. For example, just like the optimal driving route might minimize the length between the starting and ending points, a soap film spanning a wire loop can be modeled by saying that it tends to form the configuration that minimizes the surface area among all possible shapes spanning the loop. Closely related ideas include functionals from materials science that model contact between distinct phases of matter. Even though these are natural and well-studied settings, many basic questions about the shape and nature of optimal configurations remain unsolved. These projects will focus on the notion of "stability" which is related to the question of how a configuration compares to nearby-less optimal-configurations, and specifically will study the ramifications of stability for such questions about optimal configurations. One key component of these activities will involve training the next generation of researchers to tackle such problems. This will be accomplished by mentoring and teaching as well as creating publicly accessible educational materials describing cutting edge research topics.This research program will focus on stable minimal hypersurfaces and related problems. Jointly with Chao Li, the principal investigator has recently solved the stable Bernstein problem in four-dimensions: a complete stable minimal hypersurface in four-dimensional Euclidean space is flat. A series of questions will be studied that are connected to stable minimal hypersurfaces as well as related problems such as scalar curvature, with the eventual goal of understanding stable Bernstein problem in higher dimensions. Similar problems will be investigated for related areas such as the Allen-Cahn equation. These projects will also consider the relationship of stability and scalar curvature comparison geometry, as well as investigate weaker forms of stability (finite Morse index) as it relates to the min-max constructions of minimal (and other) surfaces. For example, these projects will investigate the area-spectrum (p-widths) of other surfaces, following work with Christos Mantoulidis computing the p-widths of the two-sphere. The PI will continue to mentor graduate students and postdocs, as well as continue to give classes and minicourses related to these areas of research.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.