The Ideal Circle Pattern with Boundaries via Analytic Method

Abstract

The geometric structure of three-dimensional manifolds is a fundamental problem in geometric topology. In this paper, we investigate the ideal circle pattern problem on a surface with boundaries under a cellular decomposition. Employing the variational method, we establish a series of equivalent conditions for the existence of a circle pattern in both Euclidean and hyperbolic background geometries.

Keywords

Ideal circle pattern / analytic method / generalized Gaussian curvature / 52C26

References

[1] Andreev EM. On convex polyhedra in Lobachevskii spaces. Mat. Sb. (N.S.), 1970, 81(3): 445-478

[2] Andreev EM. On convex polyhedra of finite volume in Lobachevskii space. Mat. Sb. (N.S.), 1970, 83(2): 256-260

[3] Bobenko AI, Springborn BA. Variational principles for circle patterns and Koebe’s theorem. Trans. Amer. Math. Soc., 2004, 356(2): 659-689

[4] Brägger W. Kreispackungen und triangulierungen. Enseign. Math. (2), 1992, 38(3–4): 201-217

[5] Chow B, Luo F. Combinatorial Ricci flows on surfaces. J. Differential Geom., 2003, 63(1): 97-129

[6] Colin de Verdière Y. Un principe variationnel pour les empilements de cercles. Invent. Math., 1991, 104(3): 655-669

[7] Feng K, Ge H, Hua B. Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds. Geom. Topol., 2022, 26(3): 1349-1384

[8] Feng K, Ge H, Hua B, Xu X. Combinatorial Ricci flows with applications to the hyperbolization of cusped 3-manifolds. Int. Math. Res. Not. IMRN, 2022, 2022(20): 15549-15573

[9] Ge H. Combinatorial methods and geometric equations, 2012, Beijing, Peking University

[10] Ge H, Hua B. 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. Trans. Amer. Math. Soc., 2020, 373(7): 5111-5140

[11] Ge H, Hua B, Zhou Z. Circle patterns on surfaces of finite topological type. Amer. J. Math., 2021, 143(5): 1397-1430

[12] Ge H., Hua B., Zhou Z., Combinatorial Ricci flows for ideal circle patterns. Adv. Math., 2021, 383: Paper No. 107698, 26 pp.

[13] Ge H., Jiang W., On the deformation of discrete conformal factors on surfaces. Calc. Var. Partial Differential Equations, 2016, 55(6): Art. 136, 14 pp.

[14] Ge H, Jiang W. On the deformation of inversive distance circle packings, II. J. Funct. Anal., 2017, 272(9): 3573-3595

[15] Ge H, Jiang W. On the deformation of inversive distance circle packings, III. J. Funct. Anal., 2017, 272(9): 3596-3609

[16] Ge H., Jiang W., Shen L.. On the deformation of ball packings. Adv. Math. 2022, 398: Paper No. 108192, 44 pp.

[17] Ge H., Xu X., α-curvatures and α-flows on low dimensional triangulated manifolds. Calc. Var. Partial Differential Equations, 2016, 55(1): Art. 12, 16 pp.

[18] Ge H, Xu X. A discrete Ricci flow on surfaces with hyperbolic background geometry. Int. Math. Res. Not. IMRN, 2017, 2017(11): 3510-3527

[19] Ge H, Xu X. On a combinatorial curvature for surfaces with inversive distance circle packing metrics. J. Funct. Anal., 2018, 275(3): 523-558

[20] Ge H., Xu X., A combinatorial Yamabe problem on two and three dimensional manifolds. Calc. Var. Partial Differential Equations, 2021, 60(1): Paper No. 20, 45 pp.

[21] Guo R. A note on circle patterns on surfaces. Geom. Dedicata, 2007, 125: 175-190

[22] Hamilton R. Three-manifolds with positive Ricci curvature. J. Differential Geometry, 1982, 17(2): 255-306

[23] Hamilton R. The Ricci flow on surfaces. Mathematics and General Relativity (Santa Cruz, CA, 1986), 1988, Providence, RI, Amer. Math. Soc.237262 71

[24] Hirsch MW. Differential Topology, 1976, New York–Heidelberg, Springer-Verlag33

[25] Horn RA, Johnson CR. Matrix Analysis, 1985, Cambridge, Cambridge University Press

[26] Koebe P. Kontaktprobleme der konformen abbildung. Ber. Verh. Sächs. Akad. Wiss. Leipzig, 1936, 88: 141-164

[27] Leibon G. Characterizing the Delaunay decompositions of compact hyperbolic surfaces. Geom. Topol., 2002, 6: 361-391

[28] Rivin I. Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. of Math. (2), 1994, 139(3): 553-580

[29] Thurston WP. The Geometry and Topology of Three-Manifolds, 1980

RIGHTS & PERMISSIONS

Peking University