Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space

Hoffmann, T.; Rossman, W.; Sasaki, T.; Yoshida, M.

doi:10.1090/S0002-9947-2012-05698-4

Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space
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by T. Hoffmann, W. Rossman, T. Sasaki and M. Yoshida PDF

Trans. Amer. Math. Soc. 364 (2012), 5605-5644 Request permission

Abstract:

We define discrete flat surfaces in hyperbolic $3$-space $\mathbb {H}^3$ from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature $1$ surfaces in $\mathbb {H}^3$, and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. Along the way, we also examine discrete linear Weingarten surfaces of Bryant type in $\mathbb {H}^3$, and consider an example of a discrete flat surface related to the Airy equation that exhibits swallowtail singularities and a Stokes phenomenon.

References

S. I. Agafonov, Discrete Riccati equation, hypergeometric functions and circle patterns of Schramm type, Glasg. Math. J. 47 (2005), no. A, 1–16. MR 2237186, DOI 10.1017/S0017089505002247
H. Ando, M. Hay, K. Kajiwara and T. Masuda, An explicit formula for the discrete power function associated with circle patterns of Schramm type, preprint 2010.
Alexander I. Bobenko, Discrete conformal maps and surfaces, Symmetries and integrability of difference equations (Canterbury, 1996) London Math. Soc. Lecture Note Ser., vol. 255, Cambridge Univ. Press, Cambridge, 1999, pp. 97–108. MR 1705222, DOI 10.1017/CBO9780511569432.009
Alexander I. Bobenko, Tim Hoffmann, and Boris A. Springborn, Minimal surfaces from circle patterns: geometry from combinatorics, Ann. of Math. (2) 164 (2006), no. 1, 231–264. MR 2233848, DOI 10.4007/annals.2006.164.231
A. I. Bobenko, D. Matthes, and Yu. B. Suris, Discrete and smooth orthogonal systems: $C^\infty$-approximation, Int. Math. Res. Not. 45 (2003), 2415–2459. MR 2006481, DOI 10.1155/S1073792803130991
Alexander I. Bobenko, Christian Mercat, and Yuri B. Suris, Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function, J. Reine Angew. Math. 583 (2005), 117–161. MR 2146854, DOI 10.1515/crll.2005.2005.583.117
Alexander Bobenko and Ulrich Pinkall, Discrete isothermic surfaces, J. Reine Angew. Math. 475 (1996), 187–208. MR 1396732, DOI 10.1515/crll.1996.475.187
Alexander I. Bobenko and Ulrich Pinkall, Discretization of surfaces and integrable systems, Discrete integrable geometry and physics (Vienna, 1996) Oxford Lecture Ser. Math. Appl., vol. 16, Oxford Univ. Press, New York, 1999, pp. 3–58. MR 1676682
Alexander I. Bobenko and Boris A. Springborn, A discrete Laplace-Beltrami operator for simplicial surfaces, Discrete Comput. Geom. 38 (2007), no. 4, 740–756. MR 2365833, DOI 10.1007/s00454-007-9006-1
Alexander I. Bobenko and Yuri B. Suris, Discrete differential geometry, Graduate Studies in Mathematics, vol. 98, American Mathematical Society, Providence, RI, 2008. Integrable structure. MR 2467378, DOI 10.1007/978-3-7643-8621-4
Robert L. Bryant, Surfaces of mean curvature one in hyperbolic space, Astérisque 154-155 (1987), 12, 321–347, 353 (1988) (English, with French summary). Théorie des variétés minimales et applications (Palaiseau, 1983–1984). MR 955072
José A. Gálvez, Antonio Martínez, and Francisco Milán, Flat surfaces in the hyperbolic $3$-space, Math. Ann. 316 (2000), no. 3, 419–435. MR 1752778, DOI 10.1007/s002080050337
José Antonio Gálvez, Antonio Martínez, and Francisco Milán, Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3405–3428. MR 2055739, DOI 10.1090/S0002-9947-04-03592-5
Udo Hertrich-Jeromin, Transformations of discrete isothermic nets and discrete cmc-1 surfaces in hyperbolic space, Manuscripta Math. 102 (2000), no. 4, 465–486. MR 1785326, DOI 10.1007/s002290070037
