Simplicial Ricci Flow

Abstract

We construct a discrete form of Hamilton’s Ricci flow (RF) equations for a d-dimensional piecewise flat simplicial geometry, \({{\mathcal S}}\). These new algebraic equations are derived using the discrete formulation of Einstein’s theory of general relativity known as Regge calculus. A Regge–Ricci flow (RRF) equation can be associated to each edge, ℓ, of a simplicial lattice. In defining this equation, we find it convenient to utilize both the simplicial lattice \({{\mathcal S}}\) and its circumcentric dual lattice, \({{\mathcal S}^*}\). In particular, the RRF equation associated to ℓ is naturally defined on a d-dimensional hybrid block connecting ℓ with its (d−1)-dimensional circumcentric dual cell, ℓ *. We show that this equation is expressed as the proportionality between (1) the simplicial Ricci tensor, Rc ℓ , associated with the edge \({\ell\in{\mathcal S}}\), and (2) a certain volume weighted average of the fractional rate of change of the edges, \({\lambda\in \ell^*}\), of the circumcentric dual lattice, \({{\mathcal S}^*}\), that are in the dual of ℓ. The inherent orthogonality between elements of \({\mathcal S}\) and their duals in \({{\mathcal S}^*}\) provide a simple geometric representation of Hamilton’s RF equations. In this paper we utilize the well established theories of Regge calculus, or equivalently discrete exterior calculus, to construct these equations. We solve these equations for a few illustrative examples.