Complex Legendre duality
  We introduce complex generalizations of the classical Legendre transform, operating on K\"ahler metrics on a compact complex manifold. These Legendre transforms give explicit local isometric symmetries for the Mabuchi metric on the space of K\"ahler metrics around any real analytic K\"ahler metric, answering a question originating in Semmes' work. 
Bo Berndtsson, Dario Cordero-Erausquin, Bo'az Klartag, Yanir A.   Rubinstein
math.DG math.CV