Stability in Geometric Variational Problems

在科学、工程和数学中自然产生的许多现象可以用寻求最小化某些能量的构型来描述。例如，选择最优行驶路线可能涉及最小化车辆行驶的总里程或总能量消耗。对这类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“最优配置是什么样子？”之类问题的宏观理解。或者“如果我稍微偏离了最优配置，我还要多消耗多少能量？”主要研究人员的工作集中在几何上出现的这些问题上。例如，就像最优行驶路线可以最小化起点和终点之间的长度一样，跨越铁丝环的肥皂膜可以被建模为：它倾向于形成在跨越该环的所有可能形状中使表面积最小的配置。密切相关的想法包括材料科学的泛函，这些泛函模拟了物质不同阶段之间的联系。尽管这些都是自然的和经过充分研究的环境，但许多关于最佳配置的形状和性质的基本问题仍然没有解决。这些项目将把重点放在“稳定性”的概念上，这与配置如何与附近较不理想的配置相比较的问题有关，具体而言，将研究稳定性对此类最优配置问题的影响。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描述尖端研究主题的公开可访问的教育材料来实现。本研究计划将专注于稳定的极小超曲面及其相关问题。最近，他与李超合作解决了四维空间中的稳定Bernstein问题：四维欧氏空间中的完全稳定极小超曲面是平坦的。我们将研究一系列与稳定极小超曲面相关的问题，以及与之相关的问题，如标量曲率，最终目的是理解高维稳定的Bernstein问题。将针对艾伦-卡恩方程等相关领域研究类似的问题。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Morse指数)，因为它与极小(和其他)曲面的极小-极大构造有关。例如，这些项目将调查其他表面的面积谱(p-宽度)，然后与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，并继续提供与这些研究领域相关的课程和微型课程。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。

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.