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Harmonic maps and a pinching theorem for positively curved hypersurfaces

In this paper, we establish a theorem of Liouville type for stable harmonic maps in sufficiently pinched, positively curved hypersurfaces of a space form with nonnegative constant curvature. Similar results for the Euclidean sphere S' have been proved by Y. L. Xin and P. F. Leung, respectively. 1. Statement of result. As is well known, for the Euclidean n(> 3)-sphere sn, there are no nonconstant stable harmonic maps either from Sn to any Riemannian manifold [6] or from any compact manifold to Sn [3]. Its generalizations to submanifolds have been attempted by many authors [3, 4, 5], where the additional conditions imposed on submanifolds are extrinsic. It is natural to find some condition intrinsic for submanifolds. In this paper, we prove the following theorem. THEOREM. Let Mn+I (c) be an (n+ 1)-dimensional simply connected space form with constant sectional curvature c > 0. Suppose that Mn (n > 3) is a compact hypersurface in Mn+i(c) of which the sectional curvature RiemM satisfies the following pinching condition: (1) c+a2/[(n-2)c+ (n-1)a] 0. Then there is no nonconstant stable harmonic map between Mn and any compact Riemannian manifold N, i.e., either from Mn to N or from N to Mn. COROLLARY 1. If Mn (n > 3) is a compact hypersurface in the Euclidean space En+1 with pinching condition: a/(n 1) 0, then there is no nonconstant stable harmonic map between Mn and any compact Riemannian manifold. COROLLARY 2. If Mn (n > 4) is a compact hypersurface in En+1 with ! < RiemM < 1, then the same conclusion as in Corollary 1 holds. REMARK. The condition (1) is sharp in some sense; see an example below. 2. Preliminaries. Let M and N be Riemannian manifolds of dimensions n and m, respectively, and X: M -N a smooth map. Throughout this paper, we agree on the following ranges of indices unless otherwise stated: 1 < qllk .. < n 1 < aR ^-, ..