Applications to harmonic maps

Abstract

In this section we show the usefulness of Theorem 4.5 by deriving a number of results on harmonic maps. We begin by establishing a Liouville-type theorem which compares with classical work by Schoen and Yau, [146]. Direct inspection shows that our result, emphasizing the role of a suitable Schrödinger operator related to the Ricci curvature of the domain manifold, unifies in a single statement the situations considered in [146]; see Remark 6.22 below. We also give a version of this result in case the domain manifold is Kähler and see how this allows weaker integrability conditions on the energy density of the map. From this, we derive a number of geometric conclusions. We then provide a sharp upper estimate on the growth of the energy of a harmonic map. We close the section with a Schwarz-type lemma for harmonic maps with bounded dilation, and some applications to the fundamental group which extend results by Schoen and Yau and Lemaire ([93]).