The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model $n$-dimensional hyperbolic space, complementing the results of Borisov and Freitas (\textit{Comm. Anal. Geom.} 25: 507--544, 2017). We then give a synthetic proof of Cheng's sharp eigenvalue comparison theorem in metric measure spaces satisfying a 'negatively curved' Bishop-Gromov-type volume monotonicity hypothesis. As a byproduct, we provide an example of simply connected, non-compact Finsler manifold with constant negative flag curvature whose first eigenvalue is zero; this result is in a sharp contrast with its celebrated Riemannian counterpart due to McKean (\textit{J. Differential Geom.} 4: 359--366, 1970). Our proofs are based on specific properties of the Gaussian hypergeometric function combined with intrinsic aspects of the negatively curved smooth/non-smooth spaces.

中文翻译：

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负弯曲空间第一特征值的新特征

本文致力于研究负弯曲空间上第一特征值的精细性质。首先，根据空间维度的奇偶性，我们为模型 $n$ 维双曲空间中的大测地球提供了第一个特征值的渐近锐谐型展开，补充了 Borisov 和 Freitas (\textit{Comm . Anal. Geom.} 25: 507--544, 2017)。然后，我们在满足“负弯曲”Bishop-Gromov 型体积单调性假设的度量测度空间中给出了 Cheng 的尖锐特征值比较定理的综合证明。作为副产品，我们提供了一个简单连接的、非紧致的 Finsler 流形的例子，它具有恒定的负标志曲率，其第一特征值为零；由于 McKean (\textit{J. Differential Geom.} 4: 359--366, 1970)，这个结果与其著名的黎曼对应物形成鲜明对比。我们的证明基于高斯超几何函数的特定属性以及负弯曲平滑/非平滑空间的内在方面。