1. WEAK MAXIMUM PRINCIPLES FOR SCALARS AND 2TENSORS 7
(1) Nonnegative Ricci curvature is preserved under the Ricci
flow.
Oi{j = LLij
Here we have X (t) = 0 and /% = SRRtj 
6gPqRiPRqj
+ (2
Rc2

R2)
gtj
in (10.16). If RijVi = 0 at some point and time, then Rc \R2 (since
n = 3) and /3yV*W = (2
Rc2
 R.A
\V\2
0 at that point and time. (See
Corollary 6.11 on p. 177 in Volume One.)
(2) The pinching inequality Rij sRgij is preserved (for any
£ 1/3).
We leave the verification of the nulleigenvector assumption to the reader as
an exercise (see below); see also Theorem 9.6 of
[244].3
(3) Nonnegative sectional curvature is preserved.
_ 1
Note that, in dimension 3, nonnegative sectional curvature is the same as
nonnegative curvature operator. Actually, nonnegative curvature operator
is preserved under the Ricci flow in any dimension; the proof of this fact
requires a maximum principle for sections of vector bundles which will be
presented in the next section. See [245] or Corollary 6.27 on p. 187 of Volume
One.
REMARK
10.8. Bohm and Wilking [44] proved that, in general, neither
the condition of nonnegative Ricci curvature nor the condition of nonnega
tive sectional curvature is preserved under the Ricci flow on closed manifolds
(see Theorem 13.17 in this book for a statement of their result). Earlier it
was shown that there exist complete solutions to the Ricci flow on noncom
pact manifolds which neither preserve nonnegative sectional curvature [309]
nor nonnegative Ricci curvature [379].
EXERCISE
10.9 (Curvature pinching estimates). Let
(Ms,g(t)),
t G
[0,T), be a solution to the Ricci flow on a closed 3manifold.
(1) Prove that for any £ 1/3, the inequality Rij sRgij is preserved.
(2) Prove that the inequality Rij \Rg%j is preserved.
SOLUTION
10.9. (1) If e = 1/3, then g(0) is an Einstein metric (in fact,
g(0) has constant sectional curvature) and g (t) remains Einstein under the
Ricci flow. Now assume e 1/3; then the assumption Rij sRgij at t — 0
In Lemma 6.28 on p. 189 of Volume One, a proof of the qualitatively equivalent
estimate
A (Rm) C (y (Rm) + // (Rm))
is given using the maximum principle for systems instead of the maximum principle for
2tensors (following [245]). Here A (Rm) (i (Rm) v (Rm) denote the eigenvalues of
Rm and C oo is a constant.