Bounded Discrete Holomorphic Functions on the Hyperbolic Plane

  • I. A. DynnikovEmail author


It is shown that, for the discretization of complex analysis introduced earlier by S. P. Novikov and the present author, there exists a rich family of bounded discrete holomorphic functions on the hyperbolic (Lobachevsky) plane endowed with a triangulation by regular triangles whose vertices have even valence. Namely, it is shown that every discrete holomorphic function defined in a bounded convex domain can be extended to a bounded discrete holomorphic function on the whole hyperbolic plane so that the Dirichlet energy be finite.


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  1. 1.
    I. Benjamini and O. Schramm, “Random walks and harmonic functions on infinite planar graphs using square tilings,” Ann. Probab. 24 (3), 1219–1238 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Bobenko and M. Skopenkov, “Discrete Riemann surfaces: linear discretization and its convergence,” J. Reine Angew. Math. 720, 217–250 (2016).MathSciNetzbMATHGoogle Scholar
  3. 3.
    D. Chelkak and S. Smirnov, “Discrete complex analysis on isoradial graphs,” Adv. Math. 228 (3), 1590–1630 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    R. J. Duffin, “Basic properties of discrete analytic functions,” Duke Math. J. 23, 335–363 (1956).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    R. J. Duffin, “Potential theory on a rhombic lattice,” J. Comb. Theory 5, 258–272 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    I. A. Dynnikov, “On a new discretization of complex analysis,” Russ. Math. Surv. 70 (6), 1031–1050 (2015) [transl. from Usp. Mat. Nauk 70 (6), 63–84 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    I. A. Dynnikov and S. P. Novikov, “Geometry of the triangle equation on two-manifolds,” Moscow Math. J. 3 (2), 419–438 (2003).MathSciNetzbMATHGoogle Scholar
  8. 8.
    D. V. Egorov, “The Riemann–Roch theorem for the Dynnikov–Novikov discrete complex analysis,” Sib. Math. J. 58 (1), 78–79 (2017) [transl. from Sib. Mat. Zh. 58 (1), 104–106 (2017)].MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. Ferrand, “Fonctions préharmoniques et fonctions préholomorphes,” Bull. Sci. Math., Sér. 2, 68, 152–180 (1944).zbMATHGoogle Scholar
  10. 10.
    P. G. Grinevich and R. G. Novikov, “The Cauchy kernel for the Novikov–Dynnikov DN-discrete complex analysis in triangular lattices,” Russ. Math. Surv. 62 (4), 799–801 (2007) [transl. from Usp. Mat. Nauk 62 (4), 155–156 (2007)].CrossRefzbMATHGoogle Scholar
  11. 11.
    R. P. Isaacs, “A finite difference function theory,” Rev., Univ. Nac. Tucumán, Ser. A 2, 177–201 (1941).MathSciNetzbMATHGoogle Scholar
  12. 12.
    C. Mercat, “Discrete Riemann surfaces and the Ising model,” Commun. Math. Phys. 218 (1), 177–216 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S. P. Novikov, “New discretization of complex analysis: the Euclidean and hyperbolic planes,” Proc. Steklov Inst. Math. 273, 238–251 (2011) [repr. from Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 273, 257–270 (2011)].MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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