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Transactions of the American Mathematical Society

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$ L\sp{2}$-cohomology of noncompact surfaces


Author: David R. DeBaun
Journal: Trans. Amer. Math. Soc. 284 (1984), 543-565
MSC: Primary 58A14; Secondary 30F30, 58G32, 60J15
DOI: https://doi.org/10.1090/S0002-9947-1984-0743732-3
MathSciNet review: 743732
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Abstract: This paper is motivated by the question of whether nonzero $ {L^2}$-harmonic differentials exist on coverings of a Riemann surface of genus $ \geqslant 2$. Our approach will be via an analogue of the de Rham theorem. Some results concerning the invariance of $ {L^2}$-homology and the intersection number of $ {L^2}$-cycles are demonstrated. A growth estimate for triangulations of planar coverings of the two-holed torus is derived. Finally, the equivalence between the existence of $ {L^2}$-harmonic one-cycles and the transience of random walks on a planar surface is shown.


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  • [Ah] L. V. Ahlfors, Open Riemann surfaces and extremal problems on compact subregions, Comment. Math. Helv. 24 (1950), 100-154. MR 0036318 (12:90b)
  • [Ahl] -, Zur Theorie der Überlagerungsfläachen, Acta Math. 65 (1935), 157-194. MR 1555403
  • [AhS] L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton Univ. Press, Princeton, N.J., 1960. MR 0114911 (22:5729)
  • [At] M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Asterisque No. 32-33, Soc. Math. France, Paris, 1976, pp. 43-72. MR 0420729 (54:8741)
  • [CFL] R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der Physik, Math. Ann. 100 (1928-1929), 32-74. MR 1512478
  • [D] J. Dodziuk, De Rham-Hodge theory for $ {L^2}$-cohomology of infinite coverings, Topology 15 (1977), 157-165. MR 0445560 (56:3898)
  • [D1] -, Sobolev spaces of differential forms and de Rham-Hodge isomorphism, J. Differential Geometry 16 (1981), 63-73. MR 633624 (83e:58001)
  • [D2] -, Every covering of a compact Riemann surface of genus greater than one carries a non-trivial $ {L^2}$ harmonic differential (to appear).
  • [K] S. Kakutani, Random walk and the type problem of Riemann surfaces, Contributions to the Theory of Riemann Surfaces, Ann. of Math. Stud., No. 30, Princeton Univ. Press, Princeton, N.J., 1953, pp. 95-103. MR 0056100 (15:25e)
  • [KSK] J. G. Kemeny, J. L. Snell and A. W. Knapp, Denumerable Markov chains, Springer-Verlag, New York, 1976. MR 0407981 (53:11748)
  • [L] S. Lefschetz, Introduction to topology, Princeton Univ. Press, Princeton, N.J., 1949. MR 0031708 (11:193e)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0743732-3
Article copyright: © Copyright 1984 American Mathematical Society

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