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

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Abstract for NSF proposal DMS 0104049 (Leon Simon \& Brian White joint P.I.'s)Leon Simon plans to pursue various questions related to the structureof the singular sets of minimal submanifolds and energy minimizingmaps, including the extension of his recent work on construction ofsingular examples of minimal hypersurfaces. A current aim is toconstruct examples of minimal submanifolds with singular sets whichinclude ``gaps'' and other phenomena, the possibility of which is leftopen by his previous general work on the structure of the singularset. He is also proposing to complete work on the singular set ofmod-2 currents to show that, in all dimensions, the top-dimensionalpart of the singular set locally lies in a finite union ofcontinuously differentiable submanifolds. Simon also proposes topursue several questions related to asymptotics on approach tosingularities. Brian White plans to study how regularity propertiesand singular structure for minimizing chains with coefficients in ametric group depend on the group and its metric. This includes studyof immiscible fluid interfaces as a special case. He also plans toinvestigate singularities in the mean-curvature flow. He willcontinue his investigations of 2-dimensional minimal surfaces, inparticular concerning branch points and concerning total boundarycurvature.An understanding of singularities, and how singularities are formed,is a fundamental element in our overall understanding of many physicaland geometric phenomena. For example, in cosmology singularities ofspace-time (e.g. ``black holes'') play a fundamental role. Likewisein the study of the ``canonical'' objects which arise naturally intopology and geometry, singularities arise in a very natural andunavoidable manner, and the understanding of these singularities is anabsolutely fundamental problem. As with most non-linear phenomena,there is not a single general theory which applies in a wide range ofdifferent contexts. Rather, each different context has its owncollection of effective techniques, and it is the development andapplication of such techniques in the context of the geometriccalculus of variations which is the focus of the present researchproposal. Specifically, Simon and White propose to continue theirefforts toward a complete understanding of singularities, and how theyare formed, in the context of area minimizing submanifolds and energyminimizing maps. Such techniques are likely to be applicable to thestudy of other objects of geometric and physical significance---forexample to the study of immiscible fluid interfaces, soap-films, andthe equilibrium free surfaces of fluids.