On the harmonic boundary of an open Riemann surface, II

A remark on parabolic projective foliations

“…This is a regular C^{\infty} foliation on a compact real manifold denoted M_{d}=(\mathbb{C}P(2)-\cup B_{j})_{d} and its leaves are naturally Riemarm surfaces endowed with the complex structure given by the Schwarz Reflection Principle. Moreover (as it is noticed in [1]) these leaves are still parabolic as a consequence of [14]. Fix a Riemannian metric g on (\mathbb{C}P(2)-\cup B_{j})_{d} , hermitian along the leaves of \mathcal{F}_{d} .…”

Circle Means of Green’s Functions

“…For a direct proof of Lemma 2, we refer to Nakai [4]. 3. We denote by H C B(Ω P ) the class of bounded harmonic functions on Ω p which have continuous extensions to Ω p U C p .…”

“…But in general Ώ p may not be compact. However we have the following Kuramochi theorem [2] (and an alternative proof of it was given by Kusunoki-Mori [3]):…”

On weak boundary components of a Riemann surface

“…The above constructed $\omega_{r}$ has a relation with the harmonic measure of Royden's harmonic boundary [5]. Each $\gamma_{n}=\partial S_{n}$ divides $R$ into two parts, one of which is $S_{n}$ and the other is $R-S_{n}$ .…”

“…$\{\omega_{r_{n}}\}$ decreases monotone when $n$ increases. $\{\Omega_{\Delta_{n}}\}$ also decreases by the way of construction of $\Omega_{\Delta_{n}}$ and by the fact $\Delta_{n}\supset\Delta_{n+1} [5]$ . Then $\Omega_{\gamma_{\Delta}}=\lim\Omega_{\Delta_{n}}$ is the harmonic measure of $\gamma_{\Delta}$ , and…”