On the stability of the hull(s) of an n-sphere in \(\mathbb{C}^n\) via a nonlinear Riemann-Hilbert problem

Abstract. The phenomenon of analytic continuation in several complex variables gives rise to various notions of hulls (and convexity) for compact subsets of complex manifolds. Although these hulls can be abstractly defined as maximal ideal spaces of certain uniform algebras, it is of interest to identify when these can be geometrically recovered by attaching analytic discs or varieties to the given subset. In this talk, we will focus on a specific class of subsets in \(\mathbb{C}^n\), i.e., generic spheres of real dimension \(n\). In the context of hulls, much is known about \(2\)-spheres in \(\mathbb{C}^2\). However, there are no global results of this nature in higher dimensions. We will present a stability result which is the first natural step towards this goal. Along the way, we will elaborate on the role of Riemann-Hilbert boundary problems in this study. This is joint work with Chloe U. Wawrzyniak