Combinatorial p-th Calabi Flows for Discrete Conformal Factors on Surfaces

Feng, Ke; Lin, Aijin; Zhang, Xiaoxiao

doi:10.1007/s12220-019-00224-0

Cited by 10 publications

(4 citation statements)

References 27 publications

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“…The longtime existence and convergence of the combinatorial Calabi flow with surgery for Luo's vertex scalings were proved by Zhu-Xu [44]. Feng-Lin-Zhang [8] introduced the combinatorial p-th Calabi flow for Luo's vertex scalings on surfaces, and proved the corresponding longtime existence and convergence of the combinatorial p-th Calabi flow by surgery. Recently, Wu-Xu [36] introduced the fractional combinatorial Calabi flow for Luo's vertex scalings on surfaces, and proved the corresponding longtime existence and convergence of the flow by surgery.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Curvature Flows for Generalized Circle Packings on Surfaces with Boundary

Zheng

2023

International Mathematics Research Notices

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In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the $(-1,-1,-1)$-type generalized circle packing metric introduced by Guo and Luo [ 16]. To find hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths, we introduce combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings on ideally triangulated surfaces with boundary. Then we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows, which provide effective algorithms for finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.

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Section: Introductionmentioning

confidence: 99%

Combinatorial Curvature Flows for Generalized Circle Packings on Surfaces with Boundary

Zheng

2023

International Mathematics Research Notices

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“…background geometry. After that, so-called combinatorial p-th Calabi flows are also studied in [23,13], which generalize the work in [15].…”

Section: Introductionmentioning

confidence: 84%

Combinatorial Ricci flows for ideal circle patterns

Hua

Zhou

2021

Advances in Mathematics

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“…This implies that the fractional combinatorial Calabi flow (1.10) is a nonlocal combinatorial curvature flow in general. The fractional combinatorial Calabi flow (1.10) is different from the combinatorial p-th Calabi flow defined by discrete p-Laplace operator in [7,28]. Motivated by Definition 2, we further introduce a fractional combinatorial Calabi flow for decorated and hyper-ideal hyperbolic polyhedral metrics on 3-dimensional manifolds in [42], where the basic properties of the flow are also established.…”

Section: Introductionmentioning

confidence: 99%

Fractional combinatorial Calabi flow on surfaces

Wu¹,

Xu²

2021

Preprint

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Using the fractional discrete Laplace operator for triangle meshes, we introduce a fractional combinatorial Calabi flow for discrete conformal structures on surfaces, which unifies and generalizes Chow-Luo's combinatorial Ricci flow for Thurston's circle packings, Luo's combinatorial Yamabe flow for vertex scaling and the combinatorial Calabi flow for discrete conformal structures on surfaces. For Thurston's Euclidean and hyperbolic circle packings on triangulated surfaces, we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow. For vertex scalings on polyhedral surfaces, we do surgery on the fractional combinatorial Calabi flow by edge flipping under the Delaunay condition to handle the potential singularities along the flow. Using the discrete conformal theory established in [21,22], we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow with surgery.

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Combinatorial p-th Calabi Flows for Discrete Conformal Factors on Surfaces

Cited by 10 publications

References 27 publications

Combinatorial Curvature Flows for Generalized Circle Packings on Surfaces with Boundary

Combinatorial Curvature Flows for Generalized Circle Packings on Surfaces with Boundary

Combinatorial Ricci flows for ideal circle patterns

Fractional combinatorial Calabi flow on surfaces

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