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 \(I\times _{f}M^n\) $I{\times }_f{M}^n$ ( \(m\le n\) $m\le n$ ), with base \(I\subset {\mathbb {R}}\) $I\subset \mathbb{R}$ , fiber \(M^n\) $M^n$ and warping function \(f\in C^\infty (I)\) $f\in C^{\infty }(I)$ . 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 \(m=n\) $m=n$ , we investigate complete two-sided hypersurfaces and, in particular, entire graphs constructed over the fiber \(M^n\) $M^n$ 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–Nordström spaces, are also given.