The Calabi invariant, the mean action spectrum of Hamiltonian maps, and their realization as smooth Reeb flows.

• Applicant: Abror Pirnapasov
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 518128580

Since the work of Hofer-Zehnder and Schwarz, it is known that the action spectrum of Hamiltonian diffeomorphisms on closed symplectically aspherical manifolds is a compact and nowhere dense subset of the real line. It turns out that the mean action spectrum, which is a bigger set than the action spectrum, has interesting relations with the Calabi invariant for area-preserving disc maps, as shown by Hutchings. However, topological properties of the mean action spectrum are not well understood. Thus our first goal in this direction is to discover topological properties of the mean action spectrum and reveal its relation with the Calabi invariants for Hamiltonian maps of a surface with boundary. As a tool, we use the Floer homology and properties of the spectral invariants of periodic Floer homology (PFH). Moreover, we would like to use the Calabi invariant to check the perfectness of the symplectomorphism group of the standard two-dimensional plane. Further, we aim to discover conditions under which area-preserving maps of a surface with boundary can be realized as a smooth Reeb flow on a compact manifold without boundary, which means the construction of a Reeb flow on a compact manifold which admits a global surface of section such that the first return map coincides with the given map. By the results of Colin, Honda, and Laudenbach, if a map is a first return map for a global surface of a section of a Reeb flow, then the flux of the map is equal to zero. This result implies that one of the necessary conditions for area-preserving maps for realizing as a smooth Reeb flow is zero flux. We aim to investigate necessary and sufficient conditions to realize area-preserving maps as a smooth Reeb flow. We would like to construct a Bernoulli Reeb flow on every closed contact three-manifold by constructing and realizing zero-flux Bernoulli diffeomorphisms on compact surfaces with boundary. The next aim is to investigate which area-preserving pseudo-rotations (area-preserving map with only one periodic orbit) of the disc one could realize as smooth Reeb flows of the standard three-sphere. Recent results on Reeb dynamic show that generic smooth Reeb flows admit global surface of sections. However, no explicit Reeb flows exist on a closed contact manifold, which does not have a global surface of section. Our plan is to construct a counterexample in dimension five using the Anosov-Katok conjugation method.

• DFG Programme: WBP Fellowship
• International Connection: USA
• Host: Professor Dr. Daniel Cristofaro-Gardiner