Anisotropic Energy Functionals in Geometric Analysis

各向异性能是由吉布斯在世纪引入的，用于模拟晶体的平衡形状，更一般地说，用于模拟任何两种不同材料界面处的表面张力。数学中相应的几何变分问题越来越受到人们的关注。最小化面积泛函，最简单的各向异性能量，是这类问题中最著名的例子之一。它可以追溯到工作的拉格朗日在1760年，并已产生重大影响，无论是在物理学和数学。1936年，杰西·道格拉斯因其在这方面的研究成果而获得第一届菲尔兹奖。然而，各种基本问题仍然悬而未决，尤其是在更普遍的各向异性背景下。这个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艺术状态。这个项目旨在增加我们对一般椭圆被积函数临界点的理解。与极小曲面的到达理论相反，即，临界点的面积功能，很少被理解的各向异性框架。该项目的主要主题之一是各向异性极小超曲面在闭黎曼流形中的存在性。这是几何分析中一个迷人的中心问题，80年代Pitts、Schoen、Simon和Yau在面积泛函中回答了这个问题。研究人员将解决更一般的各向异性设置，扩大最小-最大理论的范围。这就需要对变倍结构进行精细的分析，这种分析在几何流的构造中也能找到应用。本项目的另一个目标是研究一般度量空间中集合论Plateau问题的能量极小解的存在性和正则性结果。这项调查将进一步完善的大小最小整流电流。此外，研究者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，稳定性和规律性的最佳分支运输将得到解决。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.