Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere
Tóm tắt
We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere
$\hat {\mathbb {C}}$
. Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.
Tài liệu tham khảo
Bobenko, A., Pinkall, U., Springborn, B.: Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19(4), 2155–2215 (2015)
Bobenko, A.I., Günther, F.: Discrete complex analysis on planar quad-graphs. In: Advances in Discrete Differential Geometry, pp 57–132. Springer, Berlin (2016)
Bobenko, A.I., Günther, F.: Discrete Riemann surfaces based on quadrilateral cellular decompositions. Adv. Math. 311, 885–932 (2017)
Bobenko, A.I., Mercat, C., Schmies, M.: Period matrices of polyhedral surfaces. In: Bobenko, A.I., Klein, C. (eds.) Computational Approach to Riemann Surfaces, pp 213–226. Springer (2011)
Bobenko, A.I., Mercat, C., Suris, Y.B.: Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function. J. reine angew. Math. 583, 117–161 (2005)
Bobenko, A.I., Pinkall, U.: Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996)
Bobenko, A.I., Skopenkov, M.: Discrete Riemann surfaces: linear discretization and its convergence. J. Reine Angew. Math. 720, 217–250 (2016)
Bobenko, A.I., Springborn, B.A.: Variational principles for circle patterns and Koebe’s theorem. Trans. Amer. Math. Soc. 356, 659–689 (2004)
Bücking, U.: Approximation of conformal mappings by circle patterns. Geom. Dedicata 137, 163–197 (2008)
Bücking, U.: Approximation of conformal mappings on triangular lattices. In: Bobenko, A. (ed.) Advances in Discrete Differential Geometry. Springer (2016)
Chelkak, D., Smirnov, S.: Discrete complex analysis on isoradial graphs. Adv. in Math. 228(3), 1590–1630 (2011)
Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differentialgleichungen der mathematischen Physik. Math. Ann. 100, 32–74 (1928)
Deconinck, B., van Hoeij, M.: Computing Riemann matrices of algebraic curves. Physica D 152-153, 28–46 (2001). Advances in Nonlinear Mathematics and Science: A Special Issue to Honor Vladimir Zakharov
Duffin, R.J.: Discrete potential theory. Duke Math. J. 20, 233–251 (1953)
Duffin, R.J.: Basic properties of discrete analytic functions. Duke Math. J. 23, 335–363 (1956)
Duffin, R.J.: Distributed and lumped networks. J. Math. Mech. 8, 793–826 (1959)
Duffin, R.J.: Potential theory on a rhombic lattice. J. Combin. Th. 5, 258–272 (1968)
Dynnikov, I., Novikov, S.: Geometry of the triangle equation on two-manifolds. Mosc. Math. J. 3, 419–438 (2003)
Ferrand, J.: Fonctions préharmoniques et fonctions préholomorphes. Bull. Sci. Math. 68, 152–180 (1944)
Frauendiener, J., Klein, C.: Computational approach to hyperelliptic Riemann surfaces. Lett. Math. Phys. 105(3), 379–400 (2015)
Frauendiener, J., Klein, C.: Computational approach to compact Riemann surfaces. Nonlinearity 30(1), 138–172 (2017)
Gianni, P., Seppälä, M., Silhol, R., Trager, B.: Riemann surfaces, plane algebraic curves and their period matrices. J. Symbolic Comput. 26(6), 789–803 (1998)
Isaacs, R.P.: A finite difference function theory. Univ. Nac. Tucumá,n Revista A 2, 177–201 (1941)
Lazarus, F., Pocchiola, M., Vegter, G., Verroust, A.: Computing a canonical polygonal schema of an orientable triangulated surface. In: Proceedings of the Seventeenth Annual Symposium on Computational Geometry, SCG ’01, pp. 80–89, Association for Computing Machinery, New York, NY, USA (2001)
Lelong-Ferrand, J.: Représentation confrome et transformations à intégrale de Dirichlet bornée. Gauthier-Villars, Paris (1955)
Lusternik, L.: Über einige Anwendunge der direkten Methoden in der Variationsrechnung. Mat. Sb. 33(2), 173–201 (1926)
MacNeal, R.: The solution of partial differential equations by means of electrical networks. Ph.D. thesis California Institute of Technology (1949)
Matthes, D.: Convergence in discrete Cauchy problems and applications to circle patterns. Conform. Geom. Dyn. 9, 1–23 (2005)
Mercat, C.: Discrete Riemann surfaces and the Ising model. Commun. Math. Phys. 218, 177–216 (2001)
Mercat, C.: Discrete period matrices and related topics. arXiv:math-ph/0111043 (2002)
Mercat, C.: Discrete Riemann surfaces. In: Papadopoulos, A. (ed.) Handbook of Teichmüller Theory, IRMA Lectures in Mathematics and Theoretical Physics. Eur. Math. Soc., vol. 11, pp 541–575 (2007)
Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential geometry operators for triangulated 2-manifolds. In: Visualization and Mathematics III, pp 35–57. Springer (2003)
Molin, P., Neurohr, C.: Computing period matrices and the Abel-Jacobi map of superelliptic curves. arXiv:1707.07249 [math.NT] (2017)
Novikov, S.: New discretization of complex analysis: the Euclidean and Hyperbolic planes. Proc Steklov Inst Math 273, 238–251 (2011)
Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Experiment. Math. 2, 15–36 (1993)
Schramm, O.: Circle patterns with the combinatorics of the square grid. Duke Math. J. 86, 347–389 (1997)
Skopenkov, M.: The boundary value problem for discrete analytic functions. Adv. Math. 240, 61–87 (2013)
Stephenson, K.: Introduction to circle packing: the theory of discrete analytic functions. Cambridge University Press, New York (2005)
Zeng, W., Jin, M., Luo, F., Gu, X.: Canonical homotopy class representative using hyperbolic structure. 171–178 (2009)