Deforming hypersurfaces of the sphere by their mean curvature (Q580737)

Let M be a compact hypersurface without boundary immersed in a Riemannian manifold N of constant positive sectional curvature. Suppose that \(M_ 0=M\) is locally given by diffeomorphisms \(F_ 0: {\mathbb{R}}^ n\supset F_ 0(U)\subset M_ 0\subset N\). The author studies the evolution equation \(\partial_ tF(x,t)=H(x,t)\), \(F(x,0)=F_ 0(x)\) for \(x\in U\), where H(\(\cdot,t)\) denotes the mean curvature vector of F(\(\cdot,t)\). In a previous paper the author [Invent. Math. 84, 463-480 (1986; Zbl 0589.53058)] has shown that for general Riemannian manifolds N all hypersurfaces satisfying a suitable convexity condition will contract to a single point in finite time during this evolution. Here, he shows that in a spherical spaceform some convergence results can be obtained without any convexity condition. It may for example happen that the hypersurfaces straighten out and become totally geodesic.