Topological Obstructions to Dynamical Convexity
Shahnaz Shamim Shahul
TL;DR
The paper establishes topological obstructions to both dynamical and strong dynamical convexity on contact manifolds, notably ruling out strongly dynamically convex unit cotangent bundles and, in particular, ST^*M for simply connected M. It connects Reeb dynamics with algebraic geometry through McLean’s minimal discrepancy, employing Viterbo’s isomorphism and Serre spectral sequences to translate dynamical conditions into loop-space and filling-topology constraints. It further shows that subcritical/flexible surgeries, 0-surgeries, and connected sums with ST^*N obstruct dynamical convexity, via detailed symplectic homology analyses. These results collectively limit which manifolds can carry dynamically convex contact structures and highlight a deep interplay between dynamics, topology, and algebraic geometry invariants.
Abstract
We study the topological obstructions of dynamical convexity on contact manifolds focusing on fillability by cotangent bundles and subcritical surgeries. Using links to algebraic geometry, we motivate and define a stronger version of dynamical convexity, and investigate the topology of these manifolds. More precisely, we show that strongly dynamically convex contact manifolds cannot arise as a unit cotangent bundle $(ST^*M,λ_{std})$ of a closed manifold $M$ and in particular that dynamically convex contact manifolds cannot arise as the unit cotangent bundle $(ST^*M,λ_{std})$ of a simply connected manifold $M$. We also show obstructions to dynamical convexity that arises from studying different kinds of subcritical surgeries.