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3D surface matching is fundamental for shape analysis. As a powerful method in geometric analysis, Ricci flow can flexibly design metrics by prescribed target curvature. In this paper we describe a novel approach for matching surfaces with complicated topologies based on hyperbolic Ricci flow. For surfaces with negative Euler characteristics, such as a human face with holes (eye contours), the canonical hyperbolic metric is conformal to the original and can be efficiently computed. Then the surface can be canonically decomposed to hyperbolic hexagons. By matching the corresponding hyperbolic hexagons, the matching between surfaces can be easily established. Compared to existing methods, hyperbolic Ricci flow induces diffeomorphisms between surfaces with complicated topologies with negative Euler characteristics, while avoiding singularities. Furthermore, all the boundaries are intrinsically mapped to hyperbolic lines as alignment constraints. Finally, we demonstrate the applicability of this intrinsic shape representation for 3D face matching and registration.