Hulls of Surfaces

TL;DR

This paper demonstrates that every compact 2D manifold can be smoothly embedded in complex 3-space with a polynomially convex hull that contains no analytic discs, highlighting a novel geometric property.

Contribution

It introduces a new embedding technique for compact surfaces into complex space ensuring hulls lack analytic discs, advancing understanding of polynomial convexity.

Findings

Every compact 2D manifold can be embedded in a03 as a smooth submanifold.

The polynomially convex hull of the embedded surface contains no analytic disc.

The hull is strictly larger than the surface itself.

Abstract

In this paper it is shown that every compact two-dimensional manifold $S$, with or without boundary, can be embedded in $\mathbb C^3$ as a smooth submanifold $\Sigma$ in such a way that the polynomially convex hull of $\Sigma$, though strictly larger than $\Sigma$, contains no analytic disc.