BIBFRAME Work

Title

Convexity of singular affine structures and toric-focus integrable Hamiltonian systems

Type

• Monograph
• Text

Contribution

• Wacheux, Christophe (author)
• Nguyen, Tien Zung (author)
• Rațiu, Tudor S. (author)

Subject

• Convex domains (LCSH)
• Differential geometry -- Classical differential geometry -- Affine differential geometry (MSC)
• Dynamical systems and ergodic theory -- Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems -- Completely integrable systems, topological structure of phase space, integratio (MSC)
• Convex and discrete geometry -- General convexity -- Axiomatic and generalized convexity (MSC)
• Hamiltonian systems (LCSH)
• Toric varieties (LCSH)
• Affine differential geometry (LCSH)

Language

• English

Illustrative Content

• illustrations

Classification

• 53A15
• 37J35
• DDC: 516/.08 (Assigner: dlc)
• 52A01
• LCC: QA639.5 .R37 2023 (Assigner: dlc) (Status: used by assigner)

Content

• text

Supplementary content

• bibliography
• index

Summary

"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"-- Provided by publisher.

Table of Contents

A brief overview of convexity in symplectic geometry and in integrable Hamiltonian systems -- Toric-focus integrable Hamiltonian systems -- Base spaces and affine manifolds with focus singularities -- Straight lines and convexity -- Local convexity at focus points -- Global convexity.

Authorized Access Point

Rațiu, Tudor S. Convexity of singular affine structures and toric-focus integrable Hamiltonian systems

Admin Metadata

• Date: 2026-09-11T20:13:42.410286+00:00
• Agent: cbc
• Status: c
• Derived from: 23262317

Admin Metadata

• Date: 2026-09-11
• Cataloger id: Unknown
• Description level: http://id.loc.gov/ontologies/bibframe-2-6-0/
• Description language: eng
• Description authentication: pcc