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Bounded Discrete Holomorphic Functions on the Hyperbolic Plane

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To S. P. Novikov on the occasion of his 80th birthday

Abstract

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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References

  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).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Bobenko and M. Skopenkov, “Discrete Riemann surfaces: linear discretization and its convergence,” J. Reine Angew. Math. 720, 217–250 (2016).

    MathSciNet  MATH  Google Scholar 

  3. D. Chelkak and S. Smirnov, “Discrete complex analysis on isoradial graphs,” Adv. Math. 228 (3), 1590–1630 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  4. R. J. Duffin, “Basic properties of discrete analytic functions,” Duke Math. J. 23, 335–363 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  5. R. J. Duffin, “Potential theory on a rhombic lattice,” J. Comb. Theory 5, 258–272 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  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)].

    Article  MathSciNet  MATH  Google Scholar 

  7. I. A. Dynnikov and S. P. Novikov, “Geometry of the triangle equation on two-manifolds,” Moscow Math. J. 3 (2), 419–438 (2003).

    MathSciNet  MATH  Google Scholar 

  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)].

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Ferrand, “Fonctions préharmoniques et fonctions préholomorphes,” Bull. Sci. Math., Sér. 2, 68, 152–180 (1944).

    MATH  Google Scholar 

  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)].

    Article  MATH  Google Scholar 

  11. R. P. Isaacs, “A finite difference function theory,” Rev., Univ. Nac. Tucumán, Ser. A 2, 177–201 (1941).

    MathSciNet  MATH  Google Scholar 

  12. C. Mercat, “Discrete Riemann surfaces and the Ising model,” Commun. Math. Phys. 218 (1), 177–216 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  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)].

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    I. A. Dynnikov

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  1. I. A. Dynnikov
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Correspondence to I. A. Dynnikov.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2018, Vol. 302, pp. 202–213.

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Dynnikov, I.A. Bounded Discrete Holomorphic Functions on the Hyperbolic Plane. Proc. Steklov Inst. Math. 302, 186–197 (2018). https://doi.org/10.1134/S0081543818060093

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  • DOI: https://doi.org/10.1134/S0081543818060093

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