Geometric Flow Matching

We propose a flow-based generative model for 3D surface generation that leverages geometric flows within the flow matching framework. Instead of relying on generic stochastic interpolation paths, we construct data-dependent trajectories using a curvature-driven geometric evolution that progressively smooths surfaces while preserving mesh topology. This formulation defines structured transport paths between simple base shapes and complex target geometries, enabling the model to learn geometry-aware transformations. Our approach operates directly on triangle meshes and produces coherent deformation sequences governed by surface curvature. We evaluate our method on 3D hand and face mesh datasets and compare it against alternative flow construction strategies, demonstrating that curvature-driven trajectories yield more realistic, geometrically faithful meshes.