A concave holomorphic filling of an overtwisted contact $3$-sphere

TL;DR

This paper constructs non-Kähler complex structures on the 4-ball with strictly pseudoconcave boundary, inducing an overtwisted contact structure on the boundary 3-sphere, advancing understanding of complex and contact geometry.

Contribution

It demonstrates the existence of non-Kähler complex structures with overtwisted contact boundary on the 4-ball, a novel example in complex and contact topology.

Findings

Existence of non-Kähler complex structures on the 4-ball

Boundary contact structure is overtwisted

Boundary is strictly pseudoconcave

Abstract

In this paper we prove that the closed $4$-ball admits non-K\"ahler complex structures with strictly pseudoconcave boundary. Moreover, the induced contact structure on the boundary $3$-sphere is overtwisted.