Stein and Weinstein structures on disk cotangent bundles of surfaces

TL;DR

This paper proves that Gompf's Stein domain structure on the disk cotangent bundle of a surface is symplectomorphic to its canonical symplectic structure, providing a new surgery diagram for the contact boundary.

Contribution

It establishes the equivalence between Gompf's Stein domain and the canonical cotangent bundle, and derives a surgery diagram for the boundary contact structure.

Findings

Gompf's Stein domain is symplectomorphic to the canonical cotangent bundle.

The boundary contact structure is contactomorphic to the unit cotangent bundle.

A surgery diagram for the canonical contact structure is obtained.

Abstract

In 1998, Gompf described a Stein domain structure on the disk cotangent bundle of any closed surface S, by a Legendrian handlebody diagram. We prove that Gompf's Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorphic to the unit cotangent bundle of S equipped with its canonical contact structure. As a corollary, we obtain a surgery diagram for the canonical contact structure on the unit cotangent bundle of S.