- CURVATURE ESTIMATES USING RICCI SOLITONS 111
On the other hand, the ODE behaves much better when s 0 < min {r, O}, in
which case we have
s (t) - r 2: so - r.
The brief analysis above shows that we cannot obtain a good upper
bound for the curvature under the normalized Ricci fl.ow simply by applying
the maximum principle to equation (5.3). Nevertheless, our formulas for s (t)
do allow us to derive useful lower bounds. Set Rmin (t) ~ infxEM2 R (x, t).
Then we have the following estimates.
LEMMA 5.9. Let g (t) be a complete so lution with bounded curvature of
the normalized Ricci flow on a compact surface.- If r < 0, then
R - r )> ( r ) - r )> (Rm1n(O) - r) e"'.
1 - 1 - __ r _ ert
Rmin(O)- If r = 0, then
R > Rmin (0) > -~.- 1 - Rmin ( 0) t t
- If r > 0 and Rmin(O) < 0, then
R )> ( r ) )> Rm1n (0) e-rt.
1 - 1 - _ r ert
Rmin(O)
Notice that in each case, the right-hand side tends to 0 as t __., oo.
In summary, we have uniform lower bounds for the curvature under the
normalized Ricci fl.ow. If the average scalar curvature r is negative, then
Rmin (t) approaches its average r exponentially fast. If r is positive and
Rmin (t) ever becomes nonnegative, it remains so for all time. If r is positive
but Rmin (t) is negative, then Rmin (t) approaches zero exponentially fast.
Since the upper bounds for the curvature which one can derive directly
from the maximum principle blow up in finite time, we shall use other tech-
niques in the next section to obtain a uniform upper bound for the curvature
when r ::; 0 and an exponential upper bound when r > 0.
- How Ricci solitons help us estimate the curvature from above
Let 9.nv (Mn) denote the space of metrics of volume V on a manifold
Mn. There is a natural right action of the group '.Do (Mn) of volume-
preserving diffeomorphisms of Mn on 9.nv given by (g, cp) 1-t cp*g. If M^2 is
a surface, then 9.nA/'.Do is the space of geometric structures of area A in a
given conformal class.
Ricci solitons (equivalently, self-similar solutions of the Ricci fl.ow) may
be regarded as fixed points of the normalized Ricci fl.ow acting 9.nA/'.Do.
These special solutions motivate us to consider certain quantities that may
guide us in developing estimates for general solutions. It turns out to be
particularly useful to study functions which are constant in space on Ricci