Anisotropic Energy Functionals in Geometric Analysis

各向异性能量是由Gibbs在19世纪引入的，用来模拟晶体的平衡形状，更广泛地说，是对任意两种不同材料界面的表面张力进行建模。数学中相应的几何变分问题越来越受到人们的关注。最简单的各向异性能量--面积泛函的最小化就是这类问题中最著名的例子之一。它可以追溯到1760年拉格朗日的工作，对物理学和数学都产生了重大影响。1936年，杰西·道格拉斯因在这一主题上的研究成果而被授予第一个菲尔兹奖；然而，许多基本问题仍然悬而未决，特别是在更一般的各向异性背景下。该项目的目标是发展新的理论和技术，以提高各向异性几何变分问题的水平。这个项目的目的是增加我们对一般椭圆积分临界点的理解。与极小曲面的REACH理论，即面积泛函的临界点相比，在各向异性框架中理解的很少。本课题的主要内容之一是研究闭黎曼流形中各向异性极小超曲面的存在性。这是几何分析中一个引人入胜的核心问题，早在八十年代，皮茨、肖恩、西蒙和尤就对面积泛函作出了回答。研究人员将讨论更一般的各向异性设置，扩大最小-最大理论的适用范围。这将需要对各种褶皱的结构进行精细的分析，这也将在几何流的构造中找到应用。本项目的另一个目的是研究一般度量空间中集合论平台问题能量极小子的存在性和正则性结果。这项调查将进一步完善尺寸最小化可整流电流。此外，研究者还将研究具有几乎恒定的各向异性平均曲率的超曲面的行为。最后，将解决最优分支传输中的稳定性和正则性猜想。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。

Anisotropic energies were introduced by Gibbs in the 19th century to model the equilibrium shape of crystals and, more generally, the surface tension at the interfaces of any two different materials. An increasing interest has been devoted to the corresponding geometric variational problems in mathematics. Minimizing the area functional, the simplest anisotropic energy, is one of the most famous examples in this class of problems. It dates back to the work of Lagrange in 1760 and has had a major impact both in physics and in mathematics. Jesse Douglas was awarded the first Fields Medal in 1936 for his results on this topic; yet, a variety of basic questions remain open, especially in the more general anisotropic setting. The goal of this project is to develop new theories and techniques to improve the state of the art of anisotropic geometric variational problems. This project aims to increase our understanding of critical points for general elliptic integrands. In contrast to the reach theory of minimal surfaces, i.e., critical points of the area functional, very little is understood in the anisotropic framework. One of the main themes of this project is the existence of anisotropic minimal hypersurfaces in closed Riemannian manifolds. This is a fascinating and central question in geometric analysis, which has been answered for the area functional by Pitts, Schoen, Simon and Yau in the eighties. The investigator will address the more general anisotropic setting, broadening the reach of the min-max theory. This will require a refined analysis of the structure of varifolds, which will find applications also in the construction of geometric flows. Another goal of this project is to study existence and regularity results for energy minimizers of the set-theoretic Plateau problem in general metric spaces. This investigation will be then further refined for size minimizer rectifiable currents. Moreover, the investigator will study the behavior of hypersurfaces with almost constant anisotropic mean curvature. Finally, the stability and regularity conjectures in optimal branched transport will be addressed.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.