Notes on the greatest harmonic minorant (Q1101580)

On the positive real segment of the unit disk \(U=\{| z| <1\}\), take a sequence of slits \(I_ n\) tending to zero. let \(D=U-\cup I_ n- \{0\}\) and \(D_ n=U-I_ n\). Join D with \(D_ n\) cross-wise along \(I_ n\) for each n. Let R denote the resulting infinite-sheeted covering surface of U and consider the superharmonic function \(u(z)=-\log | z|\) on R. Let H denote the greatest harmonic minorant of u. The main results are as follows: If \(z=0\) is an irregular boundary point of D, then \(H>0\) on R. There exists a sequence of slits \(I_ n\) such that \(z=0\) is a regular boundary point of D and \(H\equiv 0\) on R. The proof is long and delicate.