7. When breathers and solitons are Einstein
Using the dual field
w^1 = dx1, w^2 = dx^2 - x1dx^4 , w^3 = dx3 - x2dx^4 ,
define a left-invariant metric on S by
g = wl Q9 wl + w2 Q9 w2 + w3 @ w3 + w4@ w4.
Its Ricci tensor is
1 1
Rc(g) = -2
(w^1 @ w^1 ) +
2
(w^3 @ w^3 ) - (w^4 @ w^4 ).Define a vector fieldw^4 = dx^4
'
X = -2x1F1 + (-3x2 + x1x4)F2 + (-4x3 + x2x4)F3 - x4F4.
Then one has
Cxg = -2(w^1 Q9 w^1 ) - 3(w^2 Q9 w^2 ) - 4(w^3 Q9 w^3 ) - (w^4 Q9 w^4 )
and thus
-2Rc(g) = Cxg + 3g.Therefore, (N, g, X) is a Ricci soliton structure.
- When breathers and solitons are Einstein
41In this section, we review some of the significant results to date on
classifying the breathers and Ricci soliton metrics that exist on a given
type of manifold. As noted in the introduction, a shrinking or expanding
soliton on a closed manifold will evolve purely by diffeomorphisms under
the normalized Ricci flow. More generally, we define a breather for the
(normalized or unnormalized) Ricci flow to be a solution g(t) for which
there exists a period T and a diffeomorphism </> such that
g(t + T) = </>* g(t).
So a breather solution is a periodic orbit in the space of metrics modulo
diffeomorphisms, as compared to a Ricci soliton, which is a fixed point.
The following result is the analogue of Proposition 1.13 for breather
solutions to the normalized flow (see [218]).
PROPOSITION 1.66 (Steady and expanding breathers are Einstein). Anybreather or soliton for the normalized Ricci flow on a closed manifold Mn
is either Einstein with constant scalar curvature R ::::; 0 or it has positive
scalar curvature.
PROOF. Under the normalized flow, the scalar curvature satisfies
aR^0 2- 8
= ~R + 21Rcl^2 + -R(R-r),
t n(1.67)
0
where Re denotes the traceless part of the Ricci tensor and r is the average
scalar curvature. By compactness in space and periodicity in time, there
exists a point p E M where R attains its global minimum Rmin· Since
~R 2:: 0 and 8R/8t = 0 at p, then (1.67) implies that Rmin(Rmin - r) ::::; 0,