Matching equivalent entities across Knowledge graphs is a pivotal step for knowledge fusion. Previous approaches usually study the problem in Euclidean space. However, recent works have shown that hyperbolic space has a higher capacity than Euclidean space and hyperbolic embedding can represent the hierarchical structure in a knowledge graph. In this paper, we propose a localized geometric method to find equivalent entities in hyperbolic space. Specifically, we use a hyperbolic neural network to encode the lingual information of entities and the structure of both knowledge graphs into a low-dimensional hyperbolic space. To address the asymmetry of structure on different KGs and the localized nature of relations, we learn an instance-specific geometric mapping function based on rotation to match entity pairs. A contrastive loss function is used to train the model. The experiment verifies the power of low-dimensional hyperbolic space for entity matching and shows that our method outperforms the state of the art by a large margin.