Gaussian mean curvature flow |
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Authors: | Alexander A. Borisenko Vicente Miquel |
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Affiliation: | 1. Geometry Department, Mathematics Faculty, Kharkov National University, Pl. Svobodi 4, Kharkov, 61077, Ukraine 2. Departamento de Geometría y Topología, Universidad de Valencia, Avda. Andrés Estellés, 1, 46100, Burjassot (Valencia), Spain
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Abstract: | In the Euclidean Space mathbb Rn+1{mathbb {R}^{n+1}} with a density eefrac12 n m2 |x|2, (e = ±1){e^{varepsilon frac12 n mu^2 |x|^2},} {(varepsilon =pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere) depending on the relation between the bound of its principal curvatures and μ. |
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