Curvature flows in the sphere

Abstract

We consider contracting and expanding curvature flows in $\Ss$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gauß map. The contracting hypersurfaces shrink to a point $x_0$ while the expanding hypersurfaces converge to the equator of the hemisphere $\mc H(-x_0)$. After rescaling, by the same scale factor, the rescaled hypersurfaces converge to the unit spheres with centers $x_0$ \resp $-x_0$ exponentially fast in $C^\un(\Ss[n])$.

46 pages, v4: Some typos corrected, to appear in JDG

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• Pinching estimates for dual flows in hyperbolic and de Sitter space
• The quermassintegral preserving mean curvature flow in the sphere