<p>Recommender systems have recently begun to incorporate topological and geometric structures to better capture hierarchical organization, alleviate sparsity, and model complex user–item relations. Hyperbolic embedding approaches, in particular, offer a natural way to represent tree-like and uneven interaction patterns, but most existing methods rely on static curvature, fixed boundaries, and uniform fusion strategies, which constrain their adaptivity to heterogeneous semantics. To overcome these limitations, we propose AHMRec, an adaptive hyperbolic metric recommendation framework. First, a Learnable Local Curvature mechanism assigns distinct negative curvatures to different subspaces, allowing hierarchical and Euclidean-like patterns to coexist. Second, a Boundary Stabilization module introduces radial homeomorphic compression to alleviate gradient explosion and pseudo-connectivity near the hyperbolic boundary. Third, a Hybrid Adaptive Distance Scaling strategy performs pair-wise fusion of Poincaré and Lorentz distances, selecting the most suitable geometry for each interaction. Together, these designs transform the static hyperbolic metric space into a piecewise-consistent and topology-aware space while preserving the original optimization goal. Extensive experiments on multiple real-world datasets demonstrate that AHMRec outperforms existing methods.</p>