Analysis of weighted p-harmonic forms and applications

Abstract

In this paper, we study weighted [Formula: see text]-harmonic forms on smooth metric measure space [Formula: see text] with a weighted Sobolev or a weighted Poincaré inequality. When [Formula: see text] is constant, we derive a splitting theorem for Kähler manifolds with maximal bottom spectrum for the [Formula: see text]-Laplacian. For general [Formula: see text] we also obtain various splitting and vanishing theorems when the weighted curvature operator of [Formula: see text] is bounded below. As applications, we conclude Liouville property for weighted [Formula: see text]-harmonic functions and [Formula: see text]-harmonic maps.