1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

292 35. STABILITY OF RICCI FLOW

so r maps into Y. Using the estimate

f(x(t)) - f(y(t)):::; fo

1

J' (O"x(t) + (1 - O")y(t)) dO"(x(t) - y(t)),

we have llf(x) - f(y)llY :::; CR llx - Ylly· Hence

llf(x)llY :::; llf(x) - f(O)llY + llf(O)llY

:::; ~ llxlly + 1 leAxol IY ·

Letting x be the fixed point, this implies that

llxlly :::; C llxollt:e+i,

where we have used 'TJ + w < 0. So for sufficiently small x 0 , one has

llx'llt:e + llxllt:e+1 :::; ce-rit llxollt:e+1.

2.1. Little Holder spaces.

0

We have seen how little Holder spaces arise naturally in maximal regularity
theory. They are the spaces that we use to prove stability of the Ricci- DeTurck
flow at a fiat metric. In this subsection we define them precisely for the space of
symmetric (2, 0)-tensors and summarize some results from the theory that we find
useful. The reader may reference this section as needed.
Recall that if r E N and p E (0, 1), the Holder space cr,p is the Banach space
of all er functions f : Rn -+ R for which the Holder norm II! llr+p is finite. The
subspace of C^00 functions in cr,p is not dense, and so for our purposes we require
a smaller space. We use the following formulation of the little Holder spaces:
DEFINITION 35.20. The little Holder space hr+p (Rn) of functions is the
closure of the subspace of C^00 functions with respect to the ll·llr+p-norm.

It is clear that hr+p is a Banach space. Furthermore, hr+p <--+ hs+a is a con-

tinuous and dense inclusion if s :::; r and 0 < O" < p < l. Corresponding statements

hold for cr,p (D) if n c Rn is a bounded C^00 domain. By fixing a smooth atlas,

one can extend these definitions to functions defined on Mn and taking values in
the bundle S 2 (Mn). (See [40].)

DEFINITION 35.21. The little Holder space hr+p of symmetric 2 -tensors

is the closure of
coo (Mn,S2 (Mn))

with respect to the Holder norm ll·llr+p·

Note in particular that

(35.14)

remains a continuous and dense inclusion.
An exact interpolation space Ie of exponent BE (0, 1) takes any pair Bi ~
Bo of Banach spaces to a Banach space Ie (B 0 , Bi) such that

Bi ~le (Bo, Bi) ~Bo

and such that TE .C (B 0 , Ao) n .C (Bi, Ai) only if

TE .C (Ie (Bo, Bi), le (Ao, Ai))