On the Liouville theorem for harmonic maps

Abstract

Suppose M M and N N are complete Riemannian manifolds; M M with Ricci curvature bounded below by − A - A , A ⩾ 0 A \geqslant 0 , N N with sectional curvature bounded above by a positive constant K K . Let u : M → N u:M \to N be a harmonic map such that u ( M ) ⊂ B R ( y 0 ) u(M) \subset {B_R}({y_0}) . If B R ( y 0 ) {B_R}({y_0}) lies inside the cut locus of y 0 {y_0} and R > π / 2 K R > \pi /2\sqrt K , then the energy density e ( u ) e(u) of u u is bounded by a constant depending only on A A , K K and R R . If A = 0 A = 0 , then u u is a constant map.