实空间形式中弯曲管的Cheeger常数
The Cheeger constant of curved tubes in real space forms
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中文总结 AI 辅助
本文针对实空间形式中的弯曲管，结合费米坐标、校准型论证与散度定理，计算其Cheeger常数，还证明非紧实空间形式中无界弯曲管无有限体积Cheeger集。
中文摘要 AI 辅助
受由闭曲线$P$定义的弯曲管$T(P,a)$的几何性质启发，我们在具有常截面曲率的任意维实空间形式中计算弯曲管$T(P,a)$的Cheeger常数$h(T(P,a))$。费米坐标系统的结构与性质可对弯曲管进行参数化，结合$T(P,a)$的面积与体积精确公式，能直接计算$h(T(P,a))$的上界。接着，我们将管的几何与校准型论证结合，推导其下界，核心思路是用沿基础曲线$P$移动的测地球面描述该管，这提供了自然的外方向，使估计在几何上清晰，进而可通过散度定理计算下界。最后，对于非紧实空间形式中的无界弯曲管类，我们也计算了其Cheeger常数，并证明不存在有限体积的Cheeger集。
英文摘要
Motivated by the geometric properties of curved tubes $T\left(P,a\right)$ defined by closed curves $P$, we compute the Cheeger constant $h\left(T\left(P,a\right)\right)$ in the real space forms of arbitrary dimensions with constant sectional curvature. The structure and properties of the system of Fermi coordinates allows us to parametrize the curved tube and straightforwardly compute the upper bound of $h\left(T\left(P,a\right)\right)$ using the exact formulas for the area and volume of $T\left(P,a\right)$. Next, we derive the lower bound by combining the geometry of tubes with a calibration-type argument. The key idea is to describe the tube using geodesic spheres moving along the underlying curve $P$, which provides a natural outward direction and makes the estimate geometrically transparent. This allows us to compute the lower bound via the divergence theorem. Finally, for the class of unbounded curved tubes in noncompact real space forms, we also compute the Cheeger constant and prove that there is no finite-volume Cheeger set.
