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Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space. (English) Zbl 1277.53053

Away from umbilic points of a smooth minimal (or CMC) immersed surface \(f(x,y)\) into \(\mathbb{R}^3\) or \(\mathbb{H}^3\) one can take isothermic coordinates \((x,y)\), that are isothermal and diagonalize the second fundamental form for the induced metric. In the Euclidean case, the stereographic projection of the Gauss map defines a holomorphic function \(g\) on the complex plane that, with the Hopf differential, constitutes data to define the Weierstrass representation of \(f\), and consequent expressions of the derivatives \(f_x\) and \(f_y\) give rise to the definition of the difference \(f_q-f_p\) for \(p=(m,n)\) and \(q\) either \((m+1,n)\) or \((m, n+1)\) obtaining in this way a discretization of the surface \(f\) up to translations. A discrete minimal surface \(f\) is then defined (up to translations) from a holomorphic discrete map \(g:D\to \mathbb{C}\), \(D\subset\mathbb{Z}^2\), that is, the cross ratio (in quaternionic notation) \(\operatorname{cr}_{m, n}=(g_{m+1,n}-g_{m,n})(g_{m+1,n+1}-g_{m+1,n})^{-1}(g_{m,n+1}-g_{m+1,n+1})(g_{m,n}-g_{m,n+1})^{-1}\) can be described as \(\operatorname{cr}_{m, n}=\frac{\alpha_{(m,n)(m+1,n)}}{\alpha_{(m,n)(m,n+1)}}<0\) (not necessarily equal to \(-1\)) holding for all quadrilaterals, for some symmetric discrete real function \(\alpha\) satisfying \(\alpha_{(m,n)(m+1,n)}=\alpha_{(m,n+1)(m+1,n+1)}\), \(\alpha_{(m,n)(m,n+1)}=\alpha_{(m+1,n)(m+1,n+1)}\).
The first proposal of the authors is to define discrete flat surfaces in the 3-dimensional hyperbolic space in the context of discrete integrable systems. Considering the 3-dimensional hyperbolic space as a hyperquadric of the Minkowski 4-space \(\mathbb{R}^{3,1}\) and using \( SL_2(\mathbb{C})\)-valued lifts \(F\) of a smooth isothermically-parameterized CMC 1 surface \(f_1:M\to\mathbb{H}^3\) and Bryant’s equation on \(dF\), expressed in terms of an holomorphic function \(g\) (Weierstrass data) with non-zero derivative, one obtains \(f_1=F\cdot \bar{F}^T\) as shown by R. L. Bryant in [Astérisque 154–155, 321–347 (1988; Zbl 0635.53047)].
To each \(f_1\) there is a related flat surface \(f_0\) with singularities, and a suitable change of coordinates leads to an equation satisfied by the components of the lift that in some particular cases is the Airy equation. One-parameter families of deformations \(f_t\) through linear Weingarten surfaces of Bryant-type between \(f_0\) and \(f_1\) are considered and non-uniqueness is shown for this type of deformations. A discretization of such surfaces can be built using the light cone model space of \(\mathbb{H}^3\) in \(\mathbb{R}^{4,1}\) and quaternionic notation, as shown by U. Hertrich-Jeromin in [Manuscr. Math. 102, No. 4, 465–486 (2000; Zbl 0979.53008)]. In Theorem 4.2, the authors give an alternative way to define them by \(f_1=(1/\mathrm{det} F)F\bar{F}^T\) (up to rigid motion in \( \mathbb{H}^3 \)) following Bryant’s formula \( F_q-F_p = F_p G(g_p,g_q) \frac{\lambda \alpha_{pq}}{g_q-g_p}\), where \(G(g_p,g_q)\) is a matrix depending only on \(g_p\) and \(g_q\) with a discrete holomorphic function \(g\), \(\alpha\) is the cross ratio factorization function and \(\lambda\) a free real parameter. Multiplication of \(F\) by a suitable matrix depending on \(g\) defines a new lift \(E\) that represents \(f_0\) discrete and flat in \(\mathbb{H}^3\), and it is proved in Theorem 4.6 that \(f_0\) has concircular quadrilaterals.
