A Maximum Principle for Combinatorial Yamabe Flow

In his proof of Andreev’s theorem, Thurston in [1] introduced a conformal geometric structure on two dimensional simplicial complexes which is an analogue of a Riemannian metric. He then used a version of curvature to prove the existence of circle-packings (see also Marden-Rodin [2] for more exposition). Techniques very similar to elliptic partial differential equation techniques were used by Y. Colin de Verdière [3] to study conformal structures and circle packings. Cooper and Rivin in [4] then defined a version of scalar curvature on three dimensional simplicial complexes and used it to look at rigidity of sphere packings along the lines of Colin de Verdière. Inspired by this work, Chow and Luo [5] looked at a combinatorial Ricci flow on two dimensional simplicial complexes and showed that the equation satisfies a maximum principle. The maximum principle, which says that the maximum of a solution decreases and the minimum increases, is one of the most useful concepts in the study of the heat equation and other parabolic partial differential equations. Maximum principle techniques have been used to great benefit in the smooth category. We are especially inspired by Hamilton’s work on the Ricci flow (see [6]). It is in general very difficult to prove a maximum principle, or even to prove short term existence of solutions, when you do not have a strictly parabolic equation. We shall use Cooper and Rivin’s combinatorial scalar curvature to define combinatorial Yamabe flow on three dimensional simplicial complexes with conformal structures which is a three dimensional analogue of Chow and Luo’s work. The flow turns out not to be parabolic in the usual sense of Laplacians on graphs. We shall nonetheless prove a maximum principle for this equation under certain conditions. Our treatment works on the simplest examples: the curvature flow on a double tetrahedron and the boundary of a 4-simplex. Numerical studies indicate that the maximum principle is true under less restrictive conditions, perhaps even under only topological restrictions. The combinatorial Yamabe flow is a way of studying prescribed scalar curvature on simplicial complexes, which we might call the combinatorial Yamabe