$\epsilon$-approximability and Quantitative Fatou Theorem on Riemannian Manifolds

We prove the Quantitative Fatou Theorem for Lipschitz domains on complete Riemannian manifolds. This requires extending the $\epsilon$-approximation lemma to the manifold setting. Our studies apply to harmonic functions, as well as to a class of A-harmonic functions on manifolds. The presented results extend works by Dahlberg, Garnett, Bortz and Hofmann.