Calculus of Variations and Geometric Measure Theory

M. Caldarelli - G. Catino - Z. Djadli - A. Magni - C. Mantegazza

On Perelman's Dilaton

created by root on 11 Jun 2008
modified on 22 Jun 2016


Published Paper

Inserted: 11 jun 2008
Last Updated: 22 jun 2016

Journal: Geom. Dedicata
Volume: 145
Pages: 127-137
Year: 2010


By means of a Kaluza-Klein type argument we show that the Perelman's ${\mathcal F}$-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 famous first 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.