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Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature. (English) Zbl 1277.53061

Authors’ abstract: We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximal principal curvature with limit 1 at infinity. We prove that the ratio of the circumradius to the inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one. For smooth, strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point, we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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