On Gromov Width under $C^0$ Deformations

TL;DR

This paper constructs a symplectic structure on a specific manifold that admits large ball embeddings, providing a counterexample to a conjecture, and then proves the conjecture for a broad class of cases, clarifying pseudoholomorphic curve theory.

Contribution

It presents a counterexample to Savelyev's conjecture and proves the conjecture for a wide class of examples, advancing understanding of Gromov width under $C^0$ deformations.

Findings

Counterexample to Savelyev's conjecture on Gromov width

Validation of the conjecture for a broad class of symplectic manifolds

Clarification of pseudoholomorphic curve theory in non-compact settings

Abstract

We construct a uniformly bounded symplectic structure on $S^2 \times \mathbb{R}^4$ admitting embeddings by arbitrarily large balls. This provides a counterexample to a recent conjecture of Savelyev. We then prove the conjecture holds for a wide class of examples, generalizing a result by Savelyev. Along the way, we clarify some aspects of pseudoholomorphic curve theory in non-compact manifolds.