论文标题
通用的公共指数跳跃定理，应用于星形超曲面和以后的封闭特征的应用
Generalized common index jump theorem with applications to closed characteristics on star-shaped hypersurfaces and beyond
论文作者
论文摘要
在本文中，我们首先将Longzhu的共同指数跳跃定理和2016年的Duan-long-Wang定理到这种情况下，符号路径的平均指数不需要是正面的。作为应用程序，我们研究了$ {\ bf r}^{2n} $中紧凑型星形超曲面的封闭特征，当时正均值指数可能同时出现。特别是，我们在每个紧凑的非脱位式恒星hypersurface $σ$ in $ {\ bf r}^{2n} $中至少存在至少几何不同的封闭特性。此外，在$ {\ bf r}^6 $的情况下，我们通过证明仅存在有限的许多几何闭合特征的存在来删除非零均值索引条件，这意味着它们每个人都必须具有非零均值索引。我们还概括了上述关于非排分星形超曲面上封闭特征的结果，以在广泛的固定捆绑包上的非分类接触形式的闭合旋转。
In this paper, we first generalize the common index jump theorem of Long-Zhu in 2002 and Duan-Long-Wang in 2016 to the case where the mean indices of symplectic paths are not required to be all positive. As applications, we study closed characteristics on compact star-shaped hypersurfaces in ${\bf R}^{2n}$, when both positive and negative mean indices may appear simultaneously. Specially we establish the existence of at least $n$ geometrically distinct closed characteristics on every compact non-degenerate perfect star-shaped hypersurface $Σ$ in ${\bf R}^{2n}$ provided that every prime closed characteristic possesses nonzero mean index. Furthermore, in the case of ${\bf R}^6$ we remove the nonzero mean index condition by showing that the existence of only finitely many geometrically distinct closed characteristics implies that each of them must possess nonzero mean index. We also generalize the above results about closed characteristics on non-degenerate star-shaped hypersurfaces to closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles.
