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
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