1547671870-The_Ricci_Flow__Chow

3. CURVATURE ESTIMATES USING RICCI SOLITONS 113

obtain

.. 1.
(div M)i ~ V'^1 Mji = Y'jV'i\J^1 f - 2,V'iV'jV'^1 f
k 1 1
=Rik V' f + 2 \Jit::,.f = 2 (RV'd + V'iR).

Thus when R > 0, the gradient Ricci soliton equation (5. 7) implies that

V'(log R + J) = 0, hence that log R + f is constant in space. More generally,

regardless of the sign of R, we have

0 = 2(div M)i = V'iR + (R - r) V'd + r\Jd

= V'iR + 2\Ji\Jj fV'jf + r\Jd =\Ji ( R +IV' fl^2 + r f).

Hence on any Ricci gradient soliton, there is a function C (t) of time alone

such that

F ~ R +IV' !1^2 + r f = C.

Since F is constant in space on Ricci gradient solitons, we expect quantities

related to it to satisfy nice evolution equations. In fact, we may always

assume that the potential function f itself satisfies a nice equation.

LEMMA 5.12. Let Jo (x, t) be a potential of the curvature for a solution

(M^2 , g (t)) of the normalized Ricci flow on a compact surface. Then there

is a function c (t) of time alone such that the potential function f ~ Jo+ c

satisfies the evolution equation

a

(5.10) a/= !:::,.f + r f.

PROOF. By Corollary 5.5, we have gtt::,. = (R - r)!:::,.. Thus when we

differentiate (5.8) with respect to time and use equation (5.3), we obtain

(R-r)^2 +t::,.(:/o) = :tR=/::::,./::::,.fo+R(R-r),

which implies that

/::::,. ( :tfo) =/::::,.(!:::,.Jo+ r Jo).

Since the only harmonic functions on a closed manifold are constants, there

is a function '"'( ( t) of time alone such that

a

a/o = 1::::,.fo + rfo + 'Y·

The lemma follows by choosing c (t) ~ -ert J~ e-rT'Y (T) dT. D

REMARK 5.13. Our normalization for f differs from that introduced

in [60]. Compare the result above with Definition 4 .1 and the evolution

equation derived in Section 4.2 of that paper.

Applying the maximum principle to equation (5.10) yields a useful esti-

mate.