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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.
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