Mathematical Sciences: Geometric Variational Problems and Related PDE Questions

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Leon Simon and Brian White will study problems relating to regularity and singularity of minimal surfaces and extrema of other geometric variational problems. Simon's research will proceed in three broad directions. The first of these involves problems relating to the singular set of minimal surfaces. The second concerns entire and exterior solutions of the minimal surface equation for the graph of a function. The final topic involves Willmore-type functionals. The basic questions regarding singular sets of minimal surfaces are concerned with the behaviour of such surfaces on approach to a singular point. Simon has already obtained strong results in the case where the singularity is isolated and will now investigate more general situations. The investigations of the minimal surface equation will center around the asymptotic behaviour of entire and exterior solutions. Here the emphasis will be on investigations in a high dimensional setting. The Willmore functional is a certain average of the mean curvature. The questions here relate to minimizing this functional over surfaces of a certain genus. White will investigate the dimension of the singular set of area minimizing integral currents. This may lead to the discovery of such sets of fractional dimension. A related problem is to find conditions under which almost every boundary gives rise to regular area minimizing disks. He will also investigate the existence of compact minimal submanifolds in Riemannian manifolds. One direction of research will involve an investigation of how many embedded minimal submanifolds exist.