- MORE RICCI FLOW THEORY AND ANCIENT SOLUTIONS 471
THEOREM A.53 (Strong maximum principle for Rm). Let (Mn,g(t)),
t E [O, T), be a solution of the Ricci flow with nonnegative curvature operator.
There exists 15 > 0 such that for each t E (0, 15) , the set
Image (Rm [g ( t)]) c A^2 T* M
is a smooth subbundle which is invariant under parallel translation and
constant in time. Moreover, Image (Rm [g (x, t)]) is a Lie subalgebra of
A^2 T;M ~.so (n) for all x EM and t E (0, 15).
As an application of the strong maximum principle we have the following
classification result due to W.-X. Shi.
THEOREM A.54 (Complete noncompact 3-manifolds with Re 2:: 0). If(M^3 , g( t)) , t E [O, T), is a complete solution to the Ricci flow on a 3-
manifold with nonnegative sectional (Ricci) curvature, then for t E (0, T)the universal covering solution (M^3 , g(t)) is either
(1) JR^3 with the standard fiat metric,
(2) the product (N^2 , h (t)) x IR, where h (t) is a solution to the Ricciflow with positive curvature and N^2 is diff eomorphic to either 52
or IR^2 or
(3) g(t) and g(t) have positive sectional (Ricci) curvature and henceM^3 is diffeomorphic to 53 or IR^3 (in the former case M^3 is diffeo-
morphic to a spherical space form).4.2. Hamilton's matrix Harnack estimate. Motivated by the con-
sideration of expanding gradient Ricci solitons, Hamilton proved the follow-
ing (see [181]).
THEOREM A.55 (Matrix Harnack estimate for RF). If (Mn, g( t)), t E
[O, T), is a complete solution to the Ricci flow with bounded nonnegative
curvature operator, then for any 1-form WE C^00 (A^1 M) and 2-form U E
C^00 (A^2 M), we have
(A.26) Q (U EB W) = Mij wiwj + 2PpijUPiWj + RpijqUPiUqj 2:: 0,
where
and
Ppij =!= \lp~j - \liRpj·REMARK A.56. See the discussion in Section 2 of Chapter 1 for a moti-
vation for defining Mij and Ppij.
Choosing an orthonormal basis of cotangent vectors { wa} ~=l at any
point (x, t), letting W = wa and U = wa /\ X for any fixed 1-form X,
and summing over a, yield the trace Harnack estimate for the Ricci fl.ow
(Proposition A.43).