Stability of the hull(s) of an $n$-sphere in $\mathbb{C}^n$

Abstract

We study the (global) Bishop problem for small perturbations of S n -the unit sphere of C × R n−1 -in C n . We show that if S ⊂ C n is a sufficiently-small perturbation of S n (in the C 3 -norm), then S bounds an (n + 1)-dimensional ball M ⊂ C n that is foliated by analytic disks attached to S . Furthermore, if S is either smooth or real analytic, then so is M (upto its boundary). Finally, if S is real analytic (and satisfies a mild condition), then M is both the envelope of holomorphy and the polynomially convex hull of S . This generalizes the previously known case of n = 2 (CR singularities are isolated) to higher dimensions (CR singularities are nonisolated).PURVI GUPTA AND CHLOE URBANSKI WAWRZYNIAK a submanifold S ⊂ C n and a sufficiently small neighborhood E ⊂ S of p, under what conditions will E be a smooth Levi-flat submanifold with E as part of its smooth boundary? This is known as the Bishop problem. The global version of this problem is typically studied for generic closed submanifolds of C n admitting only nondegenerate CR singularities, i.e, points where the maximal complex tangent space has nongeneric dimension. These are the so-called Bishop submanifolds. If S is totally real (at p), then S is holomorphically convex (at p). Bishop discovered that for an n-manifold S in C n , if S at p has a CR singularity of elliptic type, then S is nontrivial due to the presence of embedded complex analytic disks, or Bishop disks, attached to S near p. Further, he conjectured that these disks foliate a unique Levi-flat submanifold that serves as E for a sufficiently small neighborhood E ⊂ S of p, and contains E as part of its smooth (real analytic) boundary when S is smooth (real analytic) near p. In the case of C 2 , Bishop's conjecture was settled by ) in the smooth category, and by ), Moser ([34]) and Huang-Krantz ([27]) in the real analytic category. For general n, Bishop's conjecture was finally shown to be true in the work of Huang ([25]), partially based on the previous work of ). In contrast to the ellipitc case, Forstnerič-Stout showed in [19] that if p is a hyperbolic complex point of a surface S ⊂ C 2 , then the local envelope of holomorphy of S at p is trivial.The first major breakthrough for the global version of the Bishop problem was made by Bedford and Gaveau, when they proved in [4] that any smooth real Bishop surface S ⊂ C 2 with only two elliptic complex tangent points (hence, S is a sphere) and contained in the boundary of a bounded strongly pseudoconvex domain bounds a Levi-flat hypersurface M that is foliated by embedded analytic disks attached to S . Moreover M is precisely S (and even S , in some cases). By the aforementioned work of Kenig-Webster, Moser and Huang-Moser, S is smooth (real analytic) up to S when S is smooth (real analytic). Subsequently, this result was generalized by Bedford-Klingenberg in [5] and Kruzhilin in [32], who showed that S is a Levi-flat hypersurface bound by S , even when S is a sphere admitting hyperbolic complex tangent points (of course, one cannot expect ful...