Differential Geometry and Minimal Surfaces

最小表面是出现在数学和应用科学中的物理对象，例如它们被用来模拟黑洞或拉伸结构。直观地说，最小表面是处于某种平衡位置的形状（比如肥皂膜），它可以是稳定的，也可以是不稳定的。在过去几年中，关于不稳定最小表面的存在及其随不稳定程度增加而产生的定性行为的研究取得了巨大进展。许多长期悬而未决的问题刚刚得到解决，本项目计划继续进一步调查。预计该项目还将继续培训这一活跃领域的研究生，以及研究研讨会和会议。本项目计划继续探索最小曲面的变分理论。提出了几个问题，其中共同的主题是测地线的类似问题是由测地线是哈密顿流的封闭轨道这一事实引起的。这种观点在最小表面理论中没有直接的类比，因此它的兴趣。PI将研究最小超曲面随其不稳定度增加的渐近性质，以及当环境空间为负弯曲时大面积最小曲面的性质。该项目涉及莫尔斯理论、变分方法、几何和拓扑学的结合。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 physical objects that appear in mathematics and applied science, where for instance they are used to model black holes or tensile structures. Intuitively, minimal surfaces are shapes that are in some equilibrium position (like soap films for example), which can be either stable or unstable. Over the last years immense progress has been done regarding the existence of unstable minimal surfaces and their qualitative behavior as the degrees of instability increase. Many questions which were open for a long time have just been settled and this project plans to continue that investigation further. This project is also expected to continue the training of graduate students in this active area, as well as research seminars and conferences. This project plans to continue exploring the variational theory of minimal surfaces. Several problems are proposed where the common theme is that the analogous question for geodesics follows from the fact that geodesics are closed orbits of an Hamiltonian flow. This perspective does not have a direct analogue in minimal surface theory and hence its interest. The PI will study asymptotic properties of the minimal hypersurfaces as their degrees of instability increase and also the properties of minimal surfaces of large area when the ambient space is negatively curved. The project involves combination of Morse theory, variational methods, geometry, and topology.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.