The Gaussian Isoperimetric Problem and the Self-Shrinkers of Mean Curvature Flow

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Johns Hopkins University
We record work done by the author joint with John Ross [27] on stable smooth solutions to the gaussian isoperimetric problem, and we also record work done by the author [26] on two-dimensional self-shrinkers of any co-dimension. The guassian isoperimetric problem is related to minimizing a gaussian weighted surface area while preserving an enclosed gaussian weighted volume. The work joint with the author and John Ross [27] studies smooth, stable local minimizers to the gaussian isoperimetric problem. We show that for complete critical solutions of polynomial volume growth, the only stable solutions are the hyper-planes. As in McGonagle-Ross[27], we also consider the incomplete case of stable solutions to the gaussian isoperimetric problem contained in a ball centered at the origin with their boundary contained in the boundary of the ball. We show that for general dimension, large enough ball, and appropriate euclidean area conditions on the hyper-surface, we get integral decay estimates for the curvature. Furthermore, for the two-dimensional critical solutions in three-dimensional euclidean balls, using a de Giorgi-Moser-Nash type iteration [3, 4, 19] we get point-wise decay estimates for similar conditions. These estimates are decay estimates in the sense that when everything else is held constant, we get zero curvature as the radius of the ball goes to infinity. We also record work done by the author [26] on the application of gaussian harmonic one-forms to two dimensional self-shrinkers of the mean curvature flow in any co-dimension. Gaussian harmonic one-forms are closed one-forms minimizing a weighted L2-norm in their cohomology class. We are able to show a type of rigidity result that forces the genus of the hyper-surface to be zero if the curvature is smaller than a certain size. This bound for the curvature is larger than that given in the rigidity result of Cao-Li [7] which classifies the self-shrinker as either a sphere or cylinder if the curvature is small enough. We also show that if the self-shrinker satisfies an appropriate curvature condition, then the genus gives a lower bound on the index of the self-shrinking Jacobi operator. READERS: Professor William P. Minicozzi II (Advisor) and Professor Joel Spruck
Gaussian Isoperimetric, Geometric Analysis, Self-shrinker, Mean Curvature, MCF, Self-similar, geometry, riemannian, analysis, mathematical analysis