On singular Lagrangian fibrations and applications to symplectic embeddings I

TL;DR

This paper constructs singular Lagrangian fibrations on specific cotangent bundles and applies these to analyze symplectic embedding problems, including Gromov width and Lagrangian bidisk results.

Contribution

It introduces a method to build singular Lagrangian fibrations on cotangent bundles and applies it to solve problems in symplectic embeddings.

Findings

Constructed singular Lagrangian fibrations in 4 and 6 dimensions.

Reproduced known results on Gromov width of cotangent bundles.

Applied techniques to study symplectic embedding problems.

Abstract

In this paper, we construct singular Lagrangian fibrations on some examples of disk cotangent bundles in dimensions 4 and 6. As an application, we show how this construction can be used to obtain toric domains in some cases. In particular, we recover results from Ferreira, Ramos, and Vicente on the Gromov width of the disk cotangent bundle of spheres of revolution, as well as results from Ramos on the Lagrangian bidisk. We also briefly discuss how this technique can be used to study symplectic embedding problems.