On unique extremality *

Abstract

Let M(R) denote the unit ball of the space L∞(R) of all essentially bounded Beltrami differentials on a Riemann surface R. It is well known that μ∈M(R) is infinitesimally extremal if and only if μ/(1−|μ|2)∈L∞(R) is infinitesimally extremal. In this paper, it is proved that the analogous result holds for the unique extremality, that is, μ∈M(R) is uniquely infmitesimally extremal if and only if μ/(1−|μ|2)∈L∞(R) is uniquely infinitesimally extremal. An application is also given.