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## C OMPOSITIO M ATHEMATICA

o

### 1 (1989), p. 61-81

<http://www.numdam.org/item?id=CM_1989__69_1_61_0>

© Foundation Compositio Mathematica, 1989, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

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(2)

61

### On the space of asymptotically Euclidean metrics

Department of Mathematics, University of Stockholm, Box 6701, S-11385 Stockholm, Sweden

Received 14 April 1987; accepted 24 May 1988

Abstract. A slice theorem for the action of the group of asymptotically Euclidean diffeo- morphisms on the space of asymptotically Euclidean metrics on Rn is proved.

1. Introduction

Let M be a smooth n-dimensional manifold and let M denote the space of smooth Riemannian metrices on M. The group D of all

on

M acts on M

and the

### orbit-space M/D

describes the coordinate

### independent properties

of M. This space has been studied in the case when M is

Ebin

and

The space

### M/D

is of interest in Riemannian

### geometry

as well as in General

### Relativity,

where in the case n = 3 it is known as superspace, see for

There is an

### analogous problem

in the case where n = 4 and M is the space of solutions to the Einstein

on

### M (i.e.

a subset of the space

### L(M)

of Lorentz metrics on

In this case

### M/D

is the space of true

### dynamical degrees

of freedom of the

### gravitational

field. This case has been studied

Marsden and

The space

has in

### general

the structure of a stratified ILH Frechet manifold

### ,

with lower dimensional strata

### consisting

of the metrics with nontrivial

### isotropy

groups for the action of the

group, i.e.

### Killing

fields.

In the above studies the

of

of M

### played

an

essential role in that it allowed the

### powerful

results about Fredholm proper- ties of

on

### compact

manifolds to be used. The Fredholm

fails in

### general

if the manifold is

and

### special

assump- tions have to be made in order to extend the results to this case. In the paper Supported in part by a Post-Doctoral scholarship from the Swedish Natural Sciences Research Council (NFR).

(3)

### 

some results in this direction were

due

to the

state of the

of

on

### noncompact

manifolds at that

no

### complete

results were obtained.

The aim of this paper is to

### generalize

the results known for M

### compact

to the case where M is the space of Cet)

### asymptotically

Euclidean metrics

on a manifold M

### diffeomorphic

to R n and D is the

group of Cet)

Euclidean

### diffeomorphisms

of M. It should be

### possible

to extend these results to a wide

of

M

may have more

and several

### ends;

the metric on the ends of M may be

to a

### conical, cylindrical,

constant curvature

or other

### metric,

but in the interest of

we will

consider the

### asymptotically

Euclidean case here. This case is also one of interest in the

of

### asymptotically

flat solutions of Einstein

in

the

### present

paper may be seen as a

toward

the results in

this case.

l.l. The

Euclidean case

In the

### asymptotically

Euclidean case, an

for the

of

### ordinary

Sobolev spaces, the class of

Sobolev spaces

introduced

Cantor

### .

Let B be a scalar

constant coefficient

### elliptic operator

on Rn of order m. Then B is a continuous

from

to

and is Fredholm

### (i.e.

has finite dimensional kernel and

if and

if - £5 -

and

£5 + m -

for £5 &#x3E; -

### n/p.

The reason B fails to be Fredholm in the

case where either of these conditions fails to hold is that the index

dim

dim

### coker(B)

is a function of £5 and

as à passes such

a

This is the most

### important

différence from the

case where

a scalar

has index

which

### by

the above remark no

is true in the

case.

Here we

for

### simplicity,

restrict our attention to the case

spaces

### Hs.

These spaces were studied in detail

and

Christodoulou

of

### operators

were derived. The case of

of

### operators

with variable coefficients over Rn was studied in sufficient

for the

purposes

### by

Lockhart and McOwen

The Fredholm

holds for

an

### elliptic operator

with variable coefficients if its

### symbol

tends to that of a

constant coefficient

### operator rapidly enough

and it is in fact

to

a

### sharp

characterization of this.

