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Please use this identifier to cite or link to this item: http://hdl.handle.net/2108/661

Title: Evolution of hypersurfaces by curvature functions
Other Titles: Evoluzioni di ipersuperfici secondo funzioni delle curvature
Authors: Sinestrari, Carlo
Alessandroni, Roberta
Keywords: geometric evolution equations
fully nonlinear parabolic PDEs
convex hypersurfaces
scalar curvature
entire graphs
maximum principle
singularities formation
Issue Date: 6-Oct-2008
Abstract: We consider a smooth n-dimensional hypersurface of ℝⁿ⁺¹, with n≥2, and its evolution by a class of geometric flows. The speed of these flows has normal direction with respect to the surface and its modulus S is a symmetric function of the principal curvatures. We show some general properties of these flows and compute the evolution equation for any homogeneous function of principal curvatures. Then we apply the flow with speed S=(H/(logH)), where H is the mean curvature plus a constant, to a mean convex surface to prove some convexity estimates. Using only the maximum principle we prove that the negative part of the scalar curvature tends to zero on a limit of rescalings of the evolving surfaces near a singularity. The following part is dedicated to the study of a convex initial manifold moving by powers of scalar curvature: S=R^{p}, with p>1/2. We show that if the initial surface satisfies a pinching estimate on the principal curvatures then it shrinks to a point in finite time and ...
Description: 20. ciclo
URI: http://hdl.handle.net/2108/661
Appears in Collections:Tesi di dottorato in scienze matematiche e fisiche

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