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Conformal cochains
Author(s):
Scott
O.
Wilson
Journal:
Trans. Amer. Math. Soc.
360
(2008),
5247-5264.
MSC (2000):
Primary 57R57, 32G20;
Secondary 30F99
Posted:
April 10, 2008
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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:
-
- 1.
- Dodziuk, J. ``Finite-Difference Approach to the Hodge Theory of Harmonic Forms,'' Amer. J. of Math. 98 No. 1 (1976) 79-104. MR 0407872 (53:11642)
- 2.
- Dodziuk J. and Patodi V. K. ``Riemannian Structures and Triangulations of Manifolds,'' Journal of Indian Math. Soc. 40 (1976) 1-52. MR 0488179 (58:7742)
- 3.
- Dupont, J. ``Curvature and Characteristic Classes,'' Lecture Notes in Mathematics, vol. 640, Springer-Verlag (1978). MR 0500997 (58:18477)
- 4.
- Eckmann, B. ``Harmonische Funktionnen und Randvertanfgaben in einem Komplex,'' Commentarii Math. Helvetici, 17 (1944-45), 240-245. MR 0013318 (7:138f)
- 5.
- Farkas, H. and Kra, I. ``Riemann Surfaces,'' Springer-Verlag (1991). MR 1139765 (93a:30047)
- 6.
- Gu, X. and Yau, S.T. ``Computing Conformal Structures of Surfaces,'' Comm. Inf. Sys. 2 No. 2 (2002) 121-146. MR 1958012 (2003m:65023)
- 7.
- Manin, Y. ``The Partition Function of the Polyakov String can be Expressed in Terms of Theta-Functions,'' Phys. Lett. B 172 No. 2 (1986) 184-185. MR 844733 (87k:81120)
- 8.
- Costa-Santos, R. and McCoy, B.M. ``Finite Size Corrections for the Ising Model on Higher Genus Triangular Lattices,'' J. Statist. Phys. 112 Nos. 5-6 (2003) 889-920. MR 2000227 (2004h:82021)
- 9.
- Mercat, C. ``Discrete Riemann Surfaces and the Ising Model,'' Comm. Math. Phys. 218 (2001), No. 1, 177-216. MR 1824204 (2002c:82019)
- 10.
- Mercat, C. ``Discrete period Matrices and Related Topics,'' arxiv.org math-ph/0111043, June 2002.
- 11.
- Mercat, C. ``Discrete Polynomials and Discrete Holomorphic Approximation,'' arxiv.org math-ph/0206041.
- 12.
- Ranicki, A. and Sullivan, D. ``A Semi-local Combinatorial Formula for the Signature of a
 -manifold,'' J. Diff. Geometry, Vol. II (1976), 23-29. MR 0423366 (54:11345) - 13.
- Smits, L. `` Combinatorial Approximation to the Divergence of 1-forms on Surfaces,'' Israel J. of Math. 75 (1991) 257-271. MR 1164593 (93d:57052)
- 14.
- Spivak, M. A. ``Comprehensive Introduction to Differential Geometry,'' Vol. IV, Publish or Perish Inc., Boston, MA, 1975. MR 0394452 (52:15254a)
- 15.
- Springer, G. ``Introduction to Riemann Surfaces,'' Addison-Wesley Publ. Company (1957). MR 0092855 (19:1169g)
- 16.
- Whitney, H. ``Geometric Integration Theory,'' Princeton Univ. Press (1957). MR 0087148 (19:309c)
- 17.
- Whitney, H. ``On Products in a Complex,'' Annals of Math. (2) 39 (1938) 397-432. MR 1503416
- 18.
- Wilson, S.O. ``Cochain Algebra on Manifolds and Convergence Under Refinement,'' Topology and Its Applications 154 No. 9 (2007) 1898-1920. MR 2319262
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Additional Information:
Scott
O.
Wilson
Affiliation:
School of Mathematics, University of Minnesota-Twin Cities, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, Minnesota 55455
Email:
scottw@math.umn.edu
DOI:
10.1090/S0002-9947-08-04556-X
PII:
S 0002-9947(08)04556-X
Keywords:
Cochains,
Hodge-star,
Riemann surface,
period matrices
Received by editor(s):
August 8, 2006
Posted:
April 10, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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