Representing measures and topological type of finite bordered Riemann surfaces

Abstract

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