Kohn-Rossi Cohomology and its application to the Complex Plateau Problem III
Abstract
Let
X
be a compact connected strongly pseudoconvex
CR
manifold of real dimension 2n-1 in
C
N
. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For
n≥3
and
N=n+1
, Yau found a necessary and sufficient condition for the interior regularity of the Harvey-Lawson solution to the complex Plateau problem by means of Kohn-Rossi cohomology groups on
X
in 1981. For n=2 and
N≥n+1
, the problem has been open for over 30 years. In this paper we introduce a new CR invariant
g
(1,1)
(X)
of
X
. The vanishing of this invariant will give the interior regularity of the Harvey-Lawson solution up to normalization. In case
n=2
and N=3, the vanishing of this invariant is enough to give the interior regularity.