The Palais-Smale condition for the Hamiltonian action on a mixed regularity space of loops in cotangent bundles and applications

Rigidity of cotangent lifts and integrable systems

In this article we generalize a theorem by Palais on the rigidity of compact group actions to cotangent lifts. We use this result to prove rigidity for integrable systems on symplectic manifolds including sytems with degenerate singularities which are invariant under a torus action.

Corrigendum: The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

In lines 8-11 of \cite[pp. 2977]{Lu} we wrote: "For integer $m\ge 3$, if $M$ is $C^m$-smooth and $C^{m-1}$-smooth $L:\R\times TM\to\R$ satisfies the assumptions (L1)-(L3), then the functional ${\cal L}_τ$ is $C^2$-smooth, bounded below, satisfies the Palais-Smale condition, and all critical points of it have finite Morse indexes and nullities (see \cite[Prop.4.1, 4.2]{AbF} and \cite{Be})." However, as proved in \cite{AbSc1} the claim that ${\cal L}_τ$ is $C^2$-smooth is true if and only if for every $(t,q)$ the function $v\mapsto L(t,q,v)$ is a polynomial of degree at most 2. So the arguments …

从 Morse 函数到余切丛上的 Lefschetz 纤维化

From Morse Functions to Lefschetz Fibrations on Cotangent Bundles

本文证明紧流形上任意 Morse 函数与适配梯度可构造余切丛上同伦唯一的复值辛 Lefschetz 纤维化，并研究其正则纤维的拓扑与辛几何。