Stability of mean curvature flow solitons in warped product spaces

Abstract

In this paper we establish a natural framework for the stability of mean curvature flow solitons in warped product spaces. These solitons are regarded as stationary immersions for a weighted volume functional. Under this point of view, we are able to find geometric conditions for finiteness of the index and some characterizations of stable solitons. We also prove some non-existence results for solitons as applications of a comparison principle which suits well the structure of the diffusion elliptic operator associated to the weighted measures we are considering.