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Pinching estimates and motion of hypersurfaces by curvature functions. (English) Zbl 1129.53044

Consider as a model problem fully nonlinear scalar parabolic equations of the form: \(\partial u / \partial t = F(D^2u)\). The main result here gives conditions under which an equation will preserve uniform positivity of the second derivatives in the sense that \(D^2u \geq \varepsilon \Delta uI\) for some \(\varepsilon \in (0,1/n)\) and it is obtained via a detailed analysis of gradient terms in the equations satisfied by second derivatives. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earlier results of B. Chow and the present author. Among several applications, one concerning evolving hypersurfaces is emphasized.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C40 Global submanifolds
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