Open Access3-dimensional combinatorial Yamabe flow in hyperbolic background geometry
Author(s) -
Huabin Ge,
Bobo Hua
Publication year - 2020
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/tran/8062
Subject(s) - mathematics , vertex (graph theory) , ball (mathematics) , geometry , tetrahedron , yamabe flow , differential geometry , scalar curvature , curvature , mean curvature flow , algorithm , combinatorics , sectional curvature , graph
We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist real or virtual ball packings with vanishing (extended) combinatorial scalar curvature, i.e., the total (extended) solid angle at each vertex is equal to 4 π 4\pi . In this case, if such a ball packing is real, then the (extended) combinatorial Yamabe flow converges exponentially fast to that ball packing. Moreover, we prove that there is no real or virtual ball packing with vanishing (extended) combinatorial scalar curvature if the number of tetrahedra incident to each vertex is at most 22.