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Anisotropic flow of convex hypersurfaces by the square root of the scalar curvature. (English) Zbl 1431.53012
Summary: We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor \(\psi\) when the strictly convex initial hypersurface in Euclidean space is suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere. For a general case in dimension two, it is shown that, with a volume preserving rescaling, the limit profile satisfies a soliton equation.
MSC:
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53E10 Flows related to mean curvature
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