Projections in the Space $H^{\infty}$ and the Corona Theorem for Coverings of Bordered Riemann Surfaces

Abstract

Let $M$ be a non-compact connected Riemann surface of finite type, and $R\subset\subset M$ be a relatively compact domain such that $H_{1}(M,\Z)=H_{1}(R,\Z)$. Let $\tilde R\longrightarrow R$ be a covering. We study the algebra $H^{\infty}(U)$ of bounded holomorphic functions defined in some domains $U\subset\tilde R$. Our main result is a Forelli type theorem on projections in $H^{\infty}(\Di)$.