Liouville theorems for harmonic maps

Abstract

We prove several Liouville theorems for harmonic maps between certain classes of Riemannian manifolds. In particular, the results can be applied to harmonic maps from the Euclidean space (R", go) to a large class of Riemannian manifolds. Our assumptions on the harmonic maps concern the asymptotic behavior of the maps at ~.

Liouville theorem for steady-state solutions of simplified Ericksen-Leslie system

“…Xin [30] generalized these results. Jin [31] proved Liouville theorem under assumptions on the asymptotic behavior of the maps at infinity. Rigoli-Setti [32] considered the rotationally symmetric harmonic maps into R n , upper hemisphere and hyperbolic space.…”

Complete submanifolds of manifolds of negative curvature

“…The Liouville type properties for harmonic maps have been studied extensively in the past years (cf. [4,[10][11][12]17,18,20,[30][31][32], etc.). In 1976, Schoen and Yau showed that a harmonic map of finite energy from a complete Riemannian manifold with non-negative Ricci curvature to a complete manifold with non-positive sectional curvature is constant.…”

Gradient estimate and Liouville theorems for p-harmonic maps

“…When the target manifold N is of sectional curvature bounded from above by −a 2 for some a > 0, it was proved by Y. Shen [22] that if u is a harmonic map from a complete Riemannian manifold with non-negative Ricci curvature into N such that the image of u lies in a horoball of N, then u must be a constant. There are also various Liouville type theorems for harmonic maps and their generalizations (e.g., [5], [6], [7], [9], [11] and the references therein).…”