The $$\mathcal{F}$$-Functional and Gradient Flows

Abstract

After Ricci flow was first introduced, it appeared for many years that there was no variational characterisation of the flow as the gradient flow of a geometric quantity. In particular, Bryant and Hamilton established that the Ricci flow is not the gradient flow of any functional on Met – the space of smooth Riemannian metrics – with respect to the natural L2 inner product (with the exception of the two-dimensional case, where there is indeed such an ‘energy’). Considering the prominent role variational methods have played in geometric analysis, pde’s and mathematical physics, it seemed surprising that such a natural equation as Ricci flow should be an exception. One of the many important contributions Perel’man made was to elucidate a gradient flow structure for the Ricci flow, not on Met but on a larger augmented space. Part of this structure was already implicit in the physics literature [Fri85]. In this chapter we discuss this structure, at the centre of which is Perel’man’s F-functional [Per02]. The analysis will provide the ground work for the proof of a lower bound on injectivity radius at the end of Chap. 11.