A remark about weak fillings

TL;DR

The paper shows that small tubular neighborhoods of certain totally real submanifolds in complex Euclidean space can serve as exotic weak fillings of their unit cotangent bundles with standard contact structures.

Contribution

It introduces a new class of weak fillings arising from tubular neighborhoods of non-Lagrangian totally real submanifolds.

Findings

Small tubular neighborhoods provide exotic weak fillings.

These fillings are of the standard contact structure on the unit cotangent bundle.

The submanifolds considered are not Lagrangian, which is significant.

Abstract

Let $L$ be a closed totally real submanifold of $\mathbb{C}^{n}$, $n\ge 2$, which is not Lagrangian. We observe that small enough tubular neighborhoods of $L$ give exotic examples of weak fillings of $ST^{\ast}L$ endowed with its standard contact structure.