一个关于某些$C^0$拉格朗日量的多维Birkhoff定理
A Multidimensional Birkhoff Theorem for some $C^0$ Lagrangians
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中文总结 AI 辅助
本文证明了一类新的$C^0$拉格朗日子集的多维Birkhoff定理，推广了环面扭转映射的经典结果，并首次将其扩展到仅连续对象，结合Floer理论、变分解和弱KAM理论。
中文摘要 AI 辅助
我们在余切丛中证明了一类新的“$C^0$拉格朗日子集”的多维Birkhoff定理，这类子集是紧致精确拉格朗日子流形在Liouville原函数受控下的Hausdorff极限。如果这样的子集$L$在Tonelli哈密顿流下的连续像在正负时间方向上都存在收敛子序列，则$L$及其所有像都是底空间上的Lipschitz图。这推广了Birkhoff关于环面扭转映射的著名定理及其已知的高维推广，并为仅连续对象提供了此类结果的第一个版本。证明结合了Floer理论的图选择子、Hamilton-Jacobi方程的变分解以及弱KAM理论。我们还研究了这类新的奇异拉格朗日子集的极限原函数的刚性和唯一性，这可能对$C^0$辛拓扑具有独立意义。
英文摘要
We prove a multidimensional Birkhoff theorem for a new class of "$C^0$ Lagrangian subsets" in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset $L$ under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then $L$ and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in $C^0$ symplectic topology.
发表机构
- Université Paris-Saclay（巴黎萨克雷大学）
- ETH Zürich（苏黎世联邦理工学院）
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