Prescribed Weingarten curvature equations in warped product manifolds

TL;DR

This paper proves the existence of solutions to prescribed Weingarten curvature equations in certain warped product manifolds, ensuring the existence of specific hypersurfaces with constrained curvature properties.

Contribution

It introduces a method using degree theory and a priori estimates to establish solution existence for curvature equations in warped product manifolds.

Findings

Existence of solutions to prescribed Weingarten curvature equations proven.

Hypersurfaces with specified curvature constraints exist in certain warped product manifolds.

Method based on degree theory and a priori estimates successfully applied.

Abstract

In this paper, under suitable settings, we can obtain the existence of solutions to a class of prescribed Weingarten curvature equations in warped product manifolds of special type by the standard degree theory based on the a priori estimates for the solutions. This is to say that the existence of closed hypersurface (which is graphic with respect to the base manifold and whose $k$-th Weingarten curvature satisfies some constraint) in a given warped product manifold of special type can be assured.