Representing measures and topological type of finite bordered Riemann surfaces

Abstract:

A finite bordered Riemann surface $\mathcal {R}$ with s boundary components and interior genus g has first Betti number $r = 2g + s - 1$. Let a be any interior point of $\mathcal {R}$ and ${e_a}$ denote evaluation at a on the usual hypo-Dirichlet algebra associated with $\mathcal {R}$. We establish some connections between the topological and, more strongly, the conformal type of $\mathcal {R}$ and the geometry of ${\mathfrak {M}_a}$ the set of representing measures for ${e_a}$. For example, we show that if ${\mathfrak {M}_a}$ has an isolated extreme point, then $\mathcal {R}$ must be a planar surface. Several questions posed by Sarason are answered through exhausting the possibilities for the case $r = 2$.