Symplectic non-convexity of toric domains

TL;DR

This paper explores the symplectic convexity of star-shaped toric domains in four-dimensional space, introducing elementary operations that can disrupt convexity and producing examples that are dynamically convex but not symplectically convex.

Contribution

It introduces elementary domain operations that affect symplectic convexity and provides concrete bounds for criteria determining convexity in toric domains.

Findings

Elementary operations can destroy symplectic convexity.

Produced examples of dynamically convex but not symplectically convex domains.

Provided explicit bounds for constants in convexity criteria.

Abstract

We investigate the convexity up to symplectomorphism (called symplectic convexity) of star-shaped toric domains in $\mathbb R^4$. In particular, based on the criterion from Chaidez-Edtmair via Ruelle invariant and systolic ratio of the boundary of star-shaped toric domains, we provide elementary operations on domains that can kill the symplectic convexity. These operations only result in $C^0$-small perturbations in terms of domains' volume. Moreover, one of the operations is a systematic way to produce examples of dynamically convex but not symplectically convex toric domains. Finally, we are able to provide concrete bounds for the constants that appear in Chaidez-Edtmair's criterion.