The authors also describe a deformation family \(f_t\) of discrete linear Weingarten surfaces from \(f_0\) to \(f_1\) and with concircular quadrilaterals. A discrete caustic surface of a flat discrete one is also defined by following the smooth example in terms of the Weierstrass data. This requires to define a normal at vertices of a discrete flat surface and it is shown that negativeness of \(\lambda \alpha_{pq}\) is equivalent to the normal geodesics in \(\mathbb{H}^3\) emanating from two adjacent vertices \(f_p\) and \(f_q\) to intersect (in a unique point), or equivalently the edge \(pq\) is vertical. The set \(C_f\) of such intersection points is a discrete surface named the caustic or focal surface of \(f\), not necessarily flat. A lift \(E_{(C_f)}\) is defined for each vertical edge \(pq\) satisfying the formula \( C_f=(1/\mathrm{det}(E_{C_f})) E_{C_f}\cdot \overline{ E_{C_f}}^T\). Discrete caustics have properties similar to that of caustics in the smooth case. Many examples are described throughout the paper, with some beautiful figures, and special attention is given to the discrete flat surface for the case \(g=z^{4/3}\), related to the Airy equation.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A30 Conformal differential geometry (MSC2010)
53A35 Non-Euclidean differential geometry
52C99 Discrete geometry
52B70 Polyhedral manifolds
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References:

[1] S. I. Agafonov, Discrete Riccati equation, hypergeometric functions and circle patterns of Schramm type, Glasg. Math. J. 47 (2005), no. A, 1 – 16. · Zbl 1081.39010 · doi:10.1017/S0017089505002247
[2] 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. · Zbl 1293.30087
[3] 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. · Zbl 1001.53001 · doi:10.1017/CBO9780511569432.009
[4] 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. · Zbl 1122.53003 · doi:10.4007/annals.2006.164.231
[5] A. I. Bobenko, D. Matthes, and Yu. B. Suris, Discrete and smooth orthogonal systems: \?^{\infty }-approximation, Int. Math. Res. Not. 45 (2003), 2415 – 2459. · Zbl 1085.53008 · doi:10.1155/S1073792803130991
[6] 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. · Zbl 1099.37054 · doi:10.1515/crll.2005.2005.583.117
[7] Alexander Bobenko and Ulrich Pinkall, Discrete isothermic surfaces, J. Reine Angew. Math. 475 (1996), 187 – 208. · Zbl 0845.53005 · doi:10.1515/crll.1996.475.187
[8] 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. · Zbl 0932.53004
[9] Alexander I. Bobenko and Boris A. Springborn, A discrete Laplace-Beltrami operator for simplicial surfaces, Discrete Comput. Geom. 38 (2007), no. 4, 740 – 756. · Zbl 1144.65011 · doi:10.1007/s00454-007-9006-1
[10] Alexander I. Bobenko and Yuri B. Suris, Discrete differential geometry, Graduate Studies in Mathematics, vol. 98, American Mathematical Society, Providence, RI, 2008. Integrable structure. · Zbl 1158.53001
[11] 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).
[12] José A. Gálvez, Antonio Martínez, and Francisco Milán, Flat surfaces in the hyperbolic 3-space, Math. Ann. 316 (2000), no. 3, 419 – 435. · Zbl 1003.53047 · doi:10.1007/s002080050337
[13] 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. · Zbl 1068.53044
[14] Udo Hertrich-Jeromin, Transformations of discrete isothermic nets and discrete cmc-1 surfaces in hyperbolic space, Manuscripta Math. 102 (2000), no. 4, 465 – 486. · Zbl 0979.53008 · doi:10.1007/s002290070037
[15] Udo Hertrich-Jeromin, Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, vol. 300, Cambridge University Press, Cambridge, 2003. · Zbl 1040.53002
[16] T. Hoffmann, Software for drawing discrete flat surfaces, www.math.tu-berlin.de/\( ^\sim \)hoffmann/interactive/flatFronts/FlatFront.jnlp .
[17] 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. · Zbl 1190.33004 · doi:10.1619/fesi.53.99
[18] 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. · Zbl 1110.53044 · doi:10.2140/pjm.2005.221.303
[19] 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. · Zbl 1120.53036
[20] 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. · Zbl 1177.53059
[21] M. Kokubu and M. Umehara, Global properties of linear Weingarten surfaces of Bryant type in hyperbolic \( 3\)-space, preprint. · Zbl 1235.53063
[22] Masatoshi Kokubu, Masaaki Umehara, and Kotaro Yamada, Flat fronts in hyperbolic 3-space, Pacific J. Math. 216 (2004), no. 1, 149 – 175. · Zbl 1078.53009 · doi:10.2140/pjm.2004.216.149
[23] Steen Markvorsen, Minimal webs in Riemannian manifolds, Geom. Dedicata 133 (2008), 7 – 34. · Zbl 1172.53039 · doi:10.1007/s10711-008-9230-8
[24] 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. · Zbl 1136.30315 · doi:10.4171/029-1/14
[25] Ulrich Pinkall and Konrad Polthier, Computing discrete minimal surfaces and their conjugates, Experiment. Math. 2 (1993), no. 1, 15 – 36. · Zbl 0799.53008
[26] 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. · Zbl 1140.53004
[27] 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. · Zbl 0913.53025 · doi:10.2748/tmj/1178225055
[28] 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. · Zbl 1182.33008
[29] 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. · Zbl 1182.33009 · doi:10.1007/s00025-009-0428-3
[30] 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. · Zbl 1177.33014 · doi:10.1142/S0129167X08004923
[31] Oded Schramm, Circle patterns with the combinatorics of the square grid, Duke Math. J. 86 (1997), no. 2, 347 – 389. · Zbl 1053.30525 · doi:10.1215/S0012-7094-97-08611-7
[32] 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. · Zbl 0795.53006 · doi:10.2307/2946533
[33] Hajime Urakawa, A discrete analogue of the harmonic morphism and Green kernel comparison theorems, Glasg. Math. J. 42 (2000), no. 3, 319 – 334. · Zbl 1002.05049 · doi:10.1017/S0017089500030019
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