For fixed 03B4 E

### R, s ~ N,

let MS denote the space of Riemannian metrics on

M such that

E

E

### Hs03B4(S2T*M),

where e 13 the Euclidean

(4)

metric on Rn. MS is called the space

### of asymptotically

Euclidean metrics on

M of order

The group of

of class

will be denotèd

### by Ds .

Let

n &#x3E; 2. The space of all

### diffeomorphisms

which leave MS invariant will be denoted

### by DS+ 1.

This turns out to be

0

where the

group

is either the

group

### O(n)

or the Euclidean group

Rn

on the value of

see

### §2.4.

It turns out that the Fredholm

of the

### operator 0394K,g = 03B4g o Kg : Hs’03B4’(TM) ~ Hs’-203B4’+2

is crucial and that the

in

g - e E

with

and à &#x3E; -

### n/2

for this to hold. While the condition on s is not

### sharp (due

to the fact that we are

with

values of

### s),

the condition on £5 can

not be weakened

### (cf.

the discussion in

p.

### 245]).

We will therefore in the

consider

### only s, £5 satisfying

these conditions. See

### §2.2

for further discussion of the assump- tions used in this paper.

1.2. ILH structures

The

spaces have the

Hilbert spaces

p.

which

makes it

### possible

to make use of the

### theory

of ILH structures introduced

Omori

in

the

### quotient M/D.

Let A : Ds+1 x Ms ~ MS denote the

action

of Ds+ 1 on MS

and let

denote the orbit

### A(DS+1, g)

of g E Ms+1 under the action of D.

We let

### M,

D and D denote the

### corresponding

spaces in the limit s ~ ~.

To construct the

space

### M/D

and its ILH structure

we need

two

### types

of technical results in addition to results about the ILH structure of

D and D:

### a)

A slice for the action

slice at g is

### roughly speaking

an immersed cell transverse to the orbit

This means in

that

### Os(g)

is an

immersed submanifold of MS at g.

A

the orbits

### Os(g)

are closed embedded submanifolds.

The

in

makes use of the

### compactness

of M but it turns out

to be

to use

### essentially

the same method of

also in the

Euclidean case

### infinity.

A result similar to

was stated in

Theorem 5.5

but the

there is

and shows

### only

that the orbit is an immersed manifold.

A detailed discussion of the ILH structure of

will be

to

a future paper.

(5)

1.3. Overview

this paper

In

### §2

some of the necessary

material on

Euclidean

manifolds and

is

A detailed

### analysis

of the

group DS in terms of

### Hs-spaces

does not seem to exist in the literature and is therefore included here.

In

### §3

the structure of the orbits

### Os(g)

is studied and in Theorem

Os

### (g)

is shown to be an embedded submanifold of MS . The method of

is a

of that in

In

### §4

the slice theorem is stated and

### proved following

the ideas of Ebin and in

### §5

some remarks are made about the conclusions that can be drawn from the slice

### theorem, extending

the work of Ebin and

### Bourgignon.

The

details of this will be

### postponed

to a future paper.

2. Technical

2.1. The

Sobolev spaces

### Hb

In this section we will define the

Sobolev spaces

and state some

of their basic

### properties.

DEFINITION 2.1: Let s ~ N and 03B4 E R. The space

### Hs (Rn)

is defined to be the

of

w.r.t. the norm

where

Xo

### le

is the distance function

### given by

the Euclidean

metric e on Rn w.r.t. some

### basepoint

xo Il

The definition is a

case of that

Cantor

the work

of

and Walker

### .

The above norm makes the spaces

### Hs03B4(Rn)

into Hilbert spaces

p.

### 130]

and the inclusion

c

holds

### if s

s’ and

03B4 03B4’. For ease of

we record the

useful results.

LEMMA 2.1

inclusion

Lemma

&#x3E;

then the

inclusion

E

continuous

03B4’ 03B4 +

and s’ s -

LEMMA 2.2

Lemma

The

map

is continuous

(6)

The

case of the

will be useful.

COROLLARY 2.3:

E

&#x3E;

and

and

with

SI and £5’ e R.

continuous map

where

Since s1 &#x3E;

and s’ s,

### multiplication by fl

does not decrease the smoothness

the

Q

E

with

+ 03B4’ +

The result

2.2.

### Asymptotically

Euclidean metrics on Rn

Let M be a COO

### manifold, diffeomorphic

to Rn.

