Translating solitons over Cartan-Hadamard manifolds.

We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions. In the last section of the paper we study the solutions of the soliton equation and a capillary type equation with prescribed contact angle condition.