Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems / Tudor S. Ratiu, Christophe Wacheux, and Nguyen Tien Zung.

Format:

Book

Author/Creator:

Rațiu, Tudor S., author.

Wacheux, Christophe, author.

Zung, Nguyen Tien, 1970- author.

Series:

Memoirs of the American Mathematical Society ; Volume 287.

Memoirs of the American Mathematical Society Series ; Volume 287

Language:

English

Subjects (All):

Convex domains.

Hamiltonian systems.

Toric varieties.

Physical Description:

1 online resource (102 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

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.

Contents:

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.

Notes:

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Includes bibliographical references and index.

Other Format:

Print version: Ratiu, Tudor S. Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems

ISBN:

9781470475406

1470475405