Hyperbolic geometry offers significant promise for deep learning with hierarchical data due to its capacity for modeling complex structures. However, its practical integration faces challenges including training instability, overfitting, high computational costs, and the critical need for effective hard negative sampling in its non-uniform space. Traditional sampling methods often fail to adapt to local geometry, resulting in suboptimal embeddings. To overcome these issues, we propose Hierarchical Curvature-Adaptive Negative Mining (HCANM), a novel framework for robust and efficient hyperbolic metric learning. HCANM leverages a Dynamically Learnable Local Curvature (DLLC) module to adaptively learn local geometry, guiding a Curvature-Aware Adaptive Negative Mining (CA-ANM) strategy for selecting genuinely informative hard negative samples. Furthermore, we integrate robust hyperbolic optimization techniques, such as Riemannian AdamW, Dynamic Norm Regularization, and Differentiable Hyperbolic Scaling, to ensure stable and generalizable training. Extensive experiments on diverse fine-grained hierarchical metric learning datasets demonstrate that HCANM consistently achieves stateof-the-art retrieval performance, significantly outperforming existing methods. Our ablation studies and qualitative analysis, complemented by a user study, confirm the crucial contributions of DLLC and CA-ANM and the superior semantic coherence of our learned embeddings. HCANM represents a substantial advancement towards practical and stable hyperbolic deep learning for complex data.