Contemporary Mathematics

Volume 350, 2004

Conformal Deformations of Riemannian Metrics via

"Critical Point Theory at Infinity" : the Conformally Flat

Case with Umbilic Boundary

Mohameden

Ould

Ahmedou

Dedicated to Haim Brezis and Felix Browder, and to the memory of the victims of 9/11

ABSTRACT.

In this paper we prove that every Riemannian metric on a locally

conformally fiat manifold with umbilic boundary can be conformally deformed

to a scalar fiat metric having constant mean curvature. This result can be seen

as a generalization to higher dimensions of the well known Riemann mapping

Theorem in the plane.

1.

Introduction

In [16], Jose F. Escobar raised the following question: Given a compact Rie-

mannian manifold with boundary, when it is conformally equivalent to one that has

zero scalar curvature and whose boundary has a constant mean curvature ? This

problem can be seen as a "generalization" to higher dimensions of the well known

Riemannian mapping Theorem. The later states that an open, simply connected

proper subset of the plane is conformally diffeomorphic to the disk. In higher di-

mensions few regions are conformally diffeomorphic to the ball. However one can

still ask whether a domain is conformal to a manifold that resembles the ball into

ways : namely, it has zero scalar curvature and its boundary has constant mean

curvature. In the above the term "generalization" has to be understood in that

sens. The above problem is equivalent to finding a smooth positive solution to

the following nonlinear boundary value problem on a Riemannian manifold with

boundary

(Mn,g), n

2

3:

{

A

(n~2)

R -

0 0 •

Mo

-u9

u

+

4

(n~l)

9

u- , u

m ;

OvU

+ n2

2

h9 u

=

Q(M, 8M)un":..2, on oM.

(P)

where R is the scalar curvature of M, his the mean curvature of 8M, vis the outer

normal vector with respect

tog

and Q(M, 8M) is a constant whose sign is uniquely

4

determined by the conformal structure. Indeed if

g

=

u

n-

2

g, then the metric

g

has

Key words and phrases. Critical trace Sobolev exponent, curvature, conformal invariance,

lack of compactness, critical point at infinity .

©

2004 American Mathematical Society

http://dx.doi.org/10.1090/conm/350/06334