1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

7. SOME FURTHER CALCULATIONS RELATED TO :F AND W 277

Differentiating (5.10) again and using (6.106) and (6.107), we have

d2

ds 2 :F(g, f)

=-JM (%sVij-2VikVjk) (Rij+\i'i\i'jf)e-fdμ

• JM Vij\i'p (e-f :s I'fj) dμ


• JM Vij [ \i'i\i'j ( h - ~) - (Rij + \i'i\i'jf) (h - ~)] e-f dμ

+JM [:s (~ -h)] (2~f-l\i'fl

2

+R)e-fdμ

. +JM(~ - h) (div (divv) + (v;Rc) - 2 (divv) · \i'f + Vij\i'd\i'jf) e-f dμ

• JM ( ~ - h) ( 2 ( ~ - \7 f · \7) ( h -~) -2Vij ( Rij + \7 i \7 j !) ) e-f dμ

+JM(~ -h)

2

(2~f-l\i'fl

2

+R)e-fdμ.

Integrating by parts, we have

JM Vij [ \i'i\i'j ( h - ~) - (Rij + \i'i\i'jf) ( h - ~)] e-f dμ

=JM (h- ~) .[\i'i\i'j (vije-1)-vij(Rij+\i'i\i'jf)e-f] dμ

(6.108) = f (h _ V) ( div (divv) + (v, Re) ) e-f dμ

} M 2 -2 (div V) · \7 f + Vij \7 if\7 j f

• 2 JM Vij (~j + \i'i\i'jf) ( h-~) e-f dμ.