1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

488 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

Using the facts that

(B.28)

where XT ~ X - (X, v) v (.6.. is the Laplacian with respect to the induced

metric), we have

( H \X, v) udμ = - { \X, .6..X) udμ

}Mt }Mt

= JMt (1vx1

2

u+~(v1x1

2

, Vu)) dμ

= JMt (nu- 2~ 1xTl2 u) dμ.

Multiplying this by z1 7 and adding the difference (which is zero) into (B.27),

we have

!!:__ ( udμ = ( (H \X, v) - \X, ~)2 -H2) udμ

dt}Mt }Mt T 4r

J (

(X, v))

2

=- H--- udμ.

Mt 2r

EXERCISE B.13. Verify the formulas in (B.28).

SOLUTION TO EXERCISE B.13. We have

EPX k EJX

\7i\7jX = EJxiEJxj - rij EJxk

so that

Hence

\7i\7jX = -hijV.

Tracing this yields .6..X = -Hv.

Next we compute

IVXI^2 =lJ-.-. .. EJX EJX =lJgi· .. =n

EJxi EJxJ J

and

I

212 i. I EJX ) I EJX ) I T 2

VIXI =4gJ \EJxi'X \EJxj'X =4 X I.

D