DEFINITION 2.2: Let ô and s

conditions

and

and let e be a

### given

Euclidean metric on M. Let r : M ~ R be defined

### by r(x) = |x - xo ) for

some xo E M and for any R E

let

of

such that

1 if

1 if

R. A

Riemannian

### metric g

on M is said to be

### asymptotically

Euclidean of order

### (s, (03B4)

if there is an R e R such that

E

e

The space of

### asymptotically

Euclidean metrics of order

on M is

denoted

Il

The space

### Ms

is a smooth Hilbert

see

### .

In this paper we will consider the space

for the

two cases:

Case

£5 1 -

### n/2,

Case 2: n &#x3E; 3 and 1 -

£5 -1 +

The

is that in case

the

### asymptotic

group G is the Euclidean group

while in case

G =

the

group, see

(7)

### Further,

we will restrict our attention to values of n &#x3E;

### 2,

since the

behaviour of the fundamental solution of a second order

is

in the case n =

it contains

terms. In

### particular,

this has the consequence that there is no range of ô such that

is an

and the

### analysis

of this situation

### requires

methods different from those used here.

For the same reason, we will not consider the range of ô &#x3E; -1 +

in

which case

~

an

### injection

but has nontrivial cokernel. In this case, the characterization

more

cated.

the case ô = 1 -

is

since

### 0394K,g: Hs+1-n-2 ~ H2-n/2 is

not Fredholm and therefore

which is

the range of the

paper.

In the

### following

we will consider b e R to be fixed and when convenient suppress reference to it in our notation.

2.3.

Euclidean

The

### following

facts are well known

for

Let 11t:

-

be a curve of

### diffeomorphisms

of M with qo = id and

E

### Riem(M)

be a Riemannian metric on M. Then

where the vector field X E

is

X =

and

### L,g

denotes the Lie derivative of g w.r.t. X. We will denote the

X ~

for a

g E

### Riem(M), Kg : r(TM) - 0393(S2T*M)

is a 1:st. order linear

differential

The formal

of

is

the

The

of

is

### injective

and therefore the

defined

is 2:nd order

### elliptic.

The next result follows from the results in

and

### .

PROPOSITION 2.4: With

conditions

any g E MS the

are continuous

s’ s and any b’.

with

the scalar

w.r.t. g E

we have

(8)

and

### Ag

are Fredholm operators between the above spaces

and

b

### satisfies

the conditions

Next we consider the kernel

the operator

### OK,g in

the various cases

### of interest.

PROPOSITION 2.5: Let l5 and s

condition

### (2.1)

and let n &#x3E; 2. Assume that g E

### Ms

and that s’ s. For b’ E R consider the

### The following

statements are true.

b’ - 2 +

then

is an

1 03B4’ -

then

is a

### surjection

with n-dimensional kernel

constant

X E ker

then

where

E ker

n

and

E

2 ô’ -

1 then

is a

with kernel

1:st order

X E ker

n

then

ô 1 -

where

is a

order

in ker

n

and

E

### Hs03B4-1. Further, for

n &#x3E; 3 we have the

t5 = 1 -

then we get

E

### Hs-n/2-03B5for

some e &#x3E; 0 and

&#x3E; 1 -

then X =

+

with

a

order

and

E

### Hs03B4-1,

which in this case

that

0 as x - oo.

g E

### COO,

then in the above cases, ker

c

Part

is

the lines of

Lemma

To prove

and

we

as

+

### Ag,

where the variable coefficient

is

### given by

(9)

where we have used F to denote the

of

elements

the Christoffel

g - e E

so

E

Thus

### by Corollary

2.3 we see that for s’ and 03B4’ in the present range, the

=

and e 03B4 +

In

we can take

6 &#x3E; 0. It

that

### Ag

defines a continuous

### mapping

for some 8 &#x3E; 0.

-

### min (s’ - 1, s - 2).

Assume that 03B4’ satisfies the

of

Theorem

if

E F = ker

then

### Xo

is a constant vectorfield when

### expressed

in

terms of a Cartesian coordinate

### system

for e. It follows that

E

for any 03B4

elements of

### Xo

defines a continuous map

for

and

Recall that

2.4

we have

Let

be

and consider the

= Y.

X

we

the

and

the

of

in the

range of 03B4’

Theorem

we find

e

= min

+

Let

### X(k) = E~l=k Xl

denote the

remainder term. Assume that

### equations (2.3)

have been solved for i =

### 1, ... , k -

1. The it remains to solve

the

of

### J1,.K,e

and estimates similar to the above on

### equations (2.3),

we find there exists a finite k such that

to solve

This proves the

In

if Y =

then we

E F and

E

Part

to show that

E

exists.

is determined

the choice of

This

the

of

(10)

Let

b’ be as in

let Y e

and consider the

= Y.

