TY - JOUR
TI - Stability of the hull(s) of the n-sphere
DO - https://doi.org/doi:10.7282/t3-4p72-8w92
PY - 2020
AB - For a particular natural embedding of the real n-sphere in mathbb{C}^n, the CR singularities are elliptic and nondegenerate and form an (n-2)-sphere on the equator. In particular, for n ge 3, these singularities are non-isolated. This distinguishes the difficulty of this problem from the well-studied case of n=2. It can easily be seen that the n-sphere can be filled by an (n-1)-parameter family of attached holomorphic discs foliating towards the singularities. This family of discs forms a real (n+1)-dimensional ball, which is the holomorphic and polynomial hull of the n-sphere. This dissertation investigates whether these properties are stable under C^3-small perturbations and what regularity can be expected from the resulting manifold. We find that under such perturbations, the local and global structure of the set of singularities remains the same. We then solve a Riemann-Hilbert problem, modifying a construction by Alexander, to obtain an (n-1)-parameter family of holomorphic discs attached to the perturbed sphere, away from the set of singularities. We then use the theory of multi-indices for attached holomorphic discs and nonlinear functional analysis to study the regularity of the resulting manifold. We find that in the case that the perturbation is C^{k+2, alpha}, the construction yields a C^{k,alpha} manifold. In the case that the perturbation is mathcal{C}^infty smooth or real analytic we show that the regularity of the manifold matches the regularity of the perturbation. We then patch this construction with small discs constructed by K"enig, Webster, and Huang near nondegenerate elliptic singularities to obtain a complete filling of the perturbed sphere by attached holomorphic discs, with an additional loss of regularity near the CR singularities. This filled sphere is diffeomorphic to the (n+1)-dimensional ball and is clearly contained in the hull of holomorphy. Finally, we show that if the perturbation is real analytic and admits a uniform lower bound on its radius of convergence, this perturbed ball is in fact exactly the polynomial (and holomorphic) hull of the perturbed sphere.
KW - Polynomial hull
KW - Mathematics
LA - English
ER -