Asymptotic Plateau Problem in Hyperbolic Space

本课题的研究领域是三维双曲空间中最小面积平面的刚性问题。最小面积平面是在具有相同边界的盘中任何子盘面积最小的平面。对于球面上给定的无穷远点的简单闭曲线，利用安德森的结果，证明了在三维双曲空间中存在一个跨越该曲线的最小面积平面。然而，到目前为止，关于这种最小面积平面的数量的结果很少。特别是没有一个已知的例子，一个简单的封闭曲线在无穷大边界两个不同的最小面积平面。研究人员通过结合来自不同领域的结果，如全局分析，极小曲面和椭圆偏微分方程，获得了关于该问题的强通用唯一性结果。在这个项目中，他将研究唯一性是否在一般情况下是真实的，他将试图证明双曲3-空间中最小面积平面的刚性。这样的结果将是一个至关重要的成分解决一些问题的拓扑结构的双曲3流形，Teichmuller理论，双曲几何。本项目的第二个目标是研究双曲n-空间中绝对面积极小化超曲面的同样问题。这个问题也被称为渐近平台问题的解的个数。本计画的另一部分是利用极小曲面技术来探讨“泛覆盖猜想”。泛覆盖猜想指出任何具有无限基本群的不可约3-流形的泛覆盖是开3-球。Gabai提出了一个程序来解决这个猜想，使用极小曲面。他表明，如果普遍覆盖的3流形有一个适当嵌入的最小面积平面，那么它是同胚的开放3球。作者的目标是通过在具有无限基本群的不可约3-流形的泛覆盖中构造适当嵌入的最小面积平面来填补证明的缺失部分。作者将从事微分几何、几何分析和几何拓扑的研究。他将研究渐近高原问题，通过使用拓扑技术。这个问题吸引了许多数学家超过20年。该问题的结果在一些经典的低维拓扑问题的求解中得到了很好的应用。刚性的结果，作者试图证明将有许多应用在低维拓扑，Teichmuller理论，双曲几何，这些都是传统的领域的调查，经历了巨大的进步，在过去的20年。该项目的第二部分是解决一个长期存在的低维拓扑问题，“泛覆盖猜想”，通过使用最小曲面技术。这个猜想是三维流形拓扑学中最著名的猜想之一，它本身就很有趣。它吸引了许多拓扑超过50年。极小曲面在低维拓扑学中的一些重要问题上有着广泛的应用，研究者正试图利用这些对象的有用性质来证明这一经典猜想。

The area of research of this project is the rigidity of least area planes in hyperbolic 3-space. A least area plane is a plane where any subdisk is area minimizing among the disks with same boundary. For a given simple closed curve in sphere at infinity, the existence of a least area plane that spans the given curve in hyperbolic 3-space is known by Anderson's results. However, there are few results on the number of such least area planes so far. In particular there is no known example of a simple closed curve at infinity bounding two different least area planes. The investigator obtained strong generic uniqueness results on the problem by combining the results from very different fields, like global analysis, minimal surfaces, and elliptic PDEs. In this project, he will investigate whether the uniqueness is true in general, and he will try to prove the rigidity of least area planes in hyperbolic 3-space. Such a result will be a crucial ingredient for solving some problems in the topology of hyperbolic 3-manifolds, Teichmuller theory, and hyperbolic geometry. The second goal of the project is to investigate same problem for the absolutely area minimizing hypersurfaces in hyperbolic n-space. This problem is also known as the number of solutions to the asymptotic Plateau problem. Another part of this project is to approach the "Universal Cover Conjecture" by using minimal surface techniques. Universal Cover Conjecture states that the universal cover of any irreducible 3-manifold with infinite fundamental group is an open 3-ball. Gabai suggested a program to solve this conjecture by using minimal surfaces. He showed that if the universal cover of the 3-manifold has a properly embedded least area plane, then it is homeomorphic to an open 3-ball. The author aims to fill the missing part of the proof by constructing a properly embedded least area plane in a universal cover of an irreducible 3-manifold with infinite fundamental group.The author will undertake research in Differential Geometry, Geometrical Analysis,and Geometric Topology. He will investigate the asymptotic Plateau problem by using topological techniques. The problem attracted many mathematicians for more than 20 years. The results of this problem have produced fruitful applications in the solutions of some classical low dimensional topology problems. The rigidity result which the author is trying to prove will have many applications in low dimensional topology, Teichmuller theory, and hyperbolic geometry which are all traditional areas of investigations that have experienced a tremendous progress in the last 20 years. The second part of the project is to attack a long standing low dimensional topology problem, "The Universal Cover Conjecture", by using minimal surface techniques. This conjecture is one of the most famous conjectures in 3-manifold topology, and it is interesting in own right. It has attracted many topology for more than 50 years. Minimal surfaces had many fruitful applications to several important problems in low dimensional topology, and the investigator is trying to apply the useful properties of these objects to prove this classical conjecture.