1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

294 35. STABILITY OF RICCI FLOW

Notice that fore~ (p - O") /2 E (0, 1) , it follows from (35.15) that

(35.18) and

For fixed 0 < c « 1 and 1/2 ~ f3 < a < 1, we also define

and

9a = 9a (g, c) ~ 9 (3 n (Xo, X1)a,

where g > cg means g (X, X) > c for any vector X such that IXI~ = 1.

Now for each g E 9(3, expression (35.6) allows us to define a linear operator

A9, 9 on h^2 +(]' = E1 by

A ( ) ( A ) kfpq 8

2

§,g "( ij = a X, g , g ij f)xP[)xq "fke

• b ( X, g , A ug, <=:>A g ) kfp ij oxP [) 'Ykf + C ( X, g , A ug<=:> A , u g ,::,2 A) kf ij "fkf·

For g E 9 f3, let us denote by

the unbounded linear operator A 9 , 9 on £ 0 with dense domain D(A(g)) = £ 1. Also

we denote by

A (g) : X1 ~ Xo ---+ Xo

the unbounded linear operator A9, 9 on X 0 with dense domain D(A(g)) = X 1.

By estimating appropriate Holder norms, one can prove that for each g E 9a,

the assignment 'Y H A (g) 'Y is a bounded linear map from X 1 to X 0 and that for
each g E 9 f3, the assignment 'Y H A (g) 'Y is a bounded linear map from £ 1 to £ 0.
We summarize these results as follows:

LEMMA 35. 25. The functions g H A (g) and g H A (g) define analytic maps
9a---+ .C (X1, Xo) and 9(3---+ .C (£1,£0), respectively.

Although for every g E 9(3, A (g) is bounded ifregarded as an operator £ 1 ---+ £ 0 ,
it is unbounded if regarded as an operator £ 0 ---+ £ 0 , and it is in fact only defined

on a dense subspace D (A (g)) = £ 1. Nonetheless, it has the desirable property of

generating an analytic semigroup, which is bounded (and hence defined everywhere)
as a map Eo ---+ Eo.

LEMMA 35.26. For every g E 9(3, A (g) is the infinitesimal generator of a
strongly continuous analytic semigroup on£ (£ 0 ).

PROOF. For all g E 9 (3, the operator A (g) is strongly elliptic. So its spectrum

is discrete, having a limit point only at -oo. Hence there is .A 0 > 0 such that

.AI - A (g) is a topological linear isomorphism from £ 1 onto £ 0 so long as Re .A :'.'.: .Ao.