Fefferman?s mapping theorem on almost complex manifolds in complex dimension two

Abstract

We give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds (see Theorem 1.1). The proof is mainly based on a reflection principle for pseudoholomorphic discs, on precise estimates of the Kobayashi-Royden infinitesimal pseudometric and on the scaling method in almost complex manifolds. (2000): 32H02, 53C15 Mathematics Subject Classification

Some aspects of analysis on almost complex manifolds with boundary

“…where A is independent of k (see [24]). Since all our considerations are local we set p = p ′ = 0 ∈ C n .…”

On the geometry of model almost complex manifolds with boundary

“…This is a natural development of the previous work [3] where we proved the boundary regularity of some diffeomorphisms between strictly pseudoconvex domains in almost complex manifolds of real dimension four. The general regularity result, treated in the present paper, relies on the study of complex invariants of some special almost complex structures.…”

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

“…For this recall that the cluster set cl(u, β) of u at β consists of all limits lim k→∞ u(ζ k ) for all sequences {ζ k } ⊂ ∆ converging to β. In [5] it was proved that if the cluster set cl(u, β) of a J-holomorphic map u : ∆ → X is compactly contained in a totally real submanifold W then u smoothly extends to β. Therefore we derive the following …”

“…We use the Proposition 4.1 from [5] and observe that u is in Sobolev class L 1,p up to β for all p < 4. In particular u is C β -regular up to β with β = 1 − 2 p (this means for all β < 1 2 ).…”