On the geometry of model almost complex manifolds with boundary (Q2509053)

The authors prove the following two theorems. Theorem 1. Let \(D\) and \(D'\) be two smooth relatively compact domains in real manifolds. Assume that \(D\) admits an almost complex structure \(J\) smooth on \(\overline D\) and such that \((D,J) \) is strictly pseudoconvex. Then, a smooth diffeomorphism \(f: D \rightarrow D'\) extends to a smooth diffeomorphism between \(\overline D\) and \(\overline D'\) if and only if the direct image \(f_*(J)\) of \(J\) under \(f\) extends smoothly on \(\overline{D'}\) and \((D', f_*(J)) \) is strictly pseudoconvex. Theorem 2. Let \((M,J)\) be an almost complex manifold, not equivalent to a model domain. Let \(D= (r<0)\) be a relatively compact domain in a smooth manifold \(N\) and let \((f^{\nu})_{\nu}\) be a sequence of diffeomorphisms from \(M\) to \(D\). Assume that (i) the sequence \((J_{\nu}:= f_*^{\nu}(J))_{\nu}\) extends smoothly up to \(\overline D\) and is compact in the \(C^2\) convergence on \(\overline D\), (ii) the Levi forms of \(\partial D, {\mathcal L}^{J_{\nu}}(\partial D)\) are uniformly bounded from below (with respect to \(\nu\) ) by a positive constant. Then the sequence \((f^{\nu})_{\nu}\) is compact in the compact-open topology on \(M\).