Differential Geometry and Minimal Surfaces

微分几何和最小曲面

• 批准号: 2005468
• 负责人: Andre Neves
• 金额: $ 39万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005468&HistoricalAwards=false
• 关键词: Differential; Geometry; Minimal; Surfaces

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.

极小曲面是数学和应用科学中出现的物理对象，例如它们被用来模拟黑洞或拉伸结构。直观地说，最小曲面是处于某个平衡位置的形状（例如肥皂膜），它可以是稳定的，也可以是不稳定的。 在过去的几年里，已经取得了巨大的进展，关于存在的不稳定的极小曲面和他们的定性行为的不稳定程度的增加。 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，并举办研究研讨会和会议。本计画将继续探讨极小曲面的变分理论。提出了几个问题，其中的共同主题是，类似的问题测地线如下的事实，测地线是封闭的轨道的哈密顿流。这种观点在极小曲面理论中没有直接的类似物，因此它很有趣。PI将研究极小超曲面的渐近性质，因为它们的不稳定度增加，以及当周围空间是负弯曲时，大面积极小曲面的性质。该项目涉及到莫尔斯理论、变分方法、几何学和拓扑学的结合。该奖项反映了NSF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。