Descent and C^0-rigidity of spectral invariants on monotone symplectic manifolds

We obtain estimates showing that on monotone symplectic manifolds (asymptotic) spectral invariants of Hamiltonians which vanish on a non-empty open set, U, descend to Ham_c(M\setminus U) from its universal cover. Furthermore, we show these invariants and are continuous with respect to the C^0-topology on Ham_c(M\setminus U). We apply these results to Hofer geometry and establish unboundedness of the Hofer diameter of $Ham_c(M\setminus U)$ for stably displaceable $U$. We also answer a question of F. Le Roux about $C^0$-continuity properties of the Hofer metric.Comment: 17 page

The displaced disks problem via symplectic topology

We prove that a $C^0$--small area preserving homeomorphism of a closed surface with vanishing mass flow can not displace a topological disk of large area. This resolves the displaced disks problem posed by F. B\'eguin, S. Crovisier, and F. Le Roux.Comment: 3 page

A note on C^0 rigidity of Hamiltonian isotopies

We show that a symplectic isotopy that is a $C^0$ limit of Hamiltonian isotopies is itself Hamiltonian, if the corresponding sequence of generating Hamiltonians converge in $L^{(1, \infty)}$ topology.Comment: 7 pages, to appear in Journal of Symplectic Geometr

A Macro Policy for Poverty Eradication through Structural Change

Bangladesh, cooperatives, institutions, land, micro-credit, women

Unboundedness of the Lagrangian Hofer distance in the Euclidean ball

Let L denote the space of Lagrangians Hamiltonian isotopic to the standard Lagrangian in the unit ball in Euclidean space. We prove that the Lagrangian Hofer distance on L is unbounded.Comment: 8 page

Spectral killers and Poisson bracket invariants

We find optimal upper bounds for spectral invariants of a Hamiltonian whose support is contained in a union of mutually disjoint displaceable balls. This gives a partial answer to a question posed by Leonid Polterovich in connection with his recent work on Poisson bracket invariants of coverings.Comment: 16 pages, 2 figures. V2: to appear in Journal of Modern Dynamic

New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians

We prove a new variant of the energy-capacity inequality for closed rational symplectic manifolds (as well as certain open manifolds such as cotangent bundle of closed manifolds...) and we derive some consequences to C^0-symplectic topology. Namely, we prove that a continuous function which is a uniform limit of smooth Hamiltonians whose flows converge to the identity for the spectral (or Hofer's) distance must vanish. This gives a new proof of uniqueness of continuous generating Hamiltonian for hameomorphisms. This also allows us to improve a result by Cardin and Viterbo on the C^0-rigidity of the Poisson bracket.Comment: 18 pages. v2. Several minor changes. Reference list updated. To appear in Commentarii Mathematici Helvetic

Coisotropic rigidity and C^0-symplectic geometry

We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov-Eliashberg Theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C^0-dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C^0-Hamiltonian dynamics: Uniqueness of generators for continuous analogs of Hamiltonian flows.Comment: 27 pages. v2. Significant reorganization of the paper, several typos and inaccuracies corrected after the refeering process. A theorem (Theorem 5, completing the study of C^0 dynamical properties of coisotropics) added. To appear in Duke Mathematical Journa