2 1. INTRODUCTION
is the so called homogeneous dimension. In the case of the quaternionic Heisenberg
group, cf. Section 4.1, the equation is
Lu
n
α=1
(
Tαu
2
+ Xαu
2
+
2u
+ Zαu
2
)
=
n + 1
4(n + 2)
u2∗−1
Scal.
This is also, up to a scaling, the Euler-Lagrange equation of the non-negative ex-
tremals in the
L2
Folland-Stein embedding theorem [Fo] and [FSt], see [GV1] and
[Va2]. On the other hand, on a compact quaternionic contact manifold M with
a fixed conformal class [η] the Yamabe equation characterizes the non-negative
extremals of the Yamabe functional defined by
Υ(u) =
M
4
n + 2
n + 1
|∇u|2
+ Scal ·
u2
dvg,
M
u2∗
dvg = 1, u 0,
where dvg is the volume form associated to te natural Riemannian metric g on M,
cf. (2.11). When the Yamabe constant
λ(M)
def
= λ(M, [η]) = inf{Υ(u) :
M
u2∗
dvg = 1, u 0}
is less than the corresponding constant for the 3-Sasakian sphere the existence of
solutions can be constructed with the use of suitable coordinates see [W] and [JL2].
The present paper can be considered as a contribution towards the soluiton
of the Yamabe problem in the case when the Yamabe constant of the considered
quaternionic contact manifold is equal to the Yamabe constant of the unit sphere
with its standard quaternionic contact structure, which is induced from the embed-
ding in the quaternion (n + 1)-dimensional space. It is also natural to conjecture
that if the quaternionic contact structure is not locally equivalent to the standard
sphere then the Yamabe constant is less than that of the sphere, see [JL4] for a
proof in the CR case. The results of the present paper will be instrumental for the
analysis of these and some other questions concerning the geometric analysis on
quaternionic contact structures.
In this article we provide a partial solution of the Yamabe problem on the
quaternionic sphere with its standard contact quaternionic structure or, equiva-
lently, the quaternionic Heisenberg group. Note that according to [GV2] the ex-
tremals of the above variational problem are C∞ functions, so we will not consider
regularity questions in this paper. Furthermore, according to [Va1] or [Va2] the
infimum is achieved, and the extremals are solutions of the Yamabe equation. Let
us observe that [GV2] solves the same problem in a more general setting, but under
the assumption that the solution is invariant under a certain group of rotation. If
one is on the flat models, i.e., the groups of Iwasawa type [CDKR1] the assump-
tion in [GV2] is equivalent to the a-priori assumption that, up to a translation,
the solution is radial with respect to the variables in the first layer. The proof goes
on by using the moving plane method and showing that the solution is radial also
in the variables from the center, after which a very non-trivial identity is used to
determine all cylindrical solutions. In this paper the a-priori assumption is of a
different nature, see further below, and the method has the potential of solving the
general problem.
The strategy, following the steps of [LP] and [JL3], is to solve the Yamabe
problem on the quaternionic sphere by replacing the non-linear Yamabe equation
by an appropriate geometrical system of equations which could be solved.
Previous Page Next Page