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On the hyperbolicity of certain planar domains. (English) Zbl 0628.30043

Let \(S\) be a compact Riemann surface of genus greater than one and p: \(\tilde S\to S\) be an infinite covering. J. Dodziuk [Acta Math. 152, 49-56 (1984; Zbl 0541.30035)] proved that \(\tilde S\) admits a nonzero \(L^ 2\) harmonic 1-form. The basic step in his proof is to demonstrate the existence of a nonconstant bounded subharmonic function on \(\tilde S\) when it is planar and he uses several deep results from differential geometry to establish this. This author gives a very short and simple proof of this result.
Reviewer: D.Minda

MSC:

30F30 Differentials on Riemann surfaces
30F10 Compact Riemann surfaces and uniformization

Citations:

Zbl 0541.30035
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References:

[1] Ahlfors, L.V.: Conformal invariants: Topics in geometric function theory. New York: McGraw-Hill Book Co. 1973 · Zbl 0272.30012
[2] Ahlfors, L.V., Sario, L.: Riemann Surfaces. Princeton: Princeton University 1960 · Zbl 0196.33801
[3] Dodziuk, J.: Every covering of a compact Reiemann surface of genus greater than one carries a nontrivialL 2 harmonic differential. Acta Math.152, 49-56 (1984) · Zbl 0541.30035 · doi:10.1007/BF02392190
[4] Donnely, H., Fefferman, C.:L 2-cohomology and index theorem for the Bergman metric. Ann. Math.118, 593-618 (1983) · Zbl 0532.58027 · doi:10.2307/2006983
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