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 $H_1(M,\mathbb{Z})=H_1(R,\mathbb{Z})$. Let $\tilde{R}⟶R$ 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 $H^{\infty}(\Di)$.