Differential Geometry and Minimal Surfaces

极小曲面是处于平衡位置的形状。它们是物质科学中用来模拟黑洞、肥皂膜或聚合物的物理物体。极小曲面是从纯数学的角度来研究的，在几何学中起着核心作用。这个项目的总体目标是研究它们在给定空间中的刚性或灵活性。测地线的类似问题不仅在数学上产生了巨大的影响，而且在科学的应用领域也产生了巨大的影响。这个项目的更广泛的影响包括与学生和博士后研究人员的工作。这些项目将深化极小-极大方法与极大极小曲面族的度量存在理论之间的关系。我们将继续研究负曲线空间中极小曲面的面积增长。更准确地说，我们计划研究黎曼度量的体积谱在多大程度上表征该度量，并将极小表面的面积增长与经典量(如动力系统中的熵版本)联系起来。这一奖项反映了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。

Minimal surfaces are shapes which are in an equilibrium position. They are physical objects which serve to model black holes, soap films, or polymers in material sciences. Minimal surfaces are studied from a purely mathematical point of view and play a central role in geometry. The overall goal of this project is to study how rigid or flexible they are in a given space. The analogous problem for geodesics has had a tremendous impact not only in mathematics but also in applied fields of science. Broader impacts of this project include work with students and postdoctoral researchers.These projects will deepen the relation between min-max methods and the existence theory of metrics with maximal families of minimal surfaces. Continuing efforts will study of the area-growth of minimal surfaces in negatively curved spaces. More precisely, we plan to study to which extent the volume spectrum of a Riemannian metric characterizes the metric and to relate the area-growth of minimal surfaces with classical quantities such as versions of entropy in dynamical systems.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.