Approximation Theorems on Differentiable Submanifolds of a Complex Manifold

Stein domains and branched shadows of 4-manifolds

“…It is a standard fact that M can be equipped with a Stein domain structure with respect to which is totally real and whose Morse function φ: M → [0, 1] is equal to the distance from (in a fixed Riemannian metric on M ). The above facts hold also when ∂ = 0 with two differences: M is a neighborhood of in a Stein domain in which is totally real and M is cut by a function φ which coincides with the distance from near the interior part of and is slightly perturbed near its boundary (see [11,19] for further details).…”

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

“…If the maximal J -invariant subspace H x (M) of the tangent space T x (M) at x ∈ M, called the holomorphic tangent space at x, has constant dimension for any x ∈ M, then the submanifold M is called the Cauchy-Riemann submanifold or briefly CR submanifold and the constant complex dimension of H x (M) is called the CR dimension of M [24,30]. In this article, we assume M to be a CR submanifold of maximal C R dimension, that is, at each point…”

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex Euclidean space

“…The Riemannian metric g of M is induced from the Riemannian metricḡ of M in such a way that g(X, Y ) = g(ı X, ıY ), where X, Y ∈ T (M). Further, let the maximal J -invariant subspace of the tangent space T x (M) at x ∈ M, called the holomorphic tangent space at x, has constant dimension for any x ∈ M. Then the submanifold M is called the Cauchy-Riemann submanifold or briefly CR submanifold and the constant complex dimension of the holomorphic tangent space is called the CR dimension of M [17], [26]. Now, let M be a CR submanifold of maximal C R dimension, that is, at each point x of M, the tangent space…”