𝐿²-cohomology of noncompact surfaces

DeBaun, David R.

doi:10.1090/S0002-9947-1984-0743732-3

$L^{2}$-cohomology of noncompact surfaces
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by David R. DeBaun PDF

Trans. Amer. Math. Soc. 284 (1984), 543-565 Request permission

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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