Singularities in Geometric Variational Problems

对几何变分问题的研究——这里的“问题”意味着找到一个能够最小化某些几何定义的能量概念的对象——是数学中最古老、最令人着迷的主题之一，至少可以追溯到 1932 年 J. 道格拉斯因其在这一领域的进展而获得的第一届“诺贝尔数学奖”（菲尔兹奖）。几何变分问题的解决方案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，无论物体如何弯曲或拉伸，都被视为等价的，变分问题的解决方案可以在如此大的等价类中提供首选代表。因此，几何变分问题的研究在纯数学和应用中都具有根本重要性。该项目旨在通过解决一系列新旧问题来显着提高我们对几何变分问题的了解，这些问题的答案将需要开发新技术和想法，或者为已知方法设计新方法。这将通过与该领域许多领先专家的合作来完成。该项目旨在通过以下研究方向在几何变分问题的研究中向前迈进一步。受 L. Simon 工作的启发，研究人员建议研究接近特殊类别最小锥体的最小曲面的奇异集的最佳正则性，并构建奇异面积最小化超曲面的新示例。在自由边界设置中，遵循 Caffarelli 和 Weiss 的思想，研究者建议研究 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 的联合项目中，研究人员提出了为紧凑流形上的一组通用度量构造平滑最小超曲面的问题，概括了 Hardt-Simon 和 N. Smale 的工作。这些问题具有基本性质，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，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.