COMBINATORIAL YAMABE FLOW ON HYPERBOLIC SURFACES WITH BOUNDARY

This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The combinatorial Yamabe flow is a gradient flow of a concave function. The long-time behavior of the flow and the geometric meaning is investigated.

COMBINATORIAL YAMABE FLOW ON SURFACES

In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangulated surface. We show that the flow either develops removable singularities or converges exponentially fast to a constant combinatorial curvature metric. If the singularity develops, we show that the singularity is always removable by a surgery procedure on the triangulation. We conjecture that after finitely many such surgery changes on the triangulation, the flow converges to the constant combinatorial curvature metric as time approaches infinity.