Differential Geometry and Minimal Surfaces

最小曲面是处于平衡位置的形状。它们是用于模拟黑洞、肥皂膜或材料科学中的聚合物的物理对象。最小曲面是从纯数学的角度来研究的，在几何学中起着中心作用。这个项目的总体目标是研究它们在给定空间中的刚性或柔性。测地线的类似问题不仅在数学领域，而且在应用科学领域都产生了巨大的影响。该项目更广泛的影响包括与学生和博士后研究人员的合作。这些项目将加深最小-最大方法与最小曲面的极大族度量的存在性理论之间的关系。继续努力将研究负弯曲空间中最小曲面的面积增长。更确切地说，我们计划研究黎曼度规的体积谱在多大程度上表征了该度规，并将最小曲面的面积增长与经典量（如动力系统中的熵）联系起来。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.