Singularities of solutions to geometric variational problems

DMS-0406447题目：几何变分问题解的奇异性PI：Gang Tian和Neshan Wickramasekera，M.I.T.摘要本文的主要目的是研究任意维的（光滑的和奇异的）嵌入稳定极小超曲面的弱极限的局部结构，最终目的是推广R. Schoen和L.西蒙放松嵌入性假设允许存在额外的奇点，例如切锥是重数大于1的超平面的点。建议研究这些奇点的性质，目的是获得关于这些奇点的大小和结构的信息，第二个方案是研究某些拓扑约束和解析约束的影响（在目标流形上）关于黎曼流形间能量极小调和映射奇异集的正则性. 极小曲面和调和映射的正则性理论有着非常丰富的历史，在这些领域中发展起来的结果和思想深刻地影响了当代数学和物理研究的其他几个领域，如Yang-Mills场、自由边界问题、曲率流和广义相对论。在许多纯数学和应用数学问题中，获得奇点附近的良好结构描述是非常必要的，在这方面，极小曲面理论的任何新进展都可能对其他领域产生影响。

DMS-0406447Title: Singularities of solutions to geometric variational problemsPIs: Gang Tian and Neshan Wickramasekera, M.I.T.ABSTRACT The primary focus of this proposal is to study the localstructure of weak limits of (smooth as well as singular) immersedstable minimal hypersurfaces of arbitrary dimension, withthe ultimate goal of extending the partial regularitytheory of embedded stable minimal hypersurfaces, developedin 1981 by R. Schoen and L. Simon. Relaxing the embeddedness hypothesisallows the presence of additional singularities, for instance pointswhere the tangent cones are hyperplanes with multiplicity greater thanone. It is proposed to investigate the nature of these singularities,with the aim of obtaining information concerning the size and thestructure of these singularities, possible uniqueness of their tangentcones as well as the local nature of the hypersurface near the singularities.The second project proposed is a study of the effect of certaintopological and analytic constraints (on the targetmanifold) on the regularity propeties of the singular sets of energyminimizing harmonic maps between Riemannian manifolds. The regularity theory of minimal surfaces and harmonic maps has a veryrich history and the results and ideas developed in these areas haveprofoundly influenced several other fields of contemporary research inmathematics and physics,such as Yang-Mills fields, free boundary problems, curvature flows andgeneral relativity.Obtaining a good structural description near singularities turns out to behighly desirable in many problems in pure and applied mathematics, and anynew advances in minimal surface theory in this regard will likely havean impact on these other fields.