Gromov’s method and Bennequin’s problem. (English) Zbl 0853.32003

Alexander, H.

Invent. Math. 125, No. 1, 135-148 (1996).

An analytic disk in \(\mathbb{C}^n\) is a map \(f : D \to \mathbb{C}^n\), where \(D\) is the closed unit disk, holomorphic on the open disk \(D^0\) and smooth up to the boundary. A nearly smooth analytic disk is a bounded holomorphic map \(f : D^0 \to \mathbb{C}^n\) which extends to be smooth on all of \(D\) except for at most a single point of the boundary \(bD\). The boundary of \(f\) is said to lie in \(X\) if the image by \(f\) of \(bD\) with that single point deleted is contained in \(X\).
The author proves the following result.
Theorem: Let \(L\) be a compact totally real \(n\)-dimensional submanifold of \(\mathbb{C}^n\). Then there exists a non-constant nearly smooth analytic disk with boundary in \(L\).
This is a partial answer to a problem of D. Bennequin asserting that a non-constant analytic exists in \(\mathbb{C}^2\) with boundary in a totally real 2-torus in \(\mathbb{C}^2\).
The author’s theorem uses in its proof Gromov’s method of pseudo-holomorphic curves in symplectic manifolds, by establishing a Fredholm alternative. Gromov’s compactness theorem which is usually used to show this, is not available in the totally real case, so the author relies on an ad hoc argument that also does not use almost complex structures. This method is also used to give a proof of Gromov’s theorem that if \(L\) is a compact Lagrangian manifold in \(\mathbb{C}^n\), then there exists a non-constant analytic disk in \(\mathbb{C}^n\) with boundary in \(L\). The author proves the following intermediate result between his “nearly smooth analytic disk” and Gromov’s analytic disk.
Theorem 2: If \(X \subset \mathbb{C}^n\) is compact and \(f\) is a non-constant \(H^\infty\)-disk (that is, \(f\) is bounded holomorphic on \(D^0\)) with boundary in \(X\), and if \(f\) has finite area, then \(f : D^0 \setminus f^{-1} (X) \to \mathbb{C}^n\setminus X\) is a proper mapping.
In an addendum, the author states that he and J. Duval have independently found counterexamples to Bennequin’s question.

Reviewer: J.S.Joel (Kelly)

MSC:

32B15	Analytic subsets of affine space
32V40	Real submanifolds in complex manifolds
32H35	Proper holomorphic mappings, finiteness theorems

Keywords:

analytic disk; Gromov’s method; pseudo-holomorphic curves; symplectic manifolds