ON THE GEOMETRY OF SPACELIKE MEAN CURVATURE FLOW SOLITONS IMMERSED IN A GRW SPACETIME

Abstract

Abstract We investigate geometric aspects of complete spacelike mean curvature flow solitons of codimension one in a generalized Robertson–Walker (GRW) spacetime $-I\times _{f}M^n$ , with base $I\subset \mathbb R$ , Riemannian fiber $M^n$ and warping function $f\in C^\infty (I)$ . For this, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space and to obtain nonexistence results concerning these solitons. In particular, we deal with entire graphs constructed over the Riemannian fiber $M^n$ , which are spacelike mean curvature flow solitons, and we also explore the geometry of a conformal vector field to establish topological and further rigidity results for compact (without boundary) mean curvature flow solitons in a GRW spacetime. Moreover, we study the stability of spacelike mean curvature flow solitons with respect to an appropriate stability operator. Standard examples of spacelike mean curvature flow solitons in GRW spacetimes are exhibited, and applications related to these examples are given.