Theorem

F = ker

### 0394K,e ~ Hs’03B4’

consists of first order vectorfields w.r.t. a Cartesian coordinate

### system

for e. We will denote

and

the spaces of

### homogeneous

first order and constant

the

the

of

and estimates on

### Ag,

we first find that there exists a finite k such that

to the

Since

has

been

in this case,

follows also in

### part (3).

Now let Y = 0 and write X =

+

that X solves

the

that

E F and we

### get

We now have to consider three different cases

### depending

on the value of ô.

then

to solve

and we

=

with

E

E

includes

the constant

If 03B4 = 1 -

### n/2,

then due to the fact that

### 0394K,g

is not Fredholm as a

Theorem

### 4],

the best estimate we

for

is that

E

### Hs nl2+r

for any e &#x3E; 0

may behave like

### log (r)).

Thus we can write X =

with

E

For n &#x3E;

-

ô 2 -

for

in the

range, part

to solve

In

this

Lemma 2.1 that

0 as x - oo.

This

the

of part

Remark :

### (1)

The reason one does not run into the

discussed in

is

that

of the above

### Proposition,

the variable coefficient

is an

### isomorphism

in the same range of ô as its constant coefficient

The above

### analysis

can be extended to

small

to

### give analogous

results. It would be

### interesting

to understand the situation for

03B4’.

The so-called

### logarithmic

translations which have been studied in General

### Relativity

occurs for ô = 1 -

### n/2,

in which case the

does not

### give

any detailed information.

The

### proof given

above is related to that in

Theorem

### 5.2c],

which

covers the case of the scalar

to part

of the

above

see

2.6

(11)

After

the

of the

### operator J1.K,g, it

is easy to prove the

of the scalar

### Laplacian 0394g.

PROPOSITION 2.6: Let ô and s

condition

### (2.1)

and let n &#x3E; 2. Assume that g E

### Ms

and that s’ s. For J’ ~ R consider the

The

### following

statements are true

then

is an

then

is a

### surjection

with one dimensional kernel

constant

the maximum

where

is a

order

in ker

n

and

E

### Hs03B4-1. Further, for

n &#x3E; 3 we have the

c5 = 1

then we get

E

### Hs-n/2-03B5 for

some e &#x3E; 0 and

&#x3E; 1 -

then

+

with

a

order

and

E

### H8

which in this case

that

1 ~ 0 as x ~ 00.

g E

### Coo,

then in the above cases, ker

### 0394g ~ Cl’

~

2.4. The structure

the group

Euclidean

### diffeomorphisms

DEFINITION 2.3: For s &#x3E;

and c5 E

let

### Db

denote the group of those

~ such

id

id are

Denote

L:

x

and left

maps,

### respectively.

PROPOSITION 2.7: Let s &#x3E;

### nl2

and let c5 &#x3E; -

The space

a smooth Sobolev

maps.

is smooth

all il E

and

is

11 E

In

a

### topological

group w.r.t. the induced

(12)

Remark : Part

is

but

### part (2)

is a nontrivial result for the

£5

+ 1 due to

p.

prove the 1 = 0

### part

but the rest is a

of

the

result for M

The result was

stated

for

e R in

In the

we will

### always

assume that n &#x3E; 2. Let s &#x3E;

### n/2

+ 3 and

let £5 be in Case 1 or Case 2 as in

### §2.2.

Let Ds+1 denote the group of all

### diffeomorphisms

of M which leave MS invariant

### (we

suppress reference to £5 in our notation where no confusion can

The

on

the

one that is

induced from the

This

is not

for

however. To

the space

we

for

XS+ 1 =

1 is the

### "Lie-algebra"

of Ds+1. Let X E XS+ 1.

defini-

for g E

we have

E

so

2.4

we find that

a

### surjection

with finite dimensional kernel.

elements in Xs+1

in

be in ker

n

### -nj2-I-e

for some e &#x3E; 0. It is easy to see that

the

space we have

to

### consider,

since it contains all the first order vectorfields.

### Thus, using Proposition 2.5,

we can write X E Xs+1

= Y

where Y E

and

ker

### 0394k,e.

We will consider the cases 1 and 2 defined in

The

is that in Case

### 1, Ds+103B4-1

contains the translations while in Case 2 it does not. We will not consider the

case £5 = 1 -

### n/2. Using (2.5) we see by expressing X in

terms of a Cartesian coordinate

that in Case

### 1, X can

be chosen to be a pure infinitesimal

while in Case

### 2, X is

an infinitesimal Euclidean transformation. It is

### important

to note that in either case we can

### take X E H~loc

since e is C~.

