Conformal cochains

Wilson, Scott

doi:10.1090/S0002-9947-08-04556-X

Conformal cochains
HTML articles powered by AMS MathViewer

by Scott O. Wilson PDF

Trans. Amer. Math. Soc. 360 (2008), 5247-5264 Request permission

Addendum: Trans. Amer. Math. Soc. 365 (2013), 5033-5033.

Abstract:

In this paper we define holomorphic cochains and an associated period matrix for triangulated closed topological surfaces. We use the combinatorial Hodge star operator introduced in the author’s paper of 2007, which depends on the choice of an inner product on the simplicial 1-cochains.

We prove that for a triangulated Riemannian 2-manifold (or a Riemann surface), and a particularly nice choice of inner product, the combinatorial period matrix converges to the (conformal) Riemann period matrix as the mesh of the triangulation tends to zero.

References

Jozef Dodziuk, Finite-difference approach to the Hodge theory of harmonic forms, Amer. J. Math. 98 (1976), no. 1, 79–104. MR 407872, DOI 10.2307/2373615
J. Dodziuk and V. K. Patodi, Riemannian structures and triangulations of manifolds, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 1–52 (1977). MR 488179
Johan L. Dupont, Curvature and characteristic classes, Lecture Notes in Mathematics, Vol. 640, Springer-Verlag, Berlin-New York, 1978. MR 0500997
Beno Eckmann, Harmonische Funktionen und Randwertaufgaben in einem Komplex, Comment. Math. Helv. 17 (1945), 240–255 (German). MR 13318, DOI 10.1007/BF02566245
H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765, DOI 10.1007/978-1-4612-2034-3
Xianfeng Gu and Shing-Tung Yau, Computing conformal structures of surfaces, Commun. Inf. Syst. 2 (2002), no. 2, 121–145. MR 1958012, DOI 10.4310/CIS.2002.v2.n2.a2
Yu. I. Manin, The partition function of the Polyakov string can be expressed in terms of theta-functions, Phys. Lett. B 172 (1986), no. 2, 184–185. MR 844733, DOI 10.1016/0370-2693(86)90833-6
Ruben Costa-Santos and Barry M. McCoy, Finite size corrections for the Ising model on higher genus triangular lattices, J. Statist. Phys. 112 (2003), no. 5-6, 889–920. MR 2000227, DOI 10.1023/A:1024697307618
Christian Mercat, Discrete Riemann surfaces and the Ising model, Comm. Math. Phys. 218 (2001), no. 1, 177–216. MR 1824204, DOI 10.1007/s002200000348
Mercat, C. “Discrete period Matrices and Related Topics,” arxiv.org math-ph/0111043, June 2002.
Mercat, C. “Discrete Polynomials and Discrete Holomorphic Approximation,” arxiv.org math-ph/0206041.
Andrew Ranicki and Dennis Sullivan, A semi-local combinatorial formula for the signature of a $4k$-manifold, J. Differential Geometry 11 (1976), no. 1, 23–29. MR 423366
Lieven Smits, Combinatorial approximation to the divergence of one-forms on surfaces, Israel J. Math. 75 (1991), no. 2-3, 257–271. MR 1164593, DOI 10.1007/BF02776027
Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, Publish or Perish, Inc., Boston, Mass., 1975. MR 0394452
George Springer, Introduction to Riemann surfaces, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1957. MR 0092855
Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148
Hassler Whitney, On products in a complex, Ann. of Math. (2) 39 (1938), no. 2, 397–432. MR 1503416, DOI 10.2307/1968795
Scott O. Wilson, Cochain algebra on manifolds and convergence under refinement, Topology Appl. 154 (2007), no. 9, 1898–1920. MR 2319262, DOI 10.1016/j.topol.2007.01.017