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Analytic disks with boundaries in a maximal real submanifold of \({\mathbb C}^ 2\). (English) Zbl 0583.32038
Let M be an orientable, smoothly embedded, totally real submanifold of \({\mathbb{C}}^ 2\). A continuous map F: \(\bar D\to {\mathbb{C}}^ 2\) of the closed unit disk \(\bar D=\{| z| \leq 1\}\) into \({\mathbb{C}}^ 2\) that is holomorphic on the open disk \(D=\{| z| <1\}\) and maps its boundary \(bD=\{| z| =1\}\) into M is called an analytic disk with boundary in M. Given such a disk \(F^ 0\) we give a rather complete description of the nearby disks with boundary in M in terms of an integer \(m=Ind_{F^ 0}M\), called the index of M along \(F^ 0\), which only depends on the homology class of the path \(F^ 0: bD\to M\) and is easily calculated from the local parametrization of M along \(F^ 0(bD)\). If \(m\geq 1\), there is a \(2m+2\) parameter family of nearby disks which are stable under small deformations of M in \({\mathbb{C}}^ 2\). If \(m\leq 0\), there are no nearby disks, and the initial disk may disappear under a small deformation of M. When \(m=1\) the nearby disks form a smooth hypersurface with boundary in M [this is related to a result of E. Bedford in Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 9, 529-570 (1982; Zbl 0574.32019)]. We also prove a regularity theorem for immersed families of analytic disks with boundary in M, consider several examples, and construct a totally real three-torus in \({\mathbb{C}}^ 3\) with a bizzare polynomially convex hull.

MSC:
32B15 Analytic subsets of affine space
32V40 Real submanifolds in complex manifolds
32G10 Deformations of submanifolds and subspaces
32A40 Boundary behavior of holomorphic functions of several complex variables
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
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