Minimal Surfaces in Geometry and Topology

最小曲面是平衡态的形状，最早由拉格朗日在18世纪研究。 这样的表面局部地最小化面积，因此在某种意义上是最优的和普遍存在的-在化学，材料科学，生物学和广义相对论（在那里它们模拟黑洞的表观视界）。 在数学中，它们最近被用来解决几何和拓扑学中的问题，例如庞加莱猜想的证明。 PI将研究三维空间中极小曲面的构造、性质和应用。 一个中心问题是要了解的几何和拓扑类型的极小曲面之一，可以获得。 在拓扑学中，J.W.亚历山大在1932年是找到所有的方法来划分一个三维空间成两块一个更简单的类型。 极小曲面可以作为正则曲面来寻找这样的分裂。 PI还将研究变分原理产生的相关问题，例如阿诺德问题，该问题要求找到受磁场影响的粒子的闭合轨道。 除了这项研究外，PI还将专注于本科生和研究生的教学和培训，并通过组织研讨会，会议和编写临时材料来推进该领域。 更确切地说，这个项目的目标是开发新的技术来研究从西蒙-史密斯的光滑极大极小理论中产生的极小曲面。 当极小曲面由多参数扫掠构造时，一个基本且重要的公开问题是它们是否具有整数重数。 PI将工作，以表明多重性一般等于一个在平滑设置。 这个方向的一个目标是证明Lusternick-Schnirelman猜想，即每个黎曼三球面至少包含四个嵌入的极小二球面。 这样的工作需要发展定量版本的拓扑定理，如塞尔夫定理。 PI还将使用极小曲面继续研究三流形的Heegaard分裂分类。 PI还将研究涉及平均曲率的相关变分问题，例如获得黎曼双球面上常曲率闭合曲线的存在性，这是一个源于物理学和动力系统的问题。 这些项目中采用的技术结合了低维拓扑、分析、莫尔斯理论和最小曲面理论的联合收割机思想。该奖项反映了NSF的法定使命，并通过使用基金会的智力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 in equilibria first studied by Lagrange in the 1700s. Such surfaces locally minimize area and thus are in a sense optimal and ubiquitous - in chemistry, materials science, biology and general relativity (where they model apparent horizons of black holes). In mathematics, they have been used more recently to solve problems in Geometry and Topology, such as in the proof of the Poincare conjecture. The PI will study the construction, properties and applications of minimal surfaces in three-dimensional spaces. A central problem is to understand the geometry and topological type of the minimal surfaces one can obtain. In topology, a question asked by J.W. Alexander in 1932 is to find all the ways to divide a three-dimensional space into two pieces of a simpler type. Minimal surfaces can be used as canonical surfaces to find such splittings. The PI will also study related problems arising from variational principles, for instance the problem of Arnold which asks to find closed orbits of a particle subject to a magnetic field. In addition to this research, the PI will focus on teaching and training of undergraduate and graduate students as well as advancing the field by organizing seminars, conferences and writing expository materials. More precisely, the objectives of this project are to develop new techniques to study minimal surfaces arising from the smooth min-max theory of Simon-Smith. When minimal surfaces are constructed from multi-parameter sweep-outs, a basic and important open problem is whether they come with integer multiplicities. The PI will work to show that the multiplicities are generically equal to one in the smooth setting. One goal in this direction is to prove the Lusternick-Schnirelman Conjecture that every Riemannian three-sphere contains at least four embedded minimal two-spheres. Such work requires developing quantitative versions of topological theorems, such as Cerf's theorem. The PI will also use minimal surfaces to continue the study of classifying Heegaard splittings of three-manifolds. The PI will also investigate related variational problems involving mean curvature such as obtaining the existence of closed curves of constant curvature on Riemannian two-spheres, a problem originating from physics and dynamical systems. The techniques to be employed in these projects combine ideas from low-dimensional topology, analysis, Morse theory, and minimal surface 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.