Udo Hertrich-Jeromin, Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, vol. 300, Cambridge University Press, Cambridge, 2003. MR 2004958, DOI 10.1017/CBO9780511546693
T. Hoffmann, Software for drawing discrete flat surfaces, www.math.tu-berlin.de/ $^\sim$hoffmann/interactive/flatFronts/FlatFront.jnlp .
Tatsuya Koike, Takeshi Sasaki, and Masaaki Yoshida, Asymptotic behavior of the hyperbolic Schwarz map at irregular singular points, Funkcial. Ekvac. 53 (2010), no. 1, 99–132. MR 2668516, DOI 10.1619/fesi.53.99
Masatoshi Kokubu, Wayne Rossman, Kentaro Saji, Masaaki Umehara, and Kotaro Yamada, Singularities of flat fronts in hyperbolic space, Pacific J. Math. 221 (2005), no. 2, 303–351. MR 2196639, DOI 10.2140/pjm.2005.221.303
Masatoshi Kokubu, Wayne Rossman, Masaaki Umehara, and Kotaro Yamada, Flat fronts in hyperbolic 3-space and their caustics, J. Math. Soc. Japan 59 (2007), no. 1, 265–299. MR 2302672
Masatoshi Kokubu, Wayne Rossman, Masaaki Umehara, and Kotaro Yamada, Asymptotic behavior of flat surfaces in hyperbolic 3-space, J. Math. Soc. Japan 61 (2009), no. 3, 799–852. MR 2552916
M. Kokubu and M. Umehara, Global properties of linear Weingarten surfaces of Bryant type in hyperbolic $3$-space, preprint.
Masatoshi Kokubu, Masaaki Umehara, and Kotaro Yamada, Flat fronts in hyperbolic 3-space, Pacific J. Math. 216 (2004), no. 1, 149–175. MR 2094586, DOI 10.2140/pjm.2004.216.149
Steen Markvorsen, Minimal webs in Riemannian manifolds, Geom. Dedicata 133 (2008), 7–34. MR 2390065, DOI 10.1007/s10711-008-9230-8
Christian Mercat, Discrete Riemann surfaces, Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys., vol. 11, Eur. Math. Soc., Zürich, 2007, pp. 541–575. MR 2349680, DOI 10.4171/029-1/14
Ulrich Pinkall and Konrad Polthier, Computing discrete minimal surfaces and their conjugates, Experiment. Math. 2 (1993), no. 1, 15–36. MR 1246481, DOI 10.1080/10586458.1993.10504266
Pedro Roitman, Flat surfaces in hyperbolic space as normal surfaces to a congruence of geodesics, Tohoku Math. J. (2) 59 (2007), no. 1, 21–37. MR 2321990
Wayne Rossman, Masaaki Umehara, and Kotaro Yamada, Irreducible constant mean curvature $1$ surfaces in hyperbolic space with positive genus, Tohoku Math. J. (2) 49 (1997), no. 4, 449–484. MR 1478909, DOI 10.2748/tmj/1178225055
Takeshi Sasaki and Masaaki Yoshida, Hyperbolic Schwarz maps of the Airy and the confluent hypergeometric differential equations and their asymptotic behaviors, J. Math. Sci. Univ. Tokyo 15 (2008), no. 2, 195–218. MR 2478109
Takeshi Sasaki and Masaaki Yoshida, Singularities of flat fronts and their caustics, and an example arising from the hyperbolic Schwarz map of a hypergeometric equation, Results Math. 56 (2009), no. 1-4, 369–385. MR 2575867, DOI 10.1007/s00025-009-0428-3
Takeshi Sasaki, Kotaro Yamada, and Masaaki Yoshida, Derived Schwarz map of the hypergeometric differential equation and a parallel family of flat fronts, Internat. J. Math. 19 (2008), no. 7, 847–863. MR 2437074, DOI 10.1142/S0129167X08004923
Oded Schramm, Circle patterns with the combinatorics of the square grid, Duke Math. J. 86 (1997), no. 2, 347–389. MR 1430437, DOI 10.1215/S0012-7094-97-08611-7
Masaaki Umehara and Kotaro Yamada, Complete surfaces of constant mean curvature $1$ in the hyperbolic $3$-space, Ann. of Math. (2) 137 (1993), no. 3, 611–638. MR 1217349, DOI 10.2307/2946533
Hajime Urakawa, A discrete analogue of the harmonic morphism and Green kernel comparison theorems, Glasg. Math. J. 42 (2000), no. 3, 319–334. MR 1793801, DOI 10.1017/S0017089500030019