Aspects of Mean Curvature Flow Solitons in Warped Products

In this article, we study mean curvature flow solitons immersed in warped products and we prove some rigidity results for these immersions under assumptions such as integrability and stochastic completeness. We also obtain gap-type and nonexistence results concerning these geometric objects and some applications to self-shrinkers in the Euclidean space, to mean curvature flow solitons in the Euclidean sphere and to translating mean curvature flow solitons in the Euclidean space. Moreover, we provide a lower diameter estimate for mean curvature flow solitons.