Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces
Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces
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DOI：
10.1215/s0012-7094-07-13931-0
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发表时间：
2007-01
影响因子：
2.5
通讯作者：
Wei Wang;Xijun Hu;Y. Long
中科院分区：
文献类型：
--
作者：
Wei Wang;Xijun Hu;Y. Long
哈密顿分析中有一个长期存在的猜想，即 $\R^{2n}$ 中 $n\ge 2$ 中的每个紧凸超曲面上都存在至少 $n$ 个几何上不同的闭合特征。除了许多部分结果之外，这个猜想仅在 $n=2$ 时才被完全解决。在本文中，我们针对 $n=3$ 给出了这个猜想的确认答案。为了证明这个结果，当$\Sg$上几何上不同的闭合特征的数量有限时，我们首先为$\R^{2n}$中每个紧凸超曲面$\Sg$上的闭合特征建立一个新的共振恒等式。然后使用这个恒等式和索引迭代理论的早期技术，我们证明了 $\R^6$ 的重数结果。如果 $\R^4$ 中的紧凸超曲面上正好有两个几何上不同的闭特征，我们证明它们都必须是无理椭圆。
There is a long standing conjecture in Hamiltonian analysis which claims that there exist at least $n$ geometrically distinct closed characteristics on every compact convex hypersurface in $\R^{2n}$ with $n\ge 2$. Besides many partial results, this conjecture has been only completely solved for $n=2$. In this paper, we give a confirmed answer to this conjecture for $n=3$. In order to prove this result, we establish first a new resonance identity for closed characteristics on every compact convex hypersurface $\Sg$ in $\R^{2n}$ when the number of geometrically distinct closed characteristics on $\Sg$ is finite. Then using this identity and earlier techniques of the index iteration theory, we prove the mentioned multiplicity result for $\R^6$. If there are exactly two geometrically distinct closed characteristics on a compact convex hypersuface in $\R^4$, we prove that both of them must be irrationally elliptic.