ON COMPLETE SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE IN NEGATIVE PINCHED MANIFOLDS

【摘要】 <正>A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n + p)-dimensional manifold Nn+p with negative sectional curvature is proved. For given positive integers n(≥2), p and for a constant H satisfying H > 1 there exists a negative numberτ(n,p,H)∈(-1,0) with the property that if the sectional curvature of N is pinched in [-1,τ(n,p,H)], and if the squared length of the second fundamental form is in a certain interval, then Nn+p is isometric to the hyperbolic space Hn+p(-1). As a consequence, this submanifold M is congruent to Sn(?) or the Veronese surface in S4(?).更多还原

【Abstract】 A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n + p)-dimensional manifold Nn+p with negative sectional curvature is proved. For given positive integers n(≥2), p and for a constant H satisfying H > 1 there exists a negative numberτ(n,p,H)∈(-1,0) with the property that if the sectional curvature of N is pinched in [-1,τ(n,p,H)], and if the squared length of the second fundamental form is in a certain interval, then Nn+p is isometric to the hyperbolic space Hn+p(-1). As a consequence, this submanifold M is congruent to Sn(?) or the Veronese surface in S4(?).更多还原