On Splitting Complete Manifolds via Infinity Harmonic Functions

In this paper, we prove some splitting results for manifolds supporting a non-constant infinity harmonic function which has at most linear growth on one side. Manifolds with non-negative Ricci or sectional curvature are considered. In dimension $2$, we extend Savin's theorem on Lipschitz infinity harmonic functions in the plane to every surface with non-negative sectional curvature.