Sobolev Inequalities on Forms and L p,q -Cohomology on Complete Riemannian Manifolds

Abstract

We prove some Sobolev inequalities on differential forms over a class of complete non-compact Riemannian manifolds with suitable geometric conditions. Moreover, we establish some Lp,q-estimates and existence theorems of the Cartan-De Rham equation and the Hodge systems. As applications, we prove some vanishing theorems of the Lp,q-cohomology and prove the Lq-solvability of the nonlinear p-Laplace equation on forms on complete non-compact Riemannian manifolds with suitable geometric conditions.