Singularities of solutions to geometric variational problems

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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.