@inbook{b5be236af89a46d98ee0099c88f0056d,

title = "The 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 {\textquoteleft}energy{\textquoteright}). Considering the prominent role variational methods have played in geometric analysis, pde{\textquoteright}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{\textquoteright}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{\textquoteright}man{\textquoteright}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.",

keywords = "Gradient Flow, Injectivity Radius, Natural Equation, Ricci Flow, Riemannian Metrics",

author = "Ben Andrews and Christopher Hopper",

note = "Publisher Copyright: {\textcopyright} 2011, Springer-Verlag Berlin Heidelberg.",

year = "2011",

doi = "10.1007/978-3-642-16286-2_10",

language = "English",

isbn = "9783642159664",

series = "Lecture Notes in Mathematics",

publisher = "Springer Verlag",

pages = "161--171",

booktitle = "The Ricci Flow in Riemannian Geometry",

address = "Germany",

}