Detecting the completeness of a Finsler 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 ${\mathrm{\Delta }}_{\mathrm{\infty }}^{N}u\ge g(u)$. Also, an $\mathrm{\infty }$-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 ${\mathrm{\Delta }}_{\mathrm{\infty }}^{N}u=g(u)$ on relatively compact sets, that implies a uniform Lipschitz estimate for certain entire, bounded solutions without requiring the completeness of $M$.