三维等谱流形上的最短闭测地线
Shortest closed geodesics on three-dimensional isospectral manifolds
- Tsinghua University（清华大学）
- University of Science and Technology of China（中国科学技术大学）
- Capital Normal University（首都师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明光滑闭三维流形的谱能给出其最短非平凡闭测地线长度的正下界，并由此推出等谱族在微分同胚意义下的光滑紧致性。
AI中文摘要：
我们证明了光滑闭三维流形的谱给出了其最短非常数闭测地线长度的正下界。论证使用了三个曲率热系数和延迟波迹的第一个正奇点。热系数产生了一个曲率-长度二分法：足够大的最大曲率迫使存在一条比曲率尺度更短的闭测地线。然后我们证明，位于共轭半径之下的最短闭测地线会产生正弦迹的一个奇点。结合这些估计，得到了一个仅依赖于共同谱的下界。安德森的紧致性定理进而意味着整个等谱族在微分同胚意义下的光滑紧致性。
英文摘要：
We prove that the spectrum of a smooth closed three-manifold gives a positive lower bound for the length of its shortest nonconstant closed geodesic. The argument uses three curvature heat coefficients and the first positive singularity of the retarded wave trace. The heat coefficients yield a curvature--length dichotomy: sufficiently large maximum curvature forces a closed geodesic shorter than the curvature scale. We then show that a shortest closed geodesic lying below the conjugate radius produces a singularity of the sine trace. Combining these estimates gives a lower bound depending only on the common spectrum. The compactness theorem of Anderson then implies smooth compactness of the full isospectral family modulo diffeomorphisms.
