发表机构
Sobolev Institute of Mathematics（索博列夫数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对四面体范围内的双曲多面体，证明了在π/3≤α<arccos(1/3)时，二面角为α的等角双曲多面体中极小体积由对应正则双曲四面体实现，所用方法整合了多项几何定理与分解技术。
AI 中文摘要
我们研究所有二面角均等于固定值α的有限体积凸双曲多面体。这类非钝角等角多面体仅在π/3≤α≤π/2范围内存在。极小体积问题的端点情况已明确：当α=π/3时，极小体积由理想正则四面体实现；当α=π/2时，在直角多面体中，极小体积由三角双锥P(3,2)实现，其体积为卡塔兰常数。我们证明了四面体范围π/3≤α<arccos(1/3)内的对应结论，该范围内存在二面角为α的正则双曲四面体，即对每个此类α，所有二面角为α的等角双曲多面体中，极小体积仅由该四面体实现。证明结合了Andreev定理、Schläfli公式、Atkinson的无环与棱柱部分分解、普通棱柱及完全正交形的显式体积估计，以及Inoue边手术的直接等角版本。
英文摘要
We study the minimum-volume problem for finite-volume convex hyperbolic polyhedra whose dihedral angles are all equal to a fixed number \(α\). In the non-obtuse case one necessarily has \[ \fracπ{3}\le α\le \fracπ{2}. \] We prove that throughout the full range in which the regular hyperbolic tetrahedron with dihedral angle \(α\) exists, \[ \fracπ{3}\le α<\arccos\frac13, \] it is the unique minimum-volume equiangular hyperbolic polyhedron with prescribed angle \(α\). The left endpoint is the known ideal case, while at \(α=\arccos(1/3)\) the regular tetrahedron degenerates to the Euclidean one. The proof combines Andreev's theorem and the Schläfli formula with Atkinson's decomposition into atoroidal and prismatic parts, explicit volume estimates for ordinary prisms and complete orthoschemes, and a direct equiangular version of Inoue's edge surgery.
