In this talk, we introduce the p-parabolic ends and p-hyperbolic ends on manifolds M by p-capacities, which imply the volume estimates under the assumption of positive spectrum. If Ricci curvature of M has the lower bound depending on the spectrum, we show the Liouville property of p-harmonic functions with finite p-energy, which also infers the topological properties of M, that is, M has at most one hyperbolic end. Finally, whenever sectional curvature of M has lower bound , we prove the gradient estimate for the positive functions associated to the equation with p-Laplace operator.
