COMBINATORIAL YAMABE FLOW ON SURFACES

By Feng Luo. 2003

242 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5d21835c8964417fe67587b2bfd041862bbb51dc

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In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Discrete conformal maps and ideal hyperbolic polyhedra (2010, methodology): “The Riemann-geometric version of this notion appeared in Luo’s work on a discrete version of the Yamabe flow on surfaces [20].”
Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds (2009, methodology): “Certain conformal variations in the discrete setting have been proposed by Roček and Williams in the context of Regge calculus [38], Thurston in the setting of circle patterns [44] (see also other work on circle packing, e.g., [43]), and Luo [28] (see also [42]).”
A combinatorial Yamabe flow in three dimensions (2005, background): “The reader is also directed to some later work of Luo on how these flows evolve the conformal structure [20] .”
A discrete uniformization theorem for polyhedral surfaces (2013, methodology): “2, we use Penner’s decorated Teichumuller theory [18], the work of Bobenko-Pinkall-Springborn [4] relating PL metrics to Penner’s theory and a variational principle developed in [16].”
Conformal equivalence of triangle meshes (2008, background): “(Edge flipping was also considered by Luo [2004] , albeit during his curvature flow.)”
There exist graphs with super-exponential Ramsey multiplicity constant (2008, background): “on what comparison space is chosen [5, Prop. 2.9], [ 20 ].”
A maximum principle for combinatorial Yamabe flow (2002, methodology): “The three-dimensional combinatorial Yamabe flow was inspired by the work of Chow and Luo [3] (see also [14]).”
A discrete uniformization theorem for polyhedral surfaces II (2014, background): “This flow is a generalization of the discrete Yamabe flow introduced in [16].”
Rigidity of inversive distance circle packings revisited (2017, methodology): “Before going on, we recall the following definition and theorem of Luo in [22].”
Geometric triangulations and discrete Laplacians on manifolds (2005, methodology): “Also, the Laplacian described in [33] is actually the Laplacian described above in (14) with the same weights wij .”