Singularities in Geometric Variational Problems

几何变分问题的研究--这里的“问题”指的是找到一个使某种几何定义的能量概念最小化的对象--是数学中最古老和最迷人的课题之一，至少可以追溯到1932年授予J.道格拉斯的第一个“诺贝尔数学奖”（菲尔兹奖），以表彰他在这一领域的进步。几何变分问题的解可以描述物理系统或社会经济模型的平衡配置-我们期望现实世界系统自然达到最小能量配置的情况。 同样，在拓扑学的数学领域中--物体无论如何弯曲或拉伸都被认为是等价的--变分问题的解决方案可以在这样大的等价类中提供首选的代表。因此，几何变分问题的研究在纯数学和应用中具有根本的重要性。该项目旨在通过解决一系列新老问题来显着提高我们对几何变分问题的认识，这些问题的答案将需要开发新技术和新思想，或设计新方法来解决已知方法。这将通过与该领域的许多领先专家的合作来完成。该项目旨在通过处理以下研究路线在几何变分问题的研究中向前迈进一步。灵感来自L的工作。Simon，研究者提出研究接近特殊类极小锥的极小曲面的奇异集的最优正则性，并构造奇异面积极小化超曲面的新例子。在自由边界条件下，根据Caffarelli和韦斯的思想，研究了Alt-Caffarelli泛函和薄障碍问题的奇异解集的正则性.最后，在与Y. Liokumovich提出了紧致流形上一般度量集的光滑极小超曲面的构造问题，推广了Hardt-Simon和N。斯梅尔这些问题具有基础性质，因为它们可能导致定理的扩展，例如Colding-Minicozzi对Ricci流有限时间灭绝的证明，Marques-Neves对Willmore猜想的证明，或Schoen-Yau对正质量定理的证明。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审查标准进行评估而被认为值得支持。

The study of geometric variational problems -- the `problem' here means finding an object that minimizes some geometrically-defined notion of energy -- is one of the oldest and most fascinating topics in mathematics, dating back at least to the first "Nobel prize for mathematics" (the Fields Medal) awarded to J. Douglas in 1932 for his advances in this field. Solutions of geometric variational problems can describe equilibrium configurations of physical systems or socio-economical models -- situations where we expect real-world systems to naturally reach a minimum-energy configuration. As well, in the mathematical field of topology -- where objects are regarded as equivalent no matter how they are bent or stretched -- the solutions to variational problems can provide preferred representatives within such large equivalence classes. The investigation of geometric variational problems is thus of fundamental importance both in pure mathematics and in applications. This project is intended to significantly improve our knowledge on geometric variational problems by addressing a series of old and new questions, whose answer will require either the development of new techniques and ideas, or devising new approaches to known methods. This will be done through the collaborations with many leading experts in the field.This project intends to put a step forward in the study of geometric variational problems by dealing with the following lines of research. Inspired by work of L. Simon, the investigator proposes to study the optimal regularity of the singular set for minimal surfaces close to special classes of minimal cones and to construct new examples of singular area minimizing hypersurfaces. In the free-boundary setting, following ideas of Caffarelli and Weiss, the investigator proposes to study the regularity of the singular set of solutions to the Alt-Caffarelli functional and the thin-obstacle problem. Finally, in a joint project with Y. Liokumovich, the investigator proposes the problem of constructing smooth minimal hypersurfaces for a generic set of metrics on a compact manifold, generalizing works of Hardt-Simon and N. Smale. These problems are of fundamental nature, as they could lead to extensions of theorems such as Colding-Minicozzi's proof of the finite time extinction of the Ricci flow, Marques-Neves' proof of Willmore's conjecture, or Schoen-Yau's proof of the positive mass theorem.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.