固定秩的不可约紧致对称空间的一个直径间隙
A diameter gap for irreducible compact symmetric spaces of fixed rank
AI总结：
本文证明固定秩的单连通不可约紧致对称空间的正直径等距商空间具有一致的正归一化直径下界，并针对高秩Grassmann流形族构造了有界三角次数的不变函数以完成证明。
AI中文摘要：
对于每个正整数 $r$，我们证明秩为 $r$ 的单连通不可约紧致黎曼对称空间的正直径等距商空间，其归一化直径具有一致的正下界。该下界与维数及作用群（可能不连通）无关。秩一的情形归功于 Gorodski、Lange、Lytchak 和 Mendes \ncite{GLLM}。对于高秩的 Grassmann 流形族，我们构造了有界三角次数的非常值不变函数。实情形和复情形使用矩映射论证。四元数情形使用四次刚性论证以及紧致四元数对称空间的分类。
英文摘要：
For every positive integer $r$, we prove that positive-diameter isometric quotients of simply connected irreducible compact Riemannian symmetric spaces of rank $r$ have a uniform positive lower bound on their normalized diameter. The bound is independent of dimension and of the acting group, which may be disconnected. The rank-one case is due to Gorodski, Lange, Lytchak, and Mendes \cite{GLLM}. For the higher-rank Grassmannian families, we construct nonconstant invariant functions of bounded trigonometric degree. The real and complex cases use a moment map argument. The quaternionic case uses a quartic rigidity argument and the classification of compact quaternionic symmetric spaces.
