Almgren's multiple-valued functions and geometric measure theory

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AbstractAward: DMS-0905347Principal Investigator: Wei ZhuThis award is funded under the American Recovery and Reinvestment Act of 2009(Public Law 111-5).These research projects concern the interaction of geometricmeasure theory and partial differential equations. Particularprojects will study stationary-harmonic multiple-valuedfunctions, frequency and branch sets of multiple-valuedfunctions, and regularity of integral varifolds. Work on theseproblems will address some longstanding questions in geometricanalysis such as boundary regularity of area minimizing integralcurrents in general codimensions and the singularity estimate ofintegral varifolds under natural assumptions on the mean curvatureor the second fundamental form.Some of most useful and significant classes of submanifolds ingeometry arise are characterized as critical points of naturalenergy or area functions. For example, surfaces formed by soapbubbles are critical points of the mean curvature functional,while geodesics (shortest paths) are critical points for thelength functional. Geometric measure theory offers powerfultools for establishing that critical objects exist by expandingthe class of allowable objects from surfaces to generalizedsurfaces. These methods often leave open questions ofregularity, or smoothness, of solutions, and those issues areemphasized in these projects.