Mean curvature flow solitons in warped products: nonexistence, rigidity and stability

We deal with several aspects of the geometry of m-dimensional mean curvature flow solitons immersed in a Riemannian warped product IxfMn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\times _{f}M<^>n$$\end{document} (m <= n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\le n$$\end{document}), with base I subset of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\subset {\mathbb {R}}$$\end{document}, fiber Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M<^>n$$\end{document} and warping function f is an element of C infinity(I)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C<^>\infty (I)$$\end{document}. In this context, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space, as well as to obtain nonexistence results concerning these geometric objects. When m=n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=n$$\end{document}, we investigate complete two-sided hypersurfaces and, in particular, entire graphs constructed over the fiber Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M<^>n$$\end{document} which are mean curvature flow solitons. Furthermore, we infer the stability of closed mean curvature flow solitons with respect to an appropriate stability operator. Applications to self-shrinkers and self-expanders in the Euclidean space and to mean curvature flow solitons in important ambient spaces, like the pseudo-hyperbolic, Schwarzschild and Reissner-Nordstr & ouml;m spaces, are also given.