Almgren's multiple-valued functions and geometric measure theory

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Abstract Award: DMS-0905347 Principal Investigator: Wei Zhu This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). These research projects concern the interaction of geometric measure theory and partial differential equations. Particular projects will study stationary-harmonic multiple-valued functions, frequency and branch sets of multiple-valued functions, and regularity of integral varifolds. Work on these problems will address some longstanding questions in geometric analysis such as boundary regularity of area minimizing integral currents in general codimensions and the singularity estimate of integral varifolds under natural assumptions on the mean curvature or the second fundamental form. Some of most useful and significant classes of submanifolds in geometry arise are characterized as critical points of natural energy or area functions. For example, surfaces formed by soap bubbles are critical points of the mean curvature functional, while geodesics (shortest paths) are critical points for the length functional. Geometric measure theory offers powerful tools for establishing that critical objects exist by expanding the class of allowable objects from surfaces to generalized surfaces. These methods often leave open questions of regularity, or smoothness, of solutions, and those issues are emphasized in these projects.