Holomorphic support functions for uniformly pseudoconvex hypersurfaces, with an application to CR maps (Q6557372)

Let \(M\) be a smooth real hypersurface in \(\mathbb{C}^n\). \(M\) is said to be \textit{uniformly pseudoconvex} if it is pseudoconvex with Levi form of constant rank. For \(p\in M\), a \textit{local holomorphic support function} at \(p\) is a function \(h\) holomorphic in a neighborhood \(U\) of \(p\) such that \(h(p)=0\) and \(\mathfrak{I}(h(q))\ge 0\) for all \(q\in M\cap U\).\N\NThe main result of the paper under review (Theorem 1.1) shows the existence of a local holomorphic support function at every point of a uniformly pseudoconvex hypersurface. As an application, the author obtains a holomorphic deformation theorem (Theorem 1.2) for nowhere smooth CR maps into smooth pseudoconvex hypersurfaces with one-dimensional Levi foliation, the proof of which uses a powerful formal deformation theorem of \textit{B. Lamel} and \textit{N. Mir} [Adv. Math. 335, 696--734 (2018; Zbl 1411.32025)].