Hyper-elastic Ricci flow: Gradient flow, local existence-uniqueness, and a Perelman energy functional

Abstract

The equation of hyper-elastic Ricci flow amends classical Ricci flow by the addition of the Cauchy stress tensor which itself is derived from the a free energy. In this paper hyper-elastic Ricci flow is shown to possess three properties derived by G. Perelman for classical Ricci flow, specifically it is diffeomorphically equivalent to a gradient flow, unique smooth solutions exist locally in time, and the system possesses a non-decreasing energy function.