Approximation theorems on differentiable submanifolds of a complex manifold

Holomorphic approximation on totally real submanifolds of a complex manifold

“…In previous papers the uniform approximation (Corollary 2.2) was obtained with U a small neighborhood of K> which did not necessarily contain M. That 0(K) has dense image in C{K) was proven in [8] only for k*z4:n -l (with similar dimensional restrictions in [ó]), [September whereas we now obtain this result for k = l. 2 The techniques we use involve Ramirez integral kernels which invert d in strongly pseudoconvex domains, giving good uniform estimates (see e.g., Ramirez [9], Grauert-Lieb [3]). …”

“…Nirenberg-Wells [8]), the simplest examples being a real curve in X, the distinguished boundary of a polydomain, or R n C.C n . The geometric nature of a totally real submanifold implies that it behaves in many cases like this last example.…”

“…In particular, the Weierstrass approximation theorem tells us that holomorphic functions on O are dense in the Banach space of continuous functions on a compact subset KC.R n dC n in the supremum norm. There have been various investigations recently generalizing this type of theorem to compact subsets of totally real submanifolds (see Cirka [l], Hörmander-Wermer [6], Nirenberg-Wells [8]). In §2 we formulate our main results on holomorphic approximation, in which we improve on the previous known results by (a) extending the domain of definition of the approximating functions, (b) minimizing the differentiability requirements for the submanifold, and (c) requiring that the approximation be uniform on K along with uniform approximation of all derivatives up to the order of differentiability of the submanifold.…”

Tangential Cauchy-Riemann equations and uniform approximation

“…There is one in C 3 , a 4-sphere S 4 given as the intersection of the usual 5-sphere and a real hyperplane transverse to it. Let ft = I Si I 2 + I z 2 This result is known [3,Lemma 4.3] and [5,Lemma 3.1] when M is totally real. It is also proved in [2,Th.…”

“…These frequently have no nontrivial complex submanifolds, which is true for example of the usual 2n -1 sphere in C\ M is a generic CR manifold if its complex rank is everywhere zero, which is the totally real [5] case.…”