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用寻求最小化某些能量的构型来描述。例如，选择最佳的行驶路线可能涉及到最小化行驶的总距离或车辆消耗的总能量。这类问题的数学研究被称为变分学。数学家试图提高我们对诸如“最佳配置是什么样子的？”或“如果我稍微偏离最佳配置，我要多消耗多少能量？”等问题的宏观理解。首席研究员的工作集中在这类几何问题上。例如，就像最优的驾驶路线可能会最小化起点和终点之间的长度一样，跨越一个线圈的肥皂膜可以通过说它倾向于形成在跨越环路的所有可能形状中最小表面积的配置来建模。密切相关的思想包括材料科学的功能，它模拟物质不同相之间的接触。尽管这些都是自然的、经过充分研究的设置，但关于最佳配置的形状和性质的许多基本问题仍未得到解决。这些项目将重点关注“稳定性”的概念，这与配置与邻近最优配置的比较有关，特别是将研究关于最优配置的稳定性问题的分支。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解决这些问题。这将通过指导和教学以及创建描述前沿研究课题的可公开访问的教育材料来实现。本课题主要研究稳定极小超曲面及其相关问题。最近与李超共同解决了四维稳定的Bernstein问题：四维欧几里得空间中的完全稳定极小超曲面是平坦的。将研究一系列与稳定极小超曲面相关的问题，以及标量曲率等相关问题，最终目标是理解更高维度的稳定伯恩斯坦问题。类似的问题将在相关领域进行研究，如Allen-Cahn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，以及研究稳定性的较弱形式（有限莫尔斯指数），因为它与最小（和其他）曲面的最小-最大结构有关。例如，这些项目将研究其他表面的面积谱（p-宽度），随后与Christos 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，并继续开设与这些研究领域相关的课程和迷你课程。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。

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.