Splitting symplectic fillings

TL;DR

This paper introduces splitting surfaces, a generalization of mixed tori, enabling the decomposition of symplectic fillings of contact manifolds through Liouville surgery, extending previous decomposition results.

Contribution

It generalizes the concept of mixed tori to splitting surfaces, allowing for broader decomposition of symplectic fillings in contact geometry.

Findings

Splitting surfaces facilitate decomposition of symplectic fillings.

Decomposition results extend to strong and exact fillings.

New symplectic manifolds obtained via Liouville surgery relate to original fillings.

Abstract

We generalize the mixed tori which appear in the second author's JSJ-type decomposition theorem for symplectic fillings of contact manifolds. Mixed tori are convex surfaces in contact manifolds which may be used to decompose symplectic fillings. We call our more general surfaces splitting surfaces, and show that the decomposition of symplectic fillings continues to hold. Specifically, given a strong or exact symplectic filling of a contact manifold which admits a splitting surface, we produce a new symplectic manifold which strongly or exactly fills its boundary, and which is related to the original filling by Liouville surgery.