2 1. RICCI SOLITONS Some highlights of this chapter are a systematic description of the equa- tions obtained from differentiating the gradient Ricci soliton equation, the constructions of the Bryant steady Ricci soliton and the rotationally sym- metric expanding Ricci soliton with positive curvature operator, examples of homogeneous Ricci solitons, introduction to Perelman's energy and entropy functional via gradient Ricci solitons, and the Buscher duality transforma- tion. 1. General solitons and their canonical forms We begin by recalling the following. DEFINITION 1.1 (General Ricci soliton). A solution g(t) of the Ricci flow on Mn is a Ricci soliton (or self-similar solution) if there exist a positive function a(t) and a 1-parameter family of diffeomorphisms (p(t) : M M such that (1.1) g(t) = a(tMt)*g(0). Let dJltt denote the space of Riemannian metrics on a differentiable man- ifold M, and let 2)iff denote the group of diffeomorphisms of M. Consider the quotient map IT : OKet 9DTet/2Hff x R+, where E + acts by scalings. One verifies that a Ricci soliton is a solution g (t) of the Ricci flow for which n (g (£)) is independent of t, i.e., stationary. We start by looking at what initial conditions give rise to Ricci solitons. Differentiating (1.1) yields (1.2) - 2 Re (g(t)) = &(t)(p(t)*go + r(t)ip(t)* (Cxgo), where go ?(0), C denotes the Lie derivative, X is the time-dependent vector field such that X (p(i) (p)) = ^ (ip(t)(p)) for any p G M, and fr = 4*. DEFINITION 1.2. For obvious reasons, we say that g(t) is expanding, steady, or shrinking at a time £o if & (^o) is 0, 0, or 0, respectively. Since Rc(g(t)) = ^(£)*Rc(go)j w e c a n drop the pullbacks in (1.2) and get (1.3) -2Rc(# 0 ) = cr{t)go + £x(t)#o, where X (t) = a(t)X (t). Although go is independent of time, both &(t) and X (t) may depend on time. For example, static Euclidean space (Mn, g{t)) = (Kn,p C an) is a stationary solution to the Ricci flow and, as such, is a steady Ricci soliton however, this solution may also be considered as a Ricci soliton which expands or shrinks modulo diffeomorphisms. In particular, given any function a (t) 0 with J(0) = 1, consider the diffeomorphisms tp(t) : Rn - Rn defined by (p(t) (x) = a{t)~l/2x for x e W1. Then ^(^)*^can = cr(0_1^can. Since g(t) = ^can, we may rewrite this as (1.4) 9(t)=gCa* = (r(tMt)*g{0).
Previous Page Next Page