Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II

Xu, Xu; Zheng, Chao

doi:10.1090/tran/8572

Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II
HTML articles powered by AMS MathViewer

by Xu Xu and Chao Zheng PDF

Trans. Amer. Math. Soc. 375 (2022), 2763-2788 Request permission

Abstract:

This paper investigates the combinatorial $\alpha$-curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial $\alpha$-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu’s discrete uniformization theorem for classical combinatorial curvature [J. Differential Geom. 109 (2018), pp. 431–466]. We further introduce combinatorial $\alpha$-Yamabe flow and combinatorial $\alpha$-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial $\alpha$-curvatures. To handle the potential singularities along the combinatorial curvature flows, we do surgery along the flows by edge flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu [J. Differential Geom. 109 (2018), pp. 431–466], we prove the longtime existence and convergence of combinatorial $\alpha$-Yamabe flow and combinatorial $\alpha$-Calabi flow with surgery, which provide effective algorithms for finding piecewise hyperbolic metrics with prescribed combinatorial $\alpha$-curvatures.

References

Alexander I. Bobenko, Ulrich Pinkall, and Boris A. Springborn, Discrete conformal maps and ideal hyperbolic polyhedra, Geom. Topol. 19 (2015), no. 4, 2155–2215. MR 3375525, DOI 10.2140/gt.2015.19.2155
Bennett Chow and Feng Luo, Combinatorial Ricci flows on surfaces, J. Differential Geom. 63 (2003), no. 1, 97–129. MR 2015261
Fan R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. MR 1421568
Song Dai and Huabin Ge, Discrete Yamabe flows with $R$-curvature revisited, J. Math. Anal. Appl. 484 (2020), no. 1, 123681, 11. MR 4042611, DOI 10.1016/j.jmaa.2019.123681
H. Ge, Combinatorial methods and geometric equations, Thesis (Ph.D.)-Peking University, Beijing. 2012.
Huabin Ge, Combinatorial Calabi flows on surfaces, Trans. Amer. Math. Soc. 370 (2018), no. 2, 1377–1391. MR 3729504, DOI 10.1090/tran/7196
Huabin Ge and Bobo Hua, On combinatorial Calabi flow with hyperbolic circle patterns, Adv. Math. 333 (2018), 523–538. MR 3818085, DOI 10.1016/j.aim.2018.05.039
Huabin Ge and Wenshuai Jiang, On the deformation of inversive distance circle packings, III, J. Funct. Anal. 272 (2017), no. 9, 3596–3609. MR 3620705, DOI 10.1016/j.jfa.2016.12.020
Huabin Ge and Xu Xu, Discrete quasi-Einstein metrics and combinatorial curvature flows in 3-dimension, Adv. Math. 267 (2014), 470–497. MR 3269185, DOI 10.1016/j.aim.2014.09.011
Huabin Ge and Xu Xu, 2-dimensional combinatorial Calabi flow in hyperbolic background geometry, Differential Geom. Appl. 47 (2016), 86–98. MR 3504921, DOI 10.1016/j.difgeo.2016.03.011
Huabin Ge and Xu Xu, $\alpha$-curvatures and $\alpha$-flows on low dimensional triangulated manifolds, Calc. Var. Partial Differential Equations 55 (2016), no. 1, Art. 12, 16. MR 3449397, DOI 10.1007/s00526-016-0951-5
Huabin Ge and Xu Xu, A discrete Ricci flow on surfaces with hyperbolic background geometry, Int. Math. Res. Not. IMRN 11 (2017), 3510–3527. MR 3693657, DOI 10.1093/imrn/rnw142
Huabin Ge and Xu Xu, On a combinatorial curvature for surfaces with inversive distance circle packing metrics, J. Funct. Anal. 275 (2018), no. 3, 523–558. MR 3806817, DOI 10.1016/j.jfa.2018.04.008
Huabin Ge and Xu Xu, A combinatorial Yamabe problem on two and three dimensional manifolds, Calc. Var. Partial Differential Equations 60 (2021), no. 1, Paper No. 20, 45. MR 4201643, DOI 10.1007/s00526-020-01900-8
David Glickenstein and Joseph Thomas, Duality structures and discrete conformal variations of piecewise constant curvature surfaces, Adv. Math. 320 (2017), 250–278. MR 3709106, DOI 10.1016/j.aim.2017.08.043
Xianfeng Gu, Ren Guo, Feng Luo, Jian Sun, and Tianqi Wu, A discrete uniformization theorem for polyhedral surfaces II, J. Differential Geom. 109 (2018), no. 3, 431–466. MR 3825607, DOI 10.4310/jdg/1531188190
Xianfeng David Gu, Feng Luo, Jian Sun, and Tianqi Wu, A discrete uniformization theorem for polyhedral surfaces, J. Differential Geom. 109 (2018), no. 2, 223–256. MR 3807319, DOI 10.4310/jdg/1527040872
Gregory Leibon, Characterizing the Delaunay decompositions of compact hyperbolic surfaces, Geom. Topol. 6 (2002), 361–391. MR 1914573, DOI 10.2140/gt.2002.6.361
Feng Luo, Combinatorial Yamabe flow on surfaces, Commun. Contemp. Math. 6 (2004), no. 5, 765–780. MR 2100762, DOI 10.1142/S0219199704001501
Feng Luo, Rigidity of polyhedral surfaces, III, Geom. Topol. 15 (2011), no. 4, 2299–2319. MR 2862158, DOI 10.2140/gt.2011.15.2299
L. S. Pontryagin, Ordinary differential equations, ADIWES International Series in Mathematics, Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1962. Translated from the Russian by Leonas Kacinskas and Walter B. Counts. MR 0140742
Boris Springborn, Ideal hyperbolic polyhedra and discrete uniformization, Discrete Comput. Geom. 64 (2020), no. 1, 63–108. MR 4110530, DOI 10.1007/s00454-019-00132-8
M. Roček and R. M. Williams, The quantization of Regge calculus, Z. Phys. C 21 (1984), no. 4, 371–381. MR 734698, DOI 10.1007/BF01581603
T. Wu, X. Zhu, The convergence of discrete uniformizations for closed surfaces, arXiv:2008.06744
Xu Xu, Rigidity of inversive distance circle packings revisited, Adv. Math. 332 (2018), 476–509. MR 3810260, DOI 10.1016/j.aim.2018.05.026
Xu Xu, On the global rigidity of sphere packings on $3$-dimensional manifolds, J. Differential Geom. 115 (2020), no. 1, 175–193. MR 4081933, DOI 10.4310/jdg/1586224843
X. Xu, Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I, arXiv:1806.04516v1
X. Xu and C. Zheng, A new proof for global rigidity of vertex scaling on polyhedral surfaces, accepted by Asian J. Math. 2021.
Wei Zeng and Xianfeng David Gu, Ricci flow for shape analysis and surface registration, SpringerBriefs in Mathematics, Springer, New York, 2013. Theories, algorithms and applications. MR 3136003, DOI 10.1007/978-1-4614-8781-4
Xiang Zhu and Xu Xu, Combinatorial Calabi flow with surgery on surfaces, Calc. Var. Partial Differential Equations 58 (2019), no. 6, Paper No. 195, 20. MR 4024610, DOI 10.1007/s00526-019-1654-5