COMBINATORIAL RICCI FLOWS ON SURFACES

By Bennett Chow, Feng Luo. 2002

405 citations recorded by the source corpus.

We show that the analogue of Hamilton’s Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston’s circle packing on surfaces. As a consequence, a new proof of Thurston’s existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/76695d06807c9673efa2800b00c25f96238d2ade

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

A maximum principle for combinatorial Yamabe flow (2002, methodology): “Here we reintroduce the concepts of combinatorial Yamabe flow, based on the work on the combinatorial Ricci flow in two dimensions by Chow-Luo [ 3 ]”
A variational principle for weighted Delaunay triangulations and hyperideal polyhedra (2006, background): “Chow & Luo [10] gave a proof which is inspired by work on the Ricci flow on surfaces.”
Rigidity of Polyhedral Surfaces, III (2010, result): “1 is a generalization of an earlier result obtained by Chow and Luo [5].”
Rigidity of polyhedral surfaces, II (2006, methodology): “1; 2; 3//!R(3) is negative definite in the following cases: (i) "D 1 or (ii) (Colin de Verdiére [4], Chow–Luo [3]) .”
Geometric triangulations and discrete Laplacians on manifolds (2005, methodology): “We note that the Laplacian given by (15) is also the same as the Laplacian considered by Chow-Luo [10] in two dimensions as observed by Z.”
COMBINATORIAL YAMABE FLOW ON SURFACES (2003, methodology): “The work of [3] suggests that in general it is easier to work in hyperbolic geometry.”
Discrete conformal maps and ideal hyperbolic polyhedra (2010, background): “The Riemann-geometric version of this notion appeared in Luo’s work on a discrete version of the Yamabe flow on surfaces [20].”
Combinatorial Calabi flows on surfaces (2012, methodology): “, the metric is determined by its curvature up to a scalar multiplication ([26], [8]).”
A combinatorial Yamabe flow in three dimensions (2005, methodology): “The author would also like to thank Feng Luo and Mauro Carfora for useful conversations.”
Rigidity of inversive distance circle packings revisited (2017, positive): “Before going on, we recall the following definition and theorem of Luo in [22].”
Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds (2009, background): “See, for instance, [13] [36] [11] [9] [5] [41] [21].”