1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. BASIC RICCI FLOW

(3) The Ricci tensor Re of g evolves by

(Vl-6.7) :t Rjk = b..LRjk = b..Rjk + 2gPqgrs RpjkrRqs - 2gpq RjpRqk·

In dimension 3, this equation becomes

(Vl-6.10) :tRjk = b..Rjk + 3RRjk - 6gPqRjpRqk + (2 JRcJ^2 - R^2 ) 9jk.

(4) The scalar curvature R of g evolves by

(Vl-6.6)

(5) The volume form dμ evolves by

a

(Vl-6.5) at dμ =-Rdμ.

(6) If /t is a smooth 1-parameter family of geodesic loops, then

dt d Lt ( ) "Yt I t=T = -^1 'Yr Re ( a"'(T as ' a1T) as ds,

where s is the arc length parameter.

Part (6) is similar to Lemma 5.71 on p. 152 of Volume One.

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EXERCISE A.14. Derive the corresponding formulas for the normalized

Ricci flow. For example, under (A.19) we have

aR 2 2

at= b..R + 2 JRcJ - ;,rR.

2.1. Short-and long-time existence. Any smooth metric on a closed

manifold will flow uniquely, at least for a little while (Theorem 3.13 on p. 78

of Volume One).

THEOREM A.15 (Short-time existence for M closed). If (Mn, go) is a

closed Riemannian manifold, then there exists a unique solution g (t) to the

Ricci flow defined on some positive time interval [O, E) such that g (0) =go.

As long as the curvature stays bounded, the solution exists (Corollary

7.2 on p. 224 of Volume One).

THEOREM A.16 (Long-time existence for M closed). If (Mn, g (t)), t E

(0, T), where T < oo, is a solution to the Ricci flow on a closed manifold

with SUPMx(O,T) JRmJ < oo, then the solution g (t) can be uniquely extended

past time T.

In the theorem above the condition supMx(O,T) JRmJ < oo may be re-

placed by SUPMx(O,T) JRcJ < oo; this was proved by Sesum [321].

W.-X. Shi generalized Hamilton's short-time existence theorem to com-

plete solutions with bounded curvature on noncompact manifolds.