Some recent applications of the barycenter method in geometry

Abstract

In this paper we describe some recent applications of the barycenter method in geometry. This method was first used by Duady-Earle and later greatly extended by Besson-Courtois-Gallot in their solution of a number of long-standing problems, in particular in their proof of entropy rigidity for closed, negatively curved locally symmetric manifolds. Since there are already a number of surveys describing this work, we will concentrate here only on advances that have occured after these surveys appeared. While most of this paper is a report on results appearing in other papers, some of the material here is new.