On the geometry of model almost complex manifolds with boundary

Abstract

We study some special almost complex structures on strictly pseudoconvex domains in R 2n . They appear naturally as limits under a nonisotropic scaling procedure and play a role of model objects in the geometry of almost complex manifolds with boundary. We determine explicitely some geometric invariants of these model structures and derive necessary and sufficient conditions for their integrability. As applications we prove a boundary extension and a compactness principle for some elliptic diffeomorphisms between relatively compact domains.

Some aspects of analysis on almost complex manifolds with boundary

“…This was obtained in the paper [34]. In a neighborhood U of any totally real point of Σ J (∂D), the set S contains a wedge W U with Σ J (∂D) ∩ U as totally real edge.…”

“…This survey is for a large part an overview of different results obtained in a series of papers [33,23,24,34]. This is organized as follows.…”

Schwarz Reflection Principle, Boundary Regularity and Compactness for J-Complex Curves

“…Then using the "geometric bootstrap" from [7] we obtain C k+1,α -regularity of complex discs in C k,α -regular structures for all k 1, thus proving the Theorem 1.3.…”

“…In §3, using [10] and [7] we prove Theorem 1.3. First we establish the C 1,α -regularity of u if J ∈ C α and then, using a sort of "geometric bootstrap", we obtain the C k+1,α -regularity of u if J ∈ C k,α .…”

“…This lift is invariantly defined, see [7] for more details. After that we can use the induction on k.…”