Stability in Geometric Variational Problems

几何变分问题的稳定性

• 批准号: 2304432
• 负责人: Otis Chodosh
• 金额: $ 54.63万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304432&HistoricalAwards=false
• 关键词: Stability; Geometric; Variational; Problems

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.

科学、工程和数学中自然出现的许多现象可以通过寻求最小化能量的配置来描述。例如，选择最佳驾驶路线可能涉及最小化车辆行驶的总距离或消耗的总能量。对此类问题的数学研究被称为变分法。数学家寻求提高我们对“最佳配置是什么样的？”等问题的宏观理解。或者“如果我稍微偏离最佳配置，我会多消耗多少能量？”首席研究员的工作主要集中在几何上出现的此类问题。例如，就像最佳行驶路线可能会最小化起点和终点之间的长度一样，可以对跨越金属丝环的肥皂膜进行建模，说它倾向于形成使跨越金属丝环的所有可能形状中的表面积最小化的配置。环形。密切相关的想法包括材料科学中模拟物质不同相之间接触的泛函。尽管这些都是自然且经过充分研究的设置，但有关最佳配置的形状和性质的许多基本问题仍未解决。 这些项目将重点关注“稳定性”的概念，该概念与配置如何与邻近的最佳配置进行比较的问题相关，并且特别将研究稳定性对此类有关最佳配置的问题的影响。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此类问题。这将通过指导和教学以及创建描述前沿研究主题的公开教育材料来完成。该研究计划将重点关注稳定的最小超曲面和相关问题。课题组长与李超合作，最近解决了四维稳定伯恩斯坦问题：四维欧几里得空间中完全稳定的最小超曲面是平坦的。将研究与稳定最小超曲面以及标量曲率等相关问题相关的一系列问题，最终目标是理解高维下的稳定伯恩斯坦问题。类似的问题也将在相关领域进行研究，例如艾伦-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，并研究较弱形式的稳定性（有限莫尔斯指数），因为它与最小（和其他）表面的最小-最大结构相关。例如，在与 Christos Mantoulidis 合作计算两个球体的 p 宽度之后，这些项目将研究其他表面的面积谱（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迷你课程。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响进行评估，认为值得支持审查标准。