Regularity and Singularity in Geometric Variational Problems and in Geometric Flow Problems

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Abstract Proposal DMS-9803493 Principal Investigators: Leon Simon and Brian White Leon Simon proposes to pursue various questions related to the structure of the singular sets of minimal submanifolds and energy minimizing maps, including the extension of his recent work on smoothness of the singular set of area minimizing submanifolds to higher dimensions and to other classes of submanifolds. Specifically, he is proposing to consider the question of whether the top-dimensional part of the singular set locally lies in a finite union of smooth submanifolds, which he has recently established for 3 and 4 dimensional mod-2 minimizing submanifolds in codimension 2. Simon also proposes to continue his efforts to develop methods for generating examples of singular sets, and to pursue several questions related to asymptotics on approach to singularities. In addition he proposes to study questions related to the structure of the branching set of energy minimizing maps. Brian White plans to study how regularity properties and singular structure for minimizing chains with coefficients in a metric group depend on the group and its metric. This includes study of immiscible fluid interfaces as a special case. He also plans to investigate singularities in the mean-curvature flow and in a related hyperbolic flow that should more accurately model the dynamics of real soap films. He will also continue his investigations of branch points in 2 dimensional minimal surfaces. An understanding of singularities, and how singularities are formed, is a fundamental element in the overall understanding of many physical and geometric phenomena. For example, in cosmology singularities of space-time (e.g. "black holes") play a fundamental role. Likewise in the study of the "canonical" objects which arise naturally in topology and geometry, the understanding of singularities is absolutely fundamental. As with most non-linear phenomena, there is no single well ordered theory which a pplies in a wide range of different contexts. Rather, each different context has its own collection of effective techniques, and it is the development and application of such techniques in the context of the geometric calculus of variations which is the focus of the present research proposal. Specifically, Simon and White propose to continue their efforts toward a complete understanding of singularities, and how they are formed, in the context of area minimizing submanifolds and energy minimizing maps. Such techniques are likely to be applicable to the study of other objects of geometric and physical significance---for example to the study of immiscible fluid interfaces, soap-films, and the equilibrium free surfaces of fluids.