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Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems
About this Title
Tudor S. Ratiu, Christophe Wacheux and Nguyen Tien Zung
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 287, Number 1424
ISBNs: 978-1-4704-6439-4 (print); 978-1-4704-7540-6 (online)
DOI: https://doi.org/10.1090/memo/1424
Published electronically: June 27, 2023
Keywords: integrable system,
toric-focus,
convexity,
integral affine structure,
singularities
Table of Contents
Chapters
- 1. Introduction
- 2. A brief overview of convexity in symplectic geometry and in integrable Hamiltonian systems
- 3. Toric-focus integrable Hamiltonian systems
- 4. Base spaces and affine manifolds with focus singularities
- 5. Straight lines and convexity
- 6. Local convexity at focus points
- 7. Global convexity
Abstract
This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.- Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, MA, 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141
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