CAREER:New Development in Geometric Variational Theory

最小曲面、恒定平均曲率曲面（CMC）和规定平均曲率曲面（PMC）是肥皂膜、肥皂泡和毛细表面的数学模型。这些类型的曲面一直是几何和广义相对论中的重要主题，也激发了许多其他数学和科学学科的进步。几何变分理论是证明这类曲面存在的主要方法。在这个研究项目中，PI将通过进一步推进几何变分理论，对极小曲面、CMC曲面和PMC曲面的存在性进行一些研究项目。这项研究计划还包括对教育活动的支持。PI将开发研究生研究课题课程的新课程，招募和指导博士生和博士后，指导高级本科生的荣誉论文。PI还将为研究生和初级博士后组织与本研究计划相关的主题的夏季研讨会。其目标是促进早期职业研究人员并鼓励合作。在第一个主题中，PI将探讨最近解决多重一猜想的重要应用。特别是，PI将研究极小曲面的新的遍历性质，以及Simon-Smith设置和自由边界设置中的多重性1猜想。在第二个课题中，PI将继续研究拉格朗日乘子的几何变分理论。PI期望证明三个流形中最小-最大CMC曲面的拓扑边界，以及证明多个CMC曲面的存在性。PI还打算建立毛细表面的一般存在理论。在最后一个主题中，PI将通过使用自由边界最小-最大理论的近似研究奇异和非紧空间中最小曲面的存在性问题及其应用。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, Constant Mean Curvature (CMC) surfaces, and Prescribed Mean Curvature (PMC) surfaces are mathematical models of soap films, soap bubbles, and capillary surfaces. These types of surfaces have always been important topics in geometry and general relativity, and have also inspired advances of many other subjects in mathematics and science. Geometric Variational Theory is the major method for proving the existence of these types of surfaces. In this research program, the PI will conduct a number of research projects on the existence of minimal, CMC, and PMC surfaces by further advancing Geometric Variational Theory. This research program also includes support for educational activities. The PI will develop new curricula for graduate research topic courses, recruit and mentor Ph. D. students and postdocs, and direct advanced undergraduate students for honors theses. The PI will also organize a summer workshop for graduate students and junior postdocs on topics related to this research program. The goal is to promote early career researchers and encourage collaborations.In the first subject, the PI will explore important applications of the recent resolution of the Multiplicity One Conjecture. In particular, the PI will investigate new ergodic properties of minimal surfaces, and the Multiplicity One Conjecture in the Simon-Smith setting and the free boundary setting. In the second subject, the PI will continue the research on Geometric Variational Theory with Lagrange multipliers. The PI anticipates to prove topological bounds for the min-max CMC surfaces in three manifolds, as well as to prove the existence of multiple CMC surfaces. PI also intends to establish the general existence theory for capillary surfaces. In the last subject, the PI will investigate existence problems and applications for minimal surfaces in singular and noncompact spaces via approximations using the free boundary min-max theory.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.