On Perelman's Dilaton
  By means of a Kaluza-Klein type argument we show that the Perelman's F-functional is the Einstein-Hilbert action in a space with extra ``phantom'' dimensions.   In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his first famous paper.   As a consequence the Ricci flow (modified by a diffeomorphism and a time-dependent factor) is the evolution of the ``real'' part of the metric under a constrained gradient flow of the Einstein-Hilbert gravitational action in higher dimension. 
Marco Caldarelli, Giovanni Catino, Zindine Djadli, Annibale Magni and   Carlo Mantegazza