Vol. 36, No. 2, 1971

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Behavior of Green lines at the Kuramochi boundary of a Riemann surface

Mitsuru Nakai and Leo Sario

Vol. 36 (1971), No. 2, 447–455

We shall establish necessary and sufficient conditions, in terms of Green lines, for a point of the Kuramochi boundary Γk of a hyperbolic Riemann surface R to be of positive harmonic measure.

Explicitly, let B be the bundle of all Green lines l issuing from a fixed point of R. It forms a measure space with the Green measure. We call a subset A of B a distinguished bundle if it has positive measure and there exists a point p in Γk such that almost every l in A terminates at p. The point p will be referred to as the end of A.

Our main result is that a point p of Γk has positive measure if and only if there exists a distinguished bundle A whose end is p.

We shall also give an intrinsic characterization of the latter property, without reference to points of Γk: A bundle A is distinguished if and only if it has positive measure and for every HD-function u there exists a real number cu such that u has the limit 0u along almost every l in A.

Mathematical Subject Classification
Primary: 30.45
Received: 13 April 1970
Published: 1 February 1971
Mitsuru Nakai
Leo Sario