Asymptotic Plateau Problem in Hyperbolic Space

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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.