WEIGHTED COMBINATORIAL RICCI FLOW AND METRICS DEFINED BY DEGENERATE CIRCLE PACKINGS

Abstract

Chow and Luo [1] in 2003 had shown that the combinatorial analogue
of the Hamilton Ricci flow on surfaces under certain conditions converges
to Thruston’s circle packing metric of constant curvature. The combinatorial
setting includes weights defined for edges of a triangulation. Crucial assumption
in the paper [1] was that the weights are nonnegative. Recently we have
shown that same statement on convergence can be proved under weaker condition:
some weights can be negative and should satisfy certain inequalities, [3].
Moreover, in [6] notions of degenerate circle packing and corresponding metric
were introduced. In [6] theory of combinatorial Ricci flow for such metrics was
developed, which includes Chow–Lou theory as a partial case for nondegenerate
circle packing and nonnegative weights on edges.
On the other hand, in [2] the combinatorial Yamabe flow was introduced
and investigated. In [8, 7] we developed weighted modification of Yamabe flow.
In this paper we merge ideas from these two theories and introduce weighted
combinatorial Ricci flow on metrics defined by degenerate circle packings. We
prove that under certain conditions for any initial metric the flow converges to
a unique metric of constant curvature.