1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

472 A. BASIC RICCI FLOW THEORY

The following is a generalization of the trace Harnack estimate (see [105]

and [290]).

THEOREM A.57 (Linear trace Harnack estimate). Let (Mn,g (t)) and

h (t), t E [O, T), be a solution to the linearized Ricci flow system:

8

ot9ij = -2Rij,

8

at hij = (f}.Lh)ij

such that ( M, g ( t)) is complete with bounded and nonnegative curvature

operator, h (0) 2:: 0, and Jh (t)Jg(t) ::::; C for some constant C < oo. Then

h (t) 2:: 0 fort E [O, T) and for any vector X we have

(A.27)

where H = gij hij.

Indeed, (A.27) generalizes Hamilton's trace Harnack estimate since we

may take hij = Rij (under the Ricci fl.ow we have gtRij = (f}.L Rc)ij).

4.3. Geometry of gradient Ricci solitons. The asymptotic scalar

curvature ratio of a complete noncompact Riemannian manifold (Mn, g) is

defined by

ASCR (g) = limsup R (x) d (x, 0)^2 ,

d(x,0)--+oo

where 0 E M is a choice of origin. This definition is independent of the
choice of 0.
Theorem 9.44 on p. 354 of [111]:

THEOREM A.58 (Asymptotic scalar curvature ratio is infinite on steady

solitons, n 2:: 3). If (Mn, g, f), n 2:: 3, is a complete steady gradient Ricci

soliton with sect (g) 2:: 0, Re (g) > 0, and if R (g) attains its maximum at

some point, then ASCR (g) = oo.

Theorem 8.46 on p. 318 of [111]:

THEOREM A.59 (Dimension reduction). Let (Mn, g (t)), t E (-oo, w),
w > 0, be a complete noncompact ancient solution of the Ricci flow with
bounded nonnegative curvature operator. Suppose there exist sequences Xi E

M, ri ---7 oo, and Ai ---7 oo such that do~;xi) 2:: Ai and

(A.28) R(y, 0) ::::; r;^2 for ally E Bo(xi, Airi)·

Assume further that there exists an injectivity radius lower bound at (xi, O);

namely, inj 9 (o)(xi) 2:: <Sri for some <5 > 0. Then a subsequence of solutions

(Mn, r:;^2 g(rft), Xi) converges to a complete limit solution (M~, g 00 (t), x 00 )
which is the product of an (n - 1)-dimensional solution (with bounded non-
negative curvature operator) with a line.