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

In 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 Delta(N)(infinity) u >= g(u). Also, an infinity-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 Delta(N)(infinity) u = g(u) on relatively compact sets, that implies a uniform Lipschitz estimate for certain entire, bounded solutions without requiring the completeness of M. (C) 2021 Elsevier Inc. All rights reserved.