Stable harmonic maps from pinched manifolds

Abstract

A harmonic map is a critical point of the energy functional and a harmonic map is said to be stable if for any deformation vector field, its second variation is always non-negative. As well known, when the source or the target manifold is the Euclidean sphere Sn(n > 3), every stable harmonic map must be constant ([-4; 8]). A natural question is "Does the above fact hold too for a simply connected &pinched Riemannian manifold?". Here by a &pinched Riemannian manifold we mean a Riemannian manifold whose sectional curvatures are between the interval (~K, K] with constants K > 0 and 1 > 6 > 0 . For the case that the target manifold is a simply connected &pinched Riemannian manifold. Howard in 1985 proved that let n>3 , there is a number (n) with 1/4__ 3), then for every compact Riemannian manifold N, any stable harmonic map ~b :N ~ M" is constant. There is no result for the case that the source manifold is a simply connected 6-pinched Riemannian manifold up to now. Recently, the author in a previous paper [7] gives an affirmative answer to it with dimension-depending pinching constants. But there the pinching constants are difficult to compute. The aim of the present paper is to give a new proof of the above answer in a completely different way from which one can practically compute those pinching constants. We shall prove the following