Regularity questions in the geometric calculus of variations and in geometric flow problems

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Proposal DMS-0406209 Abstract ``Regularity questions in the geometric calculus of variationsand in geometric flow problems'' (Leon Simon and Brian White) Leon Simon plans to pursue various questions related to the structureof the singular sets of minimal submanifolds and energy minimizingmaps. More specifically Simon aims to establish first orderregularity (i.e. that the singular set lies in a locally finite unionof continuously differentiable submanifolds) for various multiplicity1 classes near points where there is a tangent cylinder with across-section which admits a calibration. Such a result would inparticular apply to mod-2 minimizers near ``top-dimensional'' singularpoints. Brian White plans to study regularity properties of meancurvature flow, including non-uniqueness properties and ``fattening.''In addition he will continue his work on the singular structure ofminimizing cones with coefficients in a metric group, and his work on2 dimensional minimal surfaces with particular emphasis on the studyof branch points. 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, and theunderstanding of singularity formation in geometric flow problems is akey ingredient in the approach of Hamilton, Perelman and others toThurston's Geometrization Conjecture. Likewise in the study of the``canonical'' objects which arise naturally in topology and geometry,singularities arise in a very natural and unavoidable manner, and theunderstanding of these singularities is an absolutely fundamentalproblem. As with most non-linear phenomena, there is not a singlegeneral theory which applies in a wide range of different contexts.Rather, each different context has its own collection of effectivetechniques, and it is the development and application of suchtechniques in the context of the geometric calculus of variationswhich is the focus of the present research proposal. Specifically,Simon and White propose to continue their efforts toward a morecomplete understanding of singularities, and how they are formed, inthe context of area minimizing submanifolds and energy minimizingmaps, and in the context of various geometric flow problems.