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**Kohn-Rossi cohomology and its application to the complex plateau problem. III.**
*(English)*
Zbl 1254.32051

The complex Plateau problem asks which odd dimensional real submanifolds of \({\mathbb C}^N\) are boundaries of complex submanifolds of \({\mathbb C}^N\). A theorem by F. R. Harvey and H. B. Lawson [Ann. Math. (2) 102, 223–290 (1975; Zbl 0317.32017)] states that an embeddable strongly pseudoconvex CR manifold \(X\), embedded in some \({\mathbb C}^N\), bounds a unique Stein variety \(V\subset {\mathbb C}^N\) with at most isolated singularities. A fundamental questions is to determine when such \(V\) is smooth. In [Ann. Math. (2) 113, 67–110 (1981; Zbl 0464.32012)] the second author proved that a compact connected strongly pseudoconvex CR manifold \(X\) of real dimension \(2n-1\), with \(n\geq 3\), in the boundary of a bounded strongly pseudoconvex domain \(D\) in \(\mathbb C^{n+1}\), is the boundary of a complex submanifold \(V\subset D\) if and only if the Kohn-Rossi cohomology groups \(H^{p,q}_{KR}(X)\) are zero, for \(1\leq q\leq n-2\).

For \(n=2\), the main difficulty comes from the fact that the Kohn-Rossi cohomology groups \(H^{p,q}_{KR}(X)\) are infinite-dimensional. In this paper the authors introduce a new CR invariant \(g^{1,1}(X)\), whose vanishing ensures the interior regularity of the Harvey Lawson solution, up to normalization. By means of this new invariant they solve, up to normalization, the intrinsic smoothness criteria for the classical Plateau problem when \(n=2\), and obtain interior regularity when \(X\) is of real codimension 3 in \({\mathbb C}^3\). More precisely, they prove the following theorems.

Theorem A. Let \(X\) be a strongly pseudoconvex compact Calabi-Yau CR manifold of dimension 3. Suppose that \(X\) is contained in the boundary of a strongly pseudoconvex bounded domain \(D\) in \({\mathbb C}^N\) with \(H^2_h(X)=0\). Then \(X\) is the boundary of a complex submanifold up to normalization \(V\subset D\) if and only if \(g^{(1,1)}=0.\)

Theorem B. Let \(X\) be a strongly pseudoconvex compact CR manifold of dimension 3. Suppose that \(X\) is contained in the boundary of a strongly pseudoconvex bounded domain \(D\) in \({\mathbb C}^3\) with \(H^2_h(X)=0\). Then \(X\) is the boundary of complex submanifold \(V\subset D\) if and only if \(g^{(1,1)}=0.\)

(Here \(H^*_h(X)=0\) denotes the holomorphic De Rham cohomology).

Editorial remark: For Part II see [the second author and H. S. Luk, J. Differ. Geom. 77, No. 1, 135–148 (2007; Zbl 1123.32020)].

For \(n=2\), the main difficulty comes from the fact that the Kohn-Rossi cohomology groups \(H^{p,q}_{KR}(X)\) are infinite-dimensional. In this paper the authors introduce a new CR invariant \(g^{1,1}(X)\), whose vanishing ensures the interior regularity of the Harvey Lawson solution, up to normalization. By means of this new invariant they solve, up to normalization, the intrinsic smoothness criteria for the classical Plateau problem when \(n=2\), and obtain interior regularity when \(X\) is of real codimension 3 in \({\mathbb C}^3\). More precisely, they prove the following theorems.

Theorem A. Let \(X\) be a strongly pseudoconvex compact Calabi-Yau CR manifold of dimension 3. Suppose that \(X\) is contained in the boundary of a strongly pseudoconvex bounded domain \(D\) in \({\mathbb C}^N\) with \(H^2_h(X)=0\). Then \(X\) is the boundary of a complex submanifold up to normalization \(V\subset D\) if and only if \(g^{(1,1)}=0.\)

Theorem B. Let \(X\) be a strongly pseudoconvex compact CR manifold of dimension 3. Suppose that \(X\) is contained in the boundary of a strongly pseudoconvex bounded domain \(D\) in \({\mathbb C}^3\) with \(H^2_h(X)=0\). Then \(X\) is the boundary of complex submanifold \(V\subset D\) if and only if \(g^{(1,1)}=0.\)

(Here \(H^*_h(X)=0\) denotes the holomorphic De Rham cohomology).

Editorial remark: For Part II see [the second author and H. S. Luk, J. Differ. Geom. 77, No. 1, 135–148 (2007; Zbl 1123.32020)].

Reviewer: Laura Geatti (Roma)