Negative Pinched Manifolds In Parallel Mean Curvature Submanifolds Shaped Rigid Theorem

J.Simons proved the famous rigidity theorem for minimal submanifolds in 1968 : Let Mⁿ be an n-dimensional manifold which is minimally immersed in a unit sphere S^n+p(1). Let S be the squared length of the second fundamental form of this immersion. If S ≤ n/2-p-1,then S (?) 0, i.e., M is totally geodesic,or S (?) n/2-p-1.Chern-do Coma-Kobayashi proved the following theorem in 1970: Let Mⁿ be an n-dimensional manifold which is minimally immersed in a unit sphere S^n+p(1).If S = n/2-p-1 ,then M is one of the following cases:1. the Veronese surface in S⁴.2. the Clifford minimal Hypersurface in Sⁿ⁺¹.H.B.Lawson proved the similar result on minimal Hypersurfaces.Shiohama-Xu proved this theorem for complete submanifolds in 1998: For given positive integers n(> 2),p and a nonnegative constant H there exists a number (?)(n,p) such that 0 < (?)(n,p) < 1 with the following properties: If Mⁿ is an oriented complete submanifold with parallel meam curvature normal field with its norm H in a complete and simply connected (n+p)-dimensional Riemannian manifold N^n+p with (?)(n,p) ≤ Kn ≤ 1, and ifwhere c := infKN,then N is isometric to S^n+p(1).Moreover,1. If supMS < α(n,H),then M is congruent to either Sⁿ(1/(?)H² + 1) or the Veronese surface in S⁴(1/(?)H² + 1).2. If M is compact,then M is congruent to one of the following:(1) Sⁿ(1/(?)H² + 1).(2) the isoparametric hypersurface .(3)one of the Clifford minimal hypersurfaces (4)the Clifford torus S¹(r1)×S¹(r2)inS³(r) with constant mean curvatureH₀, where...