Mean Curvature Flow Solitons in a GRW Spacetime and CMC Free Boundary Hypersurfaces in Rotational Domains

Abstract

In this work, we study two themes. First, we study a n-dimensional spacelike mean curvature flow solitons related to the closed conformal timelike vector field K = f(t)∂t (t ∈ I ⊂ R) which is globally defined on an generalized Robertson-Walker (GRW) spacetime −I×fMn+p with warping function f ∈ C∞(I) and Riemannian fiber Mn+p, these are particular cases of trapped submanifolds, and we obtain rigidity and non-existence results for this submanifold class via applications of suitable generalized maximum principles and under certain constraints on f and on the curvatures of Mn+p. Then, we work with the existence and uniqueness of free boundary constant mean curvature hypersurfaces in rotational domains, these are domains whose boundary is generated by a rotation of a graph. We classify the CMC free boundary hypersurfaces as topological disks or annulus, under some conditions in the generatrix function and a gap condition on the umbilicity tensor.