The circle pattern uniformization problem

Armando Rodado Amaris, Gina Lusares

Resumen


The existence of an explicit and canonical cell decomposition of the moduli space of closed Riemann surfaces of genus two shows that each Riemann surface of genus two can be parametrised by a 12-tuple of real numbers which corresponds to the  angle coordinates of a graph associated to the surface. This suggests a Circle Pattern Uniformization Problem that we have defined and solved for three classical Riemann surfaces of genus two. Although in general, finding the exact algebraic equations corresponding to a hyperbolic surface from angle coordinates is a hard problem, we prove that known numerical methods can be applied to find approximated equations of Riemann surfaces of genus two from their angle coordinates and graph data for a large family of Riemann surfaces of genus two.


Palabras clave


Uniformization Problem; Riemann Surfaces of Genus Two; Circle pattern Uniformization Problem.

Texto completo:

PDF

Referencias


A. Aigon: Transformation Hyperboliques et Courbes Algèbriques en genre 2 et 3, Université Montpellier II, thèse, (2001).

A. J. R. Rodado: Weierstrass Points and Canonical Cell Decompositions of the Moduli and Teichm"uller Spaces of Riemann Surfaces of Genus Two, University of Melbourne, PhD thesis,http://repository.unimelb.edu.au/10187/2259

A. J. R. Amaris, M.P. Cox: A Flexible Theoretical Representation of The Temporal Dynamics of Structured Populations as Paths on Polytope Complexes, Journal of Mathematical Biology, 2014, DOI 10.1007/s00285-014-0841-4

AMPL, A Modeling Language for Mathematical Programing, http://www.ampl.com

P. Buser: Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, Vol 106, Birkhauser, (1992).

Bowditch B. and B. Epstein: Natural Triangulations Associated to a Surface, Topology vol 27, No 1, pp. 91--117, (1988).

P. Buser and R. Silhol: Some remarks on the uniformization function in genus 2, EPFL Lausanne, Université Montpellier II, http://www.math.univ-montp2.fr/~rs/genus2unif.pdf, 2005.

M. Franz: Convex: a Maple package for convex geometry http://www-fourier.ujf-grenoble.fr/~franz/convex/.

H. M. Farkas and I. Kra : Riemann Surfaces,Graduate Texts in Mathematics 71, Springer-Verlag: New York, Heidelberg, Berlin (1991).

D. Griffiths: The Side Pairing Elements of Maskit's Fundamental Domain for the Modular Group in Genus Two, Annales Academiae Scientiarum Fennicae Mathematica, Vol 26, pp. 3-50, (2001).

D. Griffiths: At most 27 length inequalities define Maskit's fundamental domain for the modular group in genus two,Geometry and Topology Monographs, vol 1, The Epstein Birthday Schrift, pp. 167--180, pp. 3--50, (1998).th.-Phys. Kl. 88, 141-164, 1936.

T. Kuusalo and M. Näätänen: Geometric Uniformization in Genus 2,Annales Academiae Scientiarum Fennicae , Series A.I. Mathematica vol 20, pp. 401-418, (1995).

T. Kuusalo and M. Näätänen: Weierstrass points of extremal surfaces in genus 2, http://www.math.jyu.fi/research/pspdf/231.pdf

J. Harris and I. Morrison: Moduli of Curves, Graduate Texts in Mathematics, Springer-Verlag, (1998).

A. Hass and P. Susskind: it The Geometry of the Hyperelliptic Involutions in Genus Two,Proceeding of the American Mathematical Society, vol 105, 1, January (1989).

T. Jorgensen and M. Näätänen: Surfaces of Genus 2: Generic Fundamental Polygons, Quart. J. Math. Oxford (2), 33, pp. 451-461, (1982).

J. D. McCarthy: Weierstrass points and Z^2 homology, Topology and its Applications 63, pp. 173-188, (1995).

G. Mcshane: Weierstrass Points and Simple Geodesics, Bull. London Math. Soc. 36, pp. 181-187, (2004).

B. Maskit: New Parameters for Fuchsian Groups of Genus 2, Proceeding of The American Mathematical Society, vol 127, No 12, pp. 3643-3652, (1992).

B. Maskit: Matrices for Fenchel-Nielsen Coordinates, Annales Academiae Scientiarum Fennicae, Mathematica vol 26, pp. 267-304, (2001).

B. Maskit: A Picture of the Moduli Space, Invent. Math. 126 , pp. 341-390, (1996).

J. Milnor: Hyperbolic Geometry: The First 150 Years, Bulletin of The American Mathematical Society, Vol. 6, No 1, January (1982).

M. Näätänen: On the Stability of Identification Patterns for Dirichlet Regions, Annales Academiae Scientiarum Fennicae, Series A.I. Mathematica vol 10, pp. 411-417, (1985).

M. Näätänen: A Cellular Parametrization for Closed Surfaces with Distinguished Point, Annales Academiae Scientiarum Fennicae, Series A.I. Mathematica vol 18, pp. 46-64, (1993).

A. Rodado Amaris and G. Lusares: Parametrised databases of surfaces based on Teichmüller theory, CUBO A Mathematical Journal, Vol.18, No 01, pp. 69-88. December (2016).

I. Rivin: Combinatorial optimization in geometry, Adv. in Applied Math. 31:1, pp. 242-271, (2003).

I. Rivin and C. D. Hodgson: A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math., 111 : pp. 77-111, (1993).

B. Rodin and D. Sullivan: The convergence of circle packing to the Riemann mapping, J. Differential Geometry, 26, pp. 349-360, (1987).

K. D. Semmler and M. Seppälä: Numerical Uniformization of Hyperelliptic curves, Proc. ISSAC, (1995).

B. Springborn: Variational Principles for Circles Packing, Ph.D. thesis, arXiv:math.GT/031236 v.1, 18 Dec 2003.


Enlaces refback

  • No hay ningún enlace refback.