Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian

n this paper, we study some potential-theoretic aspects of the eikonal and infinity Laplace operator on a Finsler manifold M. Our main result shows that the forward completeness of M can be detected in terms of Liouville properties and maximum principles at infinity for subsolutions of suitable inequalities, including Δ∞Nu≥g(u). Also, an ∞-capacity criterion and a viscosity version of Ekeland principle are proved to be equivalent to the forward completeness of M. Part of the proof hinges on a new boundary-to-interior Lipschitz estimate for solutions of Δ∞Nu=g(u) on relatively compact sets, that implies a uniform Lipschitz estimate for certain entire, bounded solutions without requiring the completeness of M.