Weingarten Flows in Riemannian Manifolds

Abstract

Given orientable Riemannian manifolds $M^n$ and $\bar M^{n+1},$ we study flows $F_t:M^n\rightarrow\bar M^{n+1},$ called Weingarten flows,in which the hypersurfaces $F_t(M)$ evolve in the direction of their normal vectors with speed given by a function $W$ of their principal curvatures,called a Weingarten function, which is homogeneous, monotonic increasing with respect to any of its variables, and positive on the positive cone. We obtain existence results for flows with isoparametric initial data, in which the hypersurfaces $F_t:M^n\rightarrow\bar M^{n+1}$ are all parallel, and $\bar M^{n+1}$ is either a simply connected space form or a rank-one symmetric space of noncompact type. We prove that the avoidance principle holds for Weingarten flows defined by odd Weingarten functions, and also that such flows are embedding preserving.