AI 中文总结
本文计算ℝ³中椭球单位余切丛的Hofer-Zehnder容量，通过两种互补方法推导其上下界，为正曲率黎曼二维球面的相关容量提供了界定。
AI 中文摘要
我们计算了ℝ³中每个椭球的单位余切丛的Hofer-Zehnder容量。该容量由测地长度谱中两个特殊量的较小者决定：收缩压（systole）的两倍，以及Morse指数为3的最短简单闭测地线的长度。对于下界，我们利用椭球合适截面上的黎曼弹子球；对于上界，我们提出两种互补方法：第一种结合Hofer-Viterbo的论证与颈拉伸，更一般地，根据指定指数的闭测地线给出正曲率黎曼二维球面的上界；第二种运用辛同调中的裤衩积与Viterbo同构，将黎曼二维球面的任意余切丛的Hofer-Zehnder容量用舒张压（diastole）的两倍来界定，对于正曲率情形，舒张压与收缩压相等。
英文摘要
We compute the Hofer-Zehnder capacity of the unit disk cotangent bundle of every ellipsoid in $\mathbb R^3$. The capacity is determined by the smaller of two distinguished quantities in the geodesic length spectrum: twice the systole and the length of the shortest simple closed geodesic of Morse index 3. For the lower bound, we use Riemannian billiards on a suitable cut of the ellipsoid. For the upper bounds, we develop two complementary methods. The first combines an argument by Hofer-Viterbo with neck-stretching and yields, more generally, an upper bound for positively curved Riemannian two-spheres in terms of closed geodesics of prescribed index. The second uses the pair-of-pants product in symplectic homology and the Viterbo isomorphism to bound the Hofer-Zehnder capacity of any disk cotangent bundles of Riemannian two-spheres by twice the diastole; for positive curvature, the diastole agrees with the systole.
