Combinatorial Ricci flows for ideal circle patterns

Rigidity of bordered polyhedral surfaces

“…Similar methods have been used by Rivin [25,26], Leibon [16], Bobenko-Springborn [3] in other situations. We are also highly influenced by the works of Luo [17,18,19], Luo-Dai-Gu [20], Guo-Luo [15], Guo [14], Zhou [32,33], Xu [29,30], Ge-Hua-Zhou [8,9] and others.…”

“…The following three lemmas establish some basic properties of cosine law function. please refer to [6,20,32,8,9] for more information.…”

Combinatorial Yamabe flow on hyperbolic bordered surfaces

“…This is an elementary result. One refers to [6,7] for a proof. Now we are ready to prove Theorem 1.2.…”

“…See e.g. [6,7] for a proof. We are now ready to prove Theorem 1.1 based on the main ideas of Guo's original proof [12].…”

3-dimensional combinatorial Yamabe flow in hyperbolic background geometry

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In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under the combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case.

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