Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems

• Autore: Ratiu Tudor S
• Pubblicazione: Providence : , : American Mathematical Society, , 2023; ©2023
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (102 pages)
• Disciplina: 516/.08; 516.08
• Soggetto topico: Convex domains; Affine differential geometry; Hamiltonian systems; Toric varieties; Dynamical systems and ergodic theory -- Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems -- Completely integrable systems, topological structure of phase space, integratio; Differential geometry -- Classical differential geometry -- Affine differential geometry; Convex and discrete geometry -- General convexity -- Axiomatic and generalized convexity
• Classificazione: 37J3553A1552A01
• Altri autori: WacheuxChristophe ZungNguyen Tien
• Nota di contenuto: Cover -- Title page -- Chapter 1. Introduction -- Positive convexity results -- Negative convexity results -- Organization of the paper -- Acknowledgment -- Chapter 2. A brief overview of convexity in symplectic geometry and in integrable Hamiltonian systems -- 2.1. Kostant's Linear Convexity Theorem -- 2.2. Infinite dimensional Lie theory -- 2.3. "Linear" symplectic formulations -- 2.4. "Non-linear" symplectic formulations -- 2.5. Local-Global Convexity Principle -- 2.6. Convexity in integrable Hamiltonian systems -- Chapter 3. Toric-focus integrable Hamiltonian systems -- 3.1. Integrable systems -- 3.2. Local normal form of non-degenerate singularities -- 3.3. Semi-local structure of singularities -- 3.4. Topology and differential structure of the base space -- 3.5. Integral affine structure on the base space -- Chapter 4. Base spaces and affine manifolds with focus singularities -- 4.1. Monodromy and affine coordinates near elementary focus points -- 4.2. Affine coordinates near focus points in higher dimensions -- 4.3. Behavior of the affine structure near focus^{ } points -- 4.4. Definition of affine structures with focus points -- Chapter 5. Straight lines and convexity -- 5.1. Regular and singular straight lines -- 5.2. Singular straight lines in dimension 2 and branched extension -- 5.3. Straight lines in dimension near a focus point -- 5.4. Straight lines near a focus^{ } point -- 5.5. The notions of convexity and strong convexity -- Chapter 6. Local convexity at focus points -- 6.1. Convexity of focus boxes in dimension 2 -- 6.2. Convexity of focus boxes in higher dimensions -- 6.3. Existence of non-convex focus^{ } boxes -- Chapter 7. Global convexity -- 7.1. Local-global convexity principle -- 7.2. Angle variation of a curve on an affine surface -- 7.3. Convexity of compact affine surfaces with non-empty boundary. ; 7.4. Convexity in the non-compact proper case -- 7.5. Non-convex examples in the non-proper case -- 7.6. An affine black hole and non-convex ² -- 7.7. A globally convex ² example -- 7.8. Convexity of toric-focus base spaces in higher dimensions -- Bibliography -- Index -- Back Cover.
• Sommario/riassunto: "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"--
• Titolo autorizzato: Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems
• ISBN: 1-4704-7540-5
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910915797003321
• Lo trovi qui: Univ. Federico II
• Serie: Memoirs of the American Mathematical Society Series