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Despite recent progress in 3D head avatar generation, balancing identity preservation, i.e., reconstruction, with novel poses and expressions, i.e., animation, remains a challenge. Existing methods struggle to adapt Gaussians to varying geometrical deviations across facial regions, resulting in suboptimal quality. To address this, we propose GeoAvatar, a framework for adaptive geometrical Gaussian Splatting. GeoAvatar leverages Adaptive Pre-allocation Stage (APS), an unsupervised method that segments Gaussians into rigid and flexible sets for adaptive offset regularization. Then, based on mouth anatomy and dynamics, we introduce a novel mouth structure and the part-wise deformation strategy to enhance the animation fidelity of the mouth. Finally, we propose a regularization loss for precise rigging between Gaussians and 3DMM faces. Moreover, we release DynamicFace, a video dataset with highly expressive facial motions. Extensive experiments show the superiority of GeoAvatar compared to state-of-the-art methods in reconstruction and novel animation scenarios.