1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

7. WHEN BREATHERS AND SOLITONS ARE EINSTEIN 43

REMARK 1.69. A version of Corollary 1.68 for the Ricci tensor may be

proved directly from the evolution of the curvature operator eigenvalues,

using the maximum principle for systems. See [218] for details.

COROLLARY 1.70 (Ricci flow on 3-manifolds: R bounds Rm). For any

solution ( M^3 , g ( t)) of the Ricci flow on a closed 3-manifold, if R is uni-

formly bounded, then IRml is also uniformly bounded.

PROOF. If R ::::; C, then v 2: -C' for some constant C'. If .A 2: μ 2: v are

the eigenvalues of Rm, then R = .A + μ + v and .A ::::; R - 2v ::::; C + 20'. D

A nice application of the Hamilton-Ivey estimate is the following (see

[218]).

THEOREM 1.71 (Shrinking breathers on closed 3-manifolds are Einstein).

The only solitons (or breathers) for the normalized Ricci flow on a closed

connected 3-manifold M are constant sectional curvature metrics.

The example of Koiso's shrinking soliton (see Section 7 of Chapter 2) shows

that this result cannot be extended to dimension 4.

PROOF. By Proposition 1.66, either the metric is Einstein (which is

equivalent to constant curvature in dimension 3) or Rmin > O; in the lat-

ter case, the breather is a shrinking breather. By Corollary 1.68, for the
unnormalized Ricci flow,

Rm > -'ljJ (R) id --t 0 as R --t oo

R - R '
and Rmin (t) --t oo as t --t T, where [O, T) is the maximal time interval of
existence. So the sectional curvature becomes asymptotically nonnegative
under the unnormalized flow. Since we assume our solution to be either a
soliton or a breather, the curvature must have been nonnegative to begin
with.
In [179], Hamilton has shown that either the sectional curvature be-
comes strictly positive immediately or M splits locally as a product of a
1-dimensional flat factor and a surface with positive curvature, and this
splitting is preserved by the flow. In the former case, we know g converges
to a metric of constant positive sectional curvature under the normalized
flow. To rule out the latter case, consider the evolution equation for r under
the normalized flow:

! J Rdμ= - j ( ~;,Rc-~Rg) dμ,

where the volume is fixed at one and the pointwise inner product is given by
contraction using the metric. We can calculate the integrand on the right
as