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In this paper, we study some potential theoretic aspects of the eikonal andinfinity Laplace operator on a Finsler manifold $M$. Our main result shows thatthe forward completeness of $M$ can be detected in terms of Liouvilleproperties and maximum principles at infinity for subsolutions of suitableinequalities, including $\Delta^N_\infty u \ge g(u)$. Also, an$\infty$-capacity criterion and a viscosity version of Ekeland principle areproved to be equivalent to the forward completeness of $M$. Part of the proofhinges on a new boundary-to-interior Lipschitz estimate for solutions of$\Delta^N_\infty u = g(u)$ on relatively compact sets, that implies a uniformLipschitz estimate for certain entire, bounded solutions without requiring thecompleteness of $M$.

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