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.

在科学，工程和数学中自然产生的许多现象可以通过试图最大程度减少一些能量的配置来描述。例如，选择最佳驾驶路线可能涉及最大程度地减少行驶的总距离或车辆消耗的总能量。此类问题的数学研究称为变异的计算。数学家试图提高我们对“最佳配置是什么样的？”等问题的深刻理解。或“如果我略微偏离最佳配置，我还要使用多少能量？”首席研究人员的工作集中在几何形式出现的问题上。例如，就像最佳驾驶路线可以最大程度地减少起点和终点之间的长度一样，可以通过说出它倾向于形成构造的肥皂膜来建模，从而使所有可能的形状之间的表面积最小化。密切相关的想法包括来自材料科学的功能，这些功能模拟了物质不同阶段之间的接触。即使这些是自然而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。 这些项目将重点介绍“稳定性”的概念，该概念与配置相比与附近的最佳最佳配置相比的问题有关，特别会研究有关此类最佳配置问题的稳定性后果。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此类问题。这将通过指导和教学以及创建描述尖端研究主题的公共可访问的教育材料来实现。本研究计划将重点关注稳定的最小超级表面和相关问题。主要研究者与Chao Li共同解决了四维中稳定的伯恩斯坦问题：四维欧几里得空间中完全稳定的最小超出表面是平坦的。将研究一系列问题，这些问题与稳定的最小超曲面以及相关问题（例如标量曲率）有关，并最终是了解较高维度中稳定的伯恩斯坦问题的目标。相关领域（例如Allen-Cahn方程）将研究类似的问题。这些项目还将考虑稳定性与标态曲率比较几何形状的关系，并研究稳定性的较弱形式（有限的摩尔斯指数），因为它与最小（和其他）表面的最低最大构建体有关。例如，这些项目将研究其他表面的区域光谱（p宽度），在与Christos Mantoulidis合作计算了两杆的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评论。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛的影响来通过评估来支持的。