176 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE
LEMMA 6.7. The scalar curvature of a solution to the Ricci flow evolves
by(6.6)This is the general (n-dimensional) version of the evolution equation for
the curvature of a surface derived in Chapter 5.
Although the evolution equation (6.6) for the scalar curvature R involves
the Ricci tensor, the latter quantity enters only in a nonnegative term. Thus
the weak maximum principle for scalar equations implies that positive scalar
curvature is preserved under the Ricci flow. More generally, we have the
following result.
LEMMA 6.8. Let g (t) be a solution of the Ricci flow with g (0) =go. If
the scalar curvature of go is bounded below by some constant p, then g (t)
has scalar curvature R 2': p for as long as the solution exists.
1.2. The Ricci curvature tensor. Applying the contracted second
Bianchi identity to the second equation for the evolution of Re in Lemma
6.5 puts equation (6.3) in a nicer form.
LEMMA 6.9. Under the Ricci flow, the Ricci tensor evolves by(6.7) :t Rjk = flLRjk = flRjk + 2gPqgrs RpjkrRqs - 2gpq RjpRqk·
PROOF. We have
[)
at Rjk = flLRjk + \7 /9 kR - gpq (\7 j \7 pRqk + \7 k \7 pRjq)
1
= flLRjk + \lj\lkR- 2 (\lj\lkR + \lk \ljR).D
The presence of the Riemann tensor in the evolution equation for the
Ricci tensor is an obstacle to showing that nonnegative Ricci curvature is
preserved in arbitrary dimensions. (In fact, there exist examples [101] where
positive Ricci is not preserved.) One wants, therefore, to attain a better un-
derstanding of the contribution of the Riemann tensor to ( 6. 7). Recall that
in any dimension, the Riemann tensor admits the orthogonal decomposition(6 .8) Rm = R^1 ( o )
2 n ( n _ 1 ) (g^0 g) + n _ 2 Re^0 g + W,
where 0 denotes the Kulkarni- Nomizu product of symmetric tensors,
(P 0 Q)ijk£ ~ PieQjk + PjkQie - PikQje - PjeQiki
0
Re denotes the trace-free part of the Ricci tensor, and Wis the Weyl tensor.
Because the Weyl tensor vanishes in dimension n = 3, equation (6.8) reduces
to(6.9)