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

Author: Tudor S. Ratiu, Christophe Wacheux and Nguyen Tien Zung Series: Memoirs of the American Mathematical Society

A systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. Peculiarly, these systems’ base spaces are still smooth manifolds, analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities.

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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.

About the Author

Tudor S. Ratiu, Shanghai Jiao Tong University, China, Universite Geneve, Switzerland, and Ecole Polytechnique Federale de Lausanne, Switzerland.

Christophe Wacheux, Overflood, Lille, France.

Nguyen Tien Zung, Universite Paul Sabatier, Toulouse, France.