Negative Pinched Manifolds Submanifold In Overall Pinching Theorem

Recently, L^p — pinching problem has become an important new subject in differential geometry, which mainly studies the geometric structure and topological structure of manifolds under L^p — pinching condition. J.Simons, H.B.Lawson, S.S.Chern, M.do Carmo, O.S.Kobayashi proved the following famous rigidity theoremLet Mⁿ be an oriented closed minimal manifold in a unit sphere S^n+p(1). ifS ≤ n/(2-1/p), then S (?) 0 and hence M is totally geodesic;or S (?) n/(2-1/p). That M is one ofthe follow cases:1. the Veronese surface in S⁴.2. the Clifford minimal Hypersurface in Sⁿ⁺¹.After that, [4], [9], [14] improved the pinching condition. considering the hyperbolic space H^n+p(-1), Xu-Xiang-Gu provedLet Mⁿ be a complete submanifold with parallel mean curvature in the hyperbolic space H^n+p(-1), Denote by H and S the mean curvature and squared length. If S ≤ C(n,p,H), and sup_MS < α(n,H), where H > 1 then M is congruent to the umbilical sphere Sⁿ(1/((H²-1)¹/2)) or the Veronese surface in S⁴(1/((H²-1)¹/2)). Here the constants are given by:H.W.Xu, J.R.Gu and M.Y.He studied the L^n/2-pinching problem for n-dimensional complete submanifolds with parallel mean curvature in the hyperbolic space H^n+p(-1) with constant curvature -1, and proveLet Mⁿ(n ≥ 3) be an n-dimensional complete submanifold with parallel mean curvature in an (n+p)-dimensional hyperbolic space H^n+p(-1). Denote by H and S themean curvature satisfying H > 1 and the squared length of the second fudamental form of M respectively. If jM{S - nH2)% < C(n,H), where C(n, H) is an explicit positive constant depending on n and H, then 5 = nH2, M is a totally umbilical submanifoldSn(l/VH2 - 1).In this thesis, we proveTheorem. Let Mn(n > 3) