Growth of equivariant harmonic maps and harmonic morphisms

Abstract

The purpose of the present paper is to study the growth of certain harmonic maps in relation with the geometry of the domains and ranges. Let φ: M—>N be a harmonic map between complete noncompact Riemannian manifolds M and N. We fix a point o of M(resρ. a point o' of N) and denote by rM (resp. rN) the distance to o in N (resp. o' in N). Set μ(φ; t): = mzκ{rN(φ(x)): x&M, rM(x)=t}. We want to know the growth of φ, or the asymptotic behavior of μ(φ t) as t goes to infinity. We first recall the following result by Cheng [8] (cf. also [3] [31: Chap. 6]): Suppose that M has nonnegative Ricci curvature and N is a Hadamard manifold, namely, N is a simply connected and nonpositively curved manofod manifold. Then the energy density e{φ) of the map φ satisfies: e(φ)(o)<cmμ(φ: t) i?, where cm is a constant depending only on the dimension m of M. It follows that φ is a constant map if φ has sublinear growth, that is, lim inf μ (φ t)/t = 0. We are interested in a (nonconstant)