论文标题
稳定的封闭的大地测量学和稳定的凸出式曲面中的稳定人物
Stable closed geodesics and stable figure-eights in convex hypersurfaces
论文作者
论文摘要
对于每个奇数$ n \ geq 3 $，我们构建了一个$ \ mathbb {r}^{n+1} $的封闭凸置hyperface，其中包含一个非排级封闭的封闭地理测量，并以莫尔斯索引为零。 J. L. Synge的经典定理将禁止以$ n $ $ n $的价格进行这种结构，因此从某种意义上说，我们证明Synge的定理是“敏锐的”。我们还构建了稳定的图形：也就是说，对于每个$ n \ geq 3 $，我们将图形图嵌入了$ \ mathbb {r}^{n+1} $的封闭凸高度，以使嵌入的嵌入的小变化要么保留其图像或必须增加其长度。这些指数零的测量学和稳定的八人物主要是通过构建凸多型的“控制平行传输”的显式台球轨迹来得出的。
For each odd $n \geq 3$, we construct a closed convex hypersurface of $\mathbb{R}^{n+1}$ that contains a non-degenerate closed geodesic with Morse index zero. A classical theorem of J. L. Synge would forbid such constructions for even $n$, so in a sense we prove that Synge's theorem is "sharp." We also construct stable figure-eights: that is, for each $n \geq 3$ we embed the figure-eight graph in a closed convex hypersurface of $\mathbb{R}^{n+1}$, such that sufficiently small variations of the embedding either preserve its image or must increase its length. These index-zero geodesics and stable figure-eights are mainly derived by constructing explicit billiard trajectories with "controlled parallel transport" in convex polytopes.
