Mean curvature flow solitons in the hyperbolic space

Abstract

In this thesis, we study self-similar solutions to the mean curvature flow in the hyperbolic space. After recalling some general facts about solitons in ambient spaces endowed with a warped product metric, we focus on solitons in hyperbolic space which flow, in the expanding direction, by the conformal field whose trajectories are orthogonal to horospheres. First, we study their stability, supplying a sufficient condition. In particular, solitons which are (suitably) graphical are stable. Next, we investigate the solvability of Plateau’s problem at infinity. By means of ODE techniques, we then characterize cylindrical and rotationally symmetric examples, showing an analogy with translating solitons in Euclidean space. Indeed, the solutions are appropriate analogies of the grimreaper, bowl, and winglike translators in Euclidean space. Eventually, under some additional conditions, we characterize the grim-reaper as the only soliton whose boundary at infinity are two parallel hyperplanes. A pair of appendices contain some auxiliary material about varifolds and the boundary at infinity of Cartan-Hadamard manifolds.