Volumes of hyperbolic manifolds with geodesic boundary
Volumes of hyperbolic manifolds with geodesic boundary
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具有测地线边界的双曲流形的体积
DOI：
10.1016/0040-9383(94)90001-9
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
Yosuke Miyamoto
中科院分区：
文献类型：
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作者：
Yosuke Miyamoto
双曲流形是截面曲率恒定为-1 的黎曼流形。 Jorgensen 和 Thurston [15] 的工作表明，完全双曲 3 流形的体积形成序类型为 0”' 的 R 的良序子集。对于 n 2 4，Wang 定理 Cl63 意味着双曲 n 维流形的体积形成 R 的离散子集。此外，对于每个维度 n 2 3 ，具有相同体积的流形的等距类的数量是有限的。上述事实确保了最小的存在例如，参见 [l, 3, 7, 11, 121 以获得先前的结果。在本文中，我们关注具有非空完全测地线边界的完全双曲 n 维流形，这样的流形 N 沿着其边界 8N 具有双 DN，因此 DN 是具有有限体积的完全双曲 n 流形，并且 dN 在 DN 中是完全测地线的。 8N 是完全测地线，它是具有诱导度量的双曲 (n-l) 流形，因此定义了 8N 的 (n-l) 维体积，我们首先关注估计 M 的 n 维体积与其边界的 (n-1) 维体积的比率，然后我们将这些估计应用于最小体积问题。
A hyperbolic manifold is a Riemannian manifold of constant sectional curvature-1. Work by Jorgensen and Thurston [15] says that the volumes of complete hyperbolic 3-manifolds form a well-ordered subset of R of order type 0”‘. For n 2 4, Wang’s theorem Cl63 implies that the volumes of hyperbolic n-dimensional manifolds form a discrete subset of R. Moreover, for each dimension n 2 3 the number of the isometry classes of manifolds with the same volume is finite.The facts above ensure the existence of the minimal volumes of certain classes of hyperbolic manifolds. For instance see [l, 3, 7, 11, 121 for previous results. In this paper we focus on complete hyperbolic n-dimensional(n 2 3) manifolds of finite volume with non-empty totally geodesic boundary. Such a manifold N has double DN along its boundary 8N, so that DN is a complete hyperbolic n-manifold with finite volume and that dN is totally geodesic in DN. If 8N is totally geodesic, it is a hyperbolic (n-l)-manifold with the induced metric, and therefore the (n-l)-dimensional volume of 8N is defined. We will first be concerned with estimating the ratio of the n-dimensional volume of M to the (n-1)-dimensional volume of its boundary. We will then apply these estimates to minimal volume problems. To estimate this ratio we introduce the notion of