Some recent applications of the barycenter method in geometry
  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. 
Christopher Connell and Benson Farb