DEFINITION 2.4: Let

be as

### above,

then we define the

1

(13)

72

where the

### norm 1 in

Case 1 denotes the norm on

the Lie

of

### O(n)

and in Case 2 denotes the norm on

the Lie

of

where

R n

### O(n)

denotes the Euclidean group in n

The

Lemma is a

of

### Proposition

2.7

to the case of Ds+

LEMMA 2.8: Let

conditions

and

### (2.2).

The space DS+ 1 is a

smooth

modelled on

x

or

x

### e(n)

in Case 1 and Case

2

The

induced on Ds+1

the

via the

is

the choice

g

DS+ 1 ~

DS+

all ~ E DS+

DS+ 1 ~ DS+

~ E Ds+1+l. In

Ds+l is a

### topological

group w.r.t. the above

### topology. ~

PROPOSITION 2.9: Let

denote

and

### E(n)

in Case 1 and Case 2

### respectively. The following

statements are true.

a normal

the

Ds+

to

Remark : Part

of the

### Proposition

corrects a statement in

p.

### 10]

to the effect that in

=

where

### E(n)

is the Euclidean group in R n. This situation may be

### changed by working

with a more

restrictive set of

### asymptotic

conditions modelled on the

### quasi-isotropic

gauge introduced in General

York

2.5. A

### splitting

theorem

This is the basic result needed to show that the orbits

### Os(g)

are immersed

submanifolds of MS. The main Lemma that we will use is the

LEMMA 2.10: Let

i =

### 1, 2,

3 be Banach spaces and let A:

~

B:

### S2 ~ S3

be continuous linear operators and assume that C = BoA:

-

### S3

is Fredholm. Then ker B

and the range

is closed and

in

First note that

the Fredholm

of

and

### coker(B)

are both finite dimensional.

the

of B

that the

range of

is closed.

Now let

and

### S3/coker(B)

and define

(14)

in the natural way. Then

is

### again

Fredholm and we can

### compute

that

which are finite dimensional.

### By

the definition of

X c Y

if and

if Y = X + Z

denotes direct

### algebraic

sum and Z c Y is some closed

In

### particular,

any finite dimensional

### subspace splits. Hence,

there is a closed

Y c ker

so that ker

into

into

we can write

as a

=

Y.

if we write

ker

then we have the

sum

which

the

### proof.

Let g EMs’ for some s’ s + 1 and let

denote the Lie

of the

group

see

### §2.4. By Proposition 2.4, 0394K,g = 03B4g o Kg

is Fred-

holm as a map from

the finite

it is

### clearly

also Fredholm as a

from XS+

we can

### apply

Lemma 2.9 with A -

and B

to find that

is a

closed

We record these

in the

### following

THEOREM 2.11: The range

Remark : We are

### working

in this paper with the

Sobolev spaces

### H,8

which are Hilbert spaces, so any closed

### subspace splits,

and the Lemma 2.10 is therefore not

necessary.

### However,

the Lemma seems to be of

interest as a

### generalization

of the result of Cantor

Lemma

### 2.2]

and may be useful in

where the

spaces are not

### appropriate.

Il

(15)

3. The structure of orbits

A central

### point

in the construction of the

space

is the

### analysis

of the structure of the group of isometries of a metric g and the orbit

### 08(g)

c MS. In this section we

### provide

the necessary extensions of the results of

3.1.

and

groups

Let g E M and

be its

### isotropy

group w.r.t. A. It is well known

### that Ig

is a Lie group of dimension at most

+

and the Lie

is

the

see

It follows from

Theorem

that if

### Ig

is of maximal dimension and g E

then

is isometric to

In order to

the structure of

### orbits,

we need to construct the

space

### DS/Ig .

This

was done for the case where M is

Ebin

The technical

results in

and the

in

### [9, §5]

make it clear that in the

case, the

### only important

difference from the case treated in

is that the

group

may fail to be

LEMMA 3.1:

E M. The map i :

- DS is an

### imbedding.

Remark : The fact that i is a

onto its

is used

in the

of

Lemma

### Proof:

The smoothness of i and the

of Di are

in the same

way as in

the results

### of §2.

Let i denote the Lie

It

is clear that

is

to a

### subgroup

of the Euclidean group

m Rn.

the

L of i such that

E ker

is

first order

2.5

E L is a

of i which is

to a

of

assume

### that g

is not flat. Let X ~

for some b’

1

be a

### Killing

field for g. Then X E ker

and

2.5

+

where

### Xo is

a constant vectorfield w.r.t. Cartesian coordinates for the Euclidean metric e and

E

2.5

that

we are

E

This

### implies

that X tends to the constant

vectorfield

at

the flow of X is

For x, y E

let

### d(x, y)

be the distance function w.r.t. the metric e. Let

x E M be fixed and

E M such that

### (1) 03A6X0,t(x) = y

for some t, where

### 03A6X0

denotes the flow of

(16)

and

is

### large enough

so that

for all z E M such that

Let

### \$x,t

be the flow of X. It is now

easy to see that

for some constant 6 &#x3E; 0.

### Hence, d(x, 03A6X,t(y) ~

oo as t - oo. But

since X E

### Ig,

g is curved at oo, a contradiction to the

E M.

is

is a

### compact

Lie group and the

in

show that the

c D is an

### embedding.

It remains to consider the case

is flat.

=

### E(n),

the

Euclidean group in n dimensions. Consider

c Ds.

in DS

convergence,

### by

the Holder inclusion Lemma

We

can

and

### i(Ig) explictly

in terms of an Euclidean coordinate

### system.

The result is now obvious in view of the fact that the Holder inclusion

of the

spaces

that the

on DS is

than

Lemma

it is

to prove the

which

to

### [9, Proposion 5.10].

PROPOSITION 3.2: Let g E M. Then the inclusion

### Ig

c DS is a smooth embed-

the

is a

### manifold

and the map 03C0: Ds ~

cross

3.2.

### Proof

that orbits are embedded

e MW and let

and

denote the orbit

in MS under Ds 1

and

### D§± respectively.

Theorem 2.11 shows that the basic results on differen- tial

### topology

of infinite dimensional manifolds

to the

### present

situation and the

result is

the lines in

PROPOSITION 3.3:

conditions

and

in

Then

and

### OD(g)

are smooth immersed

### submanifolds of MS.

~

(17)

We will now prove that

and

### OsD(g)

are also embedded submanifolds

### of MS,

which is what is necessary to construct the

spaces

and

e Ms and let

### 03BAx(03B4)

denote the sectional curvature

w.r.t. the

### 2-plane

03C3 at x e M and let

The

result is

the results of

and the

LEMMA 3.4: With

as

then

g

e

e

### mapping

I: Ms ~ [R is continuous. ~

Thus we have defined a function 1: Ms ~ [R which is invariant under the action of DS+ and

### Ds+103B4-1 and

takes the value 0

### only

at the flat metric. This makes it

### possible

to extend the method used

Ebin to the

case.

THEOREM 3.5:

conditons

and

### (2.2).

Let g e Mco. Then

and

### OsD(g)

are closed embedded

We will consider

the case

the case

### OsD(g) being

similar

and easier. We know

3.2 that

### OS (g)

is an immersed submani-

### fold,

so ait we need to show is that it is closed. Let

Ds+I 1 be a

sequence

and let gm =

### 1J:g.

Assume that gm ~ g~ in the

### topology

of Ms. The Theorem is

### proved

if we can construct a

### diffeomorphism

~~ e Ds+1 such that

### ~*~g =

g~.

First assume that g is not flat.

Lemma

0 and

### I(~*mg) = I(g),

so it follows that

### I(g~) = I(g) ~

0. Let p e M and consider the sequence pm =

### ~m(p).

First we will show that this sequence has a

To

a

assume

has a

+ 3 so

### by

the Holder estimate

Lemma

### 2.4],

g and its derivatives to 2:nd order converge

### uniformly

to e

as p - oo in M and so we find

### neighborhoods Uk

of

of

Pk which are covered

coordinates

### and (!k

~ 00 as k - 00.

This means that on

### ~-1k(Uk), gk = 1Jkg

tends to a flat metric.

### By

assump- tion, gk ~ g~ which means that g~ is flat. This

that 0 =

### I(g~) ~ I(g),

a contradiction. Thus there is a compact subset K c M such that

### ~m(p)

remains in K and it follows that there is a

such that

has

a

### limit q

e M.

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