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- W2060442807 abstract "Let Wi and W2 be two open Riemann surfaces such that there is a quasi-conformal [1] mapping h of Wi onto W2. Then it is known [2 ] that either both Wi and W2 have Green's function or else neither has, i.e., quasi-conformal mapping preserves the class OG of those surfaces which have no Green's function. From this one is led to the conjecture that quasi-conformal mapping preserves the classes OHB and OHD of surfaces which have no bounded harmonic function or harmonic functions with a finite Dirichlet integral, respectively. In the present paper we shall show that the ring HD of bounded harmonic functions with a finite Dirichlet integral is the same for both Wi and W2. This has as a consequence not only the result of Pfluger on the preservation of the class OG but also the preservation of the class OHD under quasi-conformal mapping. Whether or not OHB is preserved under quasi-conformal mapping is an open question. Another interesting open question is whether or not two topologically equivalent surfaces which have the same ring HBD defined on them admit of a quasi-conformal mapping from one to the other." @default.
- W2060442807 created "2016-06-24" @default.
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- W2060442807 date "1954-02-01" @default.
- W2060442807 modified "2023-10-18" @default.
- W2060442807 title "A property of quasi-conformal mapping" @default.
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- W2060442807 doi "https://doi.org/10.1090/s0002-9939-1954-0060598-4" @default.
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