1. WEAK MAXIMUM PRINCIPLES FOR SCALARS AND 2-TENSORS 7
(1) Nonnegative Ricci curvature is preserved under the Ricci
flow.
Oi{j = -LLij
Here we have X (t) = 0 and /% = SRRtj -
6gPqRiPRqj
+ (2
|Rc|2
-
R2)
gtj
in (10.16). If RijVi = 0 at some point and time, then |Rc| \R2 (since
n = 3) and /3yV*W = (2
|Rc|2
- 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 null-eigenvector 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 3-manifold.
(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
2-tensors (following [245]). Here A (Rm) (i (Rm) v (Rm) denote the eigenvalues of
Rm and C oo is a constant.
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