Erratum to: Invent math DOI 10.1007/s00222-012-0387-2

The referee has called my attention to an error in the proof of Theorem 2.6. The Harvey–Lawson variety may be singular at M even in case this is strongly pseudoconvex (see [1, 2, 4]). However, there does exist a manifold X with boundary M (not just a strip with M as a component of its boundary as in Theorem 2.2); this is true if we do not make the unnecessary request that X stays inside ℂn.

FormalPara Theorem 1

Let M⋐ℂn be a smooth, compact, connected, CR manifold without boundary of hypersurface type, pseudoconvex-oriented. Then, there is a manifold X with boundary M equipped with a C ∞-map π:X→ℂn such that π is holomorphic on X∖M and is a smooth embedding on a neighborhood of M. Moreover, there is a weight function on X which is strictly plurisubharmonic in a neighborhood of M.

FormalPara Proof

We start from the strip on the pseudoconvex side of M, smooth up to M, of Theorem 2.2. We point out that this strip is contained in ℂn since it is given as a smooth family of discs of ℂn (attached to either M or its extension at points of local minimality and propagation respectively). We further extend the strip, again from its pseudoconvex side, to a normal variety over ℂn, according to Theorem I p. 547 of [5]. This point is also explained in detail by Yau [7] at the end of Sect. 5 and in the entire Sect. 6 which follows in turn Siu’s proof in [6] of Rothstein’s Theorem. But then the singularities of the normal variety are confined outside the initial strip and thus they are compact and hence isolated. By blowing them up, we get the manifold X with smooth boundary M and the map π:X→ℂn with the required properties.

Finally, notice that, X coinciding with the strip of ℂn in a neighborhood of M, it inherits from ℂn the strictly plurisubharmonic weight |z|2, z∈ℂn. □

With Theorem 1 in hand, the proof of Theorem 2.6 follows immediately from Kohn [3] Theorem 5.3.