Singularities of solutions to geometric variational problems

摘要本文主要研究任意维的嵌入稳定极小超曲面（光滑和奇异）弱极限的局部结构，最终目的是推广R. Schoen和L. Simon于1981年提出的嵌入稳定极小超曲面的部分正则性理论。放松嵌入性假设允许存在额外的奇点，例如切锥是多重度大于1的超平面的点。本文提出对这些奇异点的性质进行研究，目的是获得关于这些奇异点的大小和结构、切锥的可能唯一性以及奇异点附近超曲面的局部性质的信息。提出的第二个项目是研究某些拓扑和解析约束（在目标流形上）对黎曼流形之间能量最小化调和映射的奇异集的正则性的影响。极小曲面和调和图的正则性理论有着非常丰富的历史，在这些领域发展起来的结果和思想深刻地影响了当代数学和物理学的其他几个研究领域，如杨-米尔斯场、自由边界问题、曲率流和广义相对论。在纯数学和应用数学的许多问题中，在奇点附近获得良好的结构描述是非常可取的，在这方面最小曲面理论的任何新进展都可能对这些其他领域产生影响。

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.