Weingarten Flows in Riemannian Manifolds

TL;DR

This paper investigates Weingarten flows, a class of geometric evolution equations for hypersurfaces in Riemannian manifolds, establishing existence, avoidance, and embedding preservation results under specific conditions.

Contribution

It introduces existence results for Weingarten flows with isoparametric initial data in certain Riemannian manifolds and proves avoidance and embedding preservation for flows with odd Weingarten functions.

Findings

Existence of flows with isoparametric initial data in space forms and rank-one symmetric spaces.

Avoidance principle holds for flows with odd Weingarten functions.

Such flows preserve the embedding property.

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.