Abstract

In this paper we introduce a new kind of hyperbolic geometric flows — dissipative
hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear
wave equations with dissipative terms. Some interesting exact solutions are given,
in particular, a new concept — hyperbolic Ricci soliton is introduced and some of
its geometric properties are described. We also establish the short-time existence
and uniqueness theorem for the dissipative hyperbolic geometric flow, and prove the
nonlinear stability of the flow defined on the Euclidean space of dimension larger
than 2. Wave character of the evolving metrics and curvatures is illustrated and the
nonlinear wave equations satisfied by the curvatures are derived.