1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

272 6. ENTROPY AND NO LOCAL COLLAPSING

PROPOSITION 6.81 (Weaker version using the heat equation). If (.Mn, g)

is a complete noncompact Riemannian manifold with Re (g) 2:: -(n - l)g,

then .:\1 ::::; n(n4-1).

SKETCH OF PROOF. Assume that ¢ is a normalized first eigenfunction

of-~; namely, -~¢ = .:\1¢ and JM ¢^2 = 1. It is well known that¢> 0 (see

[67] or [117] for example). Now we let u: M x [O, oo)---* JR be the solution

of the heat equation

(~ at -~) u=O '

u(0)=¢^2 ,

and let f be defined as before by e-f = u. Since e-f(o)/^2 =¢and¢ is the

first eigenfunction, at t = 0 we have

and equivalently

(6.103)

Applying the above lemma, we have at t = O,

• 2 JM (l\7i\7jfl^2 + Rij\7if\7jf)udμ

= :t JM l\7fl

2

udμ

=JM (2~f-l\7fl

2

) ~udμ

= JM 4.\1~udμ = 0.

Since Re 2:: -(n - l)g and l\7i\7jfl^2 2:: ~(~f)^2 , we then have

0 2:: JM (~(~f)

2

-(n-1) l\7fl

2

) udμ

=JM(~ (4.xi+2.:\1l\7fl

2

+~1\7fl

4

)-(n-l)l\7fl

2

)udμ.

Noting that at t = 0,

4.:\1=4 JM l\7¢1

2

dμ =JM l\7fl

2

udμ,

we obtain

(6.104) 0 2:: -.:\1^12 2 - 4 (n -1) .:\1 + -^1 1 l\7 fl^4 u.dμ.

n 4n M