AI 中文总结
研究具有周期正等变辛同调的闭接触流形，在非退化接触形式下给出简单闭Reeb轨道数量的下界，并刻画达到下界时的lacunary条件，证明该下界是接触不变量。
AI 中文摘要
我们考虑具有周期正等变辛同调的闭接触流形$(M,\xi)$。这是一类非常广泛的接触流形，据我们所知，包括所有当前已知的、允许具有有限多个闭轨道的Reeb流且等变辛同调是良好定义不变量的例子。在$M$上非退化接触形式$\alpha$的弱且同调自然的指标假设下，我们建立了$\alpha$的简单闭Reeb轨道数量的锐利下界$r_M$。此外，我们证明该下界达到当且仅当$\alpha$是lacunary的，即所有闭轨道的Conley-Zehnder指标具有相同的奇偶性。下界$r_M$具有清晰的动力学刻画：只要$M$上存在非退化lacunary接触形式，$r_M$就等于其简单闭Reeb轨道的数量，因此与这种形式的选择无关。特别地，在lacunary情形下，$r_M$是一个完全由正等变辛同调决定的接触不变量。我们计算了一类广泛例子中的$r_M$，包括几个辛轨形的前量子化，并表明在这种情况下$r_M = \dim H_*(M/S^1;\mathbb{Q})$，从而给出了这个不变量的拓扑刻画。受这些结果启发，我们猜想任何具有有限多个闭Reeb轨道的接触形式必然是非退化和lacunary的，且底层的接触流形是这种类型的前量子化。
英文摘要
We consider closed contact manifolds $(M,ξ)$ with periodic positive equivariant symplectic homology. This is a very large class of contact manifolds and, to the best of our knowledge, includes all currently known examples admitting Reeb flows with finitely many closed orbits for which this homology is well defined. Under weak and homologically natural index assumptions on a non-degenerate contact form $α$ on $M$, we establish a sharp lower bound $r_M$ for the number of simple closed Reeb orbits of $α$. The quantity $r_M$ is completely defined in terms of the positive equivariant symplectic homology of $M$. Moreover, we show that this bound is attained if and only if $α$ is lacunary, i.e., the Conley-Zehnder indices of all closed orbits have the same parity. Consequently, the invariant $r_M$ admits a clean dynamical characterization: whenever $M$ admits a non-degenerate lacunary contact form, $r_M$ equals the number of its simple closed Reeb orbits and is therefore independent of the choice of such a form. In particular, in the lacunary case the number of such orbits is a contact invariant completely determined by Floer theory. We compute $r_M$ for a broad class of examples, including several prequantizations of symplectic orbifolds, and show that in this case $r_M=\dim H_*(M/S^1;\mathbb{Q})$, thereby giving a topological characterization of the invariant. Motivated by these results, we conjecture that any contact form with finitely many closed Reeb orbits is necessarily non-degenerate and lacunary, and that the underlying contact manifold is a prequantization of this type.
Comments72 pages. Version 2: minor changes
