On a combinatorial curvature for surfaces with inversive distance circle packing metrics

Abstract

In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evolves almost in the same way as that of scalar curvature along the surface Ricci flow obtained by Hamilton [21]. Then we study the long time behavior of the combinatorial Ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergence of the flow on triangulated surfaces with nonpositive Euler number. We further generalize the combinatorial curvature to α-curvature and prove that it is also globally rigid, which is in fact a generalized Bower-Stephenson conjecture [6]. We also use the combinatorial Ricci flow to study the corresponding α-Yamabe problem.

Combinatorial Calabi flow with surgery on surfaces

“…Note that ∇ u C = 2LK = −2∆ E,T K. The Euclidean combinatorial Calabi flow (1.2) can be written as Similar to the stability results in [23,24,26,27,28], we have the following result for Euclidean combinatorial Calabi flow (1.2). Proof.…”

“…On the other hand, the following lemma is a well-known fact from analysis. The reader could refer to [28] (Lemma 4.6) for a proof. Set ψ(t) = W * (u(t)).…”

Rigidity of inversive distance circle packings revisited

“…In this paper, based on an observation of Zhou [37], we prove this conjecture for inversive distance in (−1, +∞) by variational principles. We also study the global rigidity of a combinatorial curvature introduced in [14,16,19] with respect to the inversive distance circle packing metrics where the inversive distance is in (−1, +∞).…”

Discrete Calabi Flow: A Unified Conformal Parameterization Method

“…We will look into [SYGS05, CKPS18] and so on for a better spherical embedding algorithm. We will also consider spherical Calabi flow (like spherical Ricci flow [JKLG08]), and R‐curvature [GX15,GX18] Calabi flow that has better precision than angle defect Gaussian curvature.…”

“…On the other hand, flow‐based methods like discrete Ricci flow [CL*03, JKLG08, YGL*09, ZGZ*14], Yamabe flow [Gli05] and Calabi flow [Ge18, GX18, GH18, ZX18, ZLG*18] are inspired by corresponding smooth geometric flows that deform the metric of a Riemannian manifold in differential geometry. They do not operate the coordinates in the parameter domain directly and are intrinsically independent of topology.…”