Strong symplectic fillability of contact torus bundles

TL;DR

This paper investigates the conditions under which certain tight contact structures on negative parabolic and hyperbolic torus bundles are strongly symplectically or Stein fillable, providing complete classifications and necessary conditions.

Contribution

It offers a complete classification of fillability for specific tight contact structures on negative parabolic torus bundles and establishes necessary conditions for hyperbolic cases, partially addressing a conjecture.

Findings

Complete fillability classification for negative parabolic torus bundles.

Necessary conditions for hyperbolic torus bundle fillability.

Proof that certain virtually overtwisted structures are Stein fillable.

Abstract

In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting $\pi$ in $S^1$-direction on a negative parabolic torus bundle, we completely determine its strong symplectic fillability and Stein fillability. For the universally tight contact structure with twisting $\pi$ in $S^1$-direction on a negative hyperbolic torus bundle, we give a necessary condition for it being strongly symplectically fillable. For the virtually overtwisted tight contact structure on the negative parabolic torus bundle with monodromy $-T^n$ ($n<0$), we prove that it is Stein fillable. By the way, we give a partial answer to a conjecture of Golla and Lisca.