On the geometry of analytic discs attached to real manifolds

Filling hypersurfaces by discs in almost complex manifolds of dimension 2

“…We were unable to find an analogue of this phenomenon in our case of complex dimension 2. In higher dimension no such analogue is possible due to an example by Ivashkovich and Rosay [4], in which all Bishop discs through a fixed point of a hypersurface E lie in E and cover all of E. Baouendi, Rothschild, and Trépreau [1] interpret the defect of a disc in terms of its lifts attached to the conormal bundle of E in the cotangent bundle T * I C n . Although an almost complex structure admits natural lifts to the cotangent bundle of an almost complex manifold (see, e.g., [13]), seemingly, they don't give rise to a correct notion of the defect of Bishop discs.…”

Filling Real Hypersurfaces by Pseudoholomorphic Discs

“…Baouendi, Rothschild, and Trépreau [2] interpret the defect of a disc in terms of its lifts to the conormal bundle of E in the cotangent bundle T * I C n . Although an almost complex structure admits natural lifts to the cotangent bundle of an almost complex manifold (see, e.g., [16]), obviously, they don't give rise to the correct notions of the defect of Bishop discs.…”

Analytic regularity of CR-mappings

“…By applying Tumanov's theorem again to the open subset M 0 ⊂ M, we obtain another (standard) wedge W 2 with edge M 0 at 0, where all CR-functions on M 0 holomorphically extend. We shall need more precise information about the wedge W 2 that can be obtained from the theory of defects of analytic discs [Tu88] (see also the geometric definition due to Baouendi-RothschildTrépreau [BRT94] that has been used in [Tu96]). Here the C 2,α -smoothness of M 0 is required to guarantee the C 1,α -smoothness of the conormal bundle to M 0 needed for Bishop's equation.…”