Nonexistence and rigidity of spacelike mean curvature flow solitons immersed in a GRW spacetime

Abstract

We study the nonexistence and rigidity of an important class of particular cases of trapped submanifolds, more precisely, n-dimensional spacelike mean curvature flow solitons related to the closed conformal timelike vector field \(\mathcal K=f(t)\partial _t\) (\(t\in I\subset \mathbb R\)) which is globally defined on an \((n+p+1)\)-dimensional generalized Robertson–Walker (GRW) spacetime \(-I\times _fM^{n+p}\) with warping function \(f\in C^\infty (I)\) and Riemannian fiber \(M^{n+p}\), via applications of suitable generalized maximum principles and under certain constraints on f and on the curvatures of \(M^{n+p}\). In codimension 1, we also obtain new Calabi–Bernstein-type results concerning the spacelike mean curvature flow soliton equation in a GRW spacetime.