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A smooth analytic disc valued in C is a map A : ∆ → C, where ∆ denotes the unit disc in C, with A smooth in ∆ and holomorphic in ∆. We shall say that the disc A is attached to a manifold M ⊂ C if A(S) ⊂ M , where S = ∂∆. Similarly, A is partially attached to M if A(I) ⊂ M , where I ⊂ S is an open arc with 1 ∈ I. In [1] the authors proved the following unique continuation result. If A is partially attached to a totally real manifold M and flat at 1, i.e. A(1) = 0, j = 1, 2, . . . , then A is constant. On the other hand, it is well known that if M is a strictly pseudoconvex hypersurface and A is a sufficiently small nonconstant analytic disc attached to M , then A′(1) = 0. (See Remark (4.3).) It should be mentioned that starting with the work of Bishop [5] and Lewy [9], analytic discs attached to hypersurfaces and their boundary derivatives have played an important role in local and global CR geometry for such problems as holomorphic extension of CR functions and propagation of holomorphic extension. For recent work see e.g. Trépreau [10], Tumanov [11], and [3]. It has been an open question whether there can exist a smooth nonconstant analytic disc A attached to a hypersurface of finite type [8] in C with A flat at 1. In this paper we first characterize flat discs valued in C in terms of the Fourier transform of their pullbacks to the real line. (See Boutet de Monvel [4] for a similar approach.) We then use this characterization to construct arbitrarily small discs flat at 1 attached to a large class of hypersurfaces, including any homogeneous hypersurface of odd type. We conclude the paper with some remarks about sufficient conditions on a hypersurface M which imply that A′(1) = 0 for any smooth analytic disc attached to M .

Flat analytic discs attached to real hypersurfaces of finite type