Tiểu Luận Existence of conformal metrics with constant Qcurvature

Existence of conformal metrics
with constant Q-curvature
By Zindine Djadli and Andrea Malchiodi
Abstract
Given a compact four dimensional manifold, we prove existence of con-formal metrics with constant Q-curvature under generic assumptions. The
problem amounts to solving a fourth-order nonlinear elliptic equation with
variational structure. Since the corresponding Euler functional is in general
unbounded from above and from below, we employ topological methods and
min-max schemes, jointly with the compactness result of [35].
1. Introduction
In recent years, much attention has been devoted to the study of partial
dierential equations on manifolds, in order to understand some connections
between analytic and geometric properties of these objects.
A basic example is the Laplace-Beltrami operator on a compact surface
(; g ). Under the conformal change of metric ~ g = e
2w
g, we have
(1)

~ g = e
2w

g
;
g w + Kg = K~ g
e
2w
;
where
g
and Kg (resp.

~ g
and K~ g
) are the Laplace-Beltrami operator and
the Gauss curvature of (; g ) (resp. of (; ~ g)). From the above equations one
recovers in particular the conformal invariance of
R

Kg
dV
g
, which is related
to the topology of  through the Gauss-Bonnet formula
(2)
Z

Kg
dV
g
= 2();
where () is the Euler characteristic of . Of particular interest is the classi-cal Uniformization Theorem, which asserts that every compact surface carries
a (conformal) metric with constant curvature.
On four-dimensional manifolds there exists a conformally covariant oper-ator, the Paneitz operator, which enjoys analogous properties to the Laplace-Beltrami operator on surfaces, and to which is associated a natural concept
of curvature. This operator, introduced by Paneitz, [38], [39], and the cor-responding Q-curvature, introduced in [6], are dened in terms of the Ricci
814 ZINDINE DJADLI AND ANDREA MALCHIODI
tensor Ric
g
and the scalar curvature Rg
of the manifold (M; g) as
Pg
(')=

2
g
' + divg

2
3
Rg
g 2Ric
g

d';(3) Qg =
1
12

g Rg R
2
g
+ 3jRicg
j
2


; (4)
where ' is any smooth function on M . The behavior (and the mutual relation)
of Pg
and Qg
under a conformal change of metric ~ g = e
2w
g is given by
(5) P~ g = e
4w
Pg
; Pg w + 2Qg
= 2Q~ g
e
4w
:
Apart from the analogy with (1), we have an extension of the Gauss-Bonnet
formula which is the following:
(6)
Z
M

Qg +
jWg
j
2
8

dV
g
= 4
2
(M );
where Wg
denotes the Weyl tensor of (M; g) and (M ) the Euler characteristic.
In particular, since jWg
j
2
dV
g
is a pointwise conformal invariant, it follows that
the integral of Qg
over M is also a conformal invariant, which is usually denoted
with the symbol
(7) k
P =
Z
M
Qg
dV
g
:
We refer for example to the survey [18] for more details.
To mention some rst geometric properties of Pg
and Qg
, we discuss some
results of Gursky, [29] (see also [28]). If a manifold of nonnegative Yamabe class
Y (g) (this means that there is a conformal metric with nonnegative constant
scalar curvature) satises k
P  0, then the kernel of Pg
are only the constants
and Pg  0, namely Pg
is a nonnegative operator. If in addition Y (g) > 0, then
the rst Betti number of M vanishes, unless (M; g) is conformally equivalent
to a quotient of S
3
 R. On the other hand, if Y (g)  0, then k
P  8
2
,
with the equality holding if and only if (M; g) is conformally equivalent to the
standard sphere.
As for the Uniformization Theorem, one can ask whether every four-manifold (M; g) carries a conformal metric ~ g for which the corresponding
Q-curvature Q~ g
is a constant. When ~ g = e
2w
g, by (5) the problem amounts to
nding a solution of the equation
(8) Pg w + 2Qg
= 2Qe
4w
;
where Q is a real constant. By the regularity results in [43], critical points of
the following functional
(9) II (u) = hPg
u; ui + 4
Z
M
Qg
udV
g k
P
log
Z
M
e
4u
dV
g
; u 2 H
2
(M );