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We prove that several types of open Riemann surfaces, including the finitely connected planar domains, embed properly into C 2 such that the values on any given discrete sequence can be arbitrarily prescribed. Mathematics Subject Classification (2000)

We construct a family of integral kernels for solving the ∂-equation with C k and Hölder estimates in thin tubes around totally real submanifolds in C n (theorems 1.1 and 3.1). Combining this with the proof of a theorem of Serre we solve the d-equation with estimates for holomorphic forms in such tubes (theorem 5.1). We apply these techniques and a method of Moser to approximate C k -diffeomorphisms between totally real submanifolds in C n in the C k -topology by biholomorphic mappings in tubes, by unimodular and symplectic biholomorphic mappings, and by automorphisms of C n .

Let M be a smooth compact manifold and n ≥ 2. Given a smooth isotopy of embeddings f t : M → C n (0 ≤ t ≤ 1) such that f t (M) ⊂ C n is a totally real and polynomially convex submanifold of C n for each fixed t, we construct a sequence Φ j of holomorphic automorphisms of C n such that Φ j • f 0 converges to f 1 and Φ −1 j • f 1 converges to f 0 in C ∞ (M) as j → ∞. We also obtain results on approximating flows of asymptotically holomorphic vector fields by holomorphic automorphisms of C n .

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