$L^{2}$-cohomology of noncompact surfaces
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- by David R. DeBaun
- Trans. Amer. Math. Soc. 284 (1984), 543-565
- DOI: https://doi.org/10.1090/S0002-9947-1984-0743732-3
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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.References
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Bibliographic Information
- © Copyright 1984 American Mathematical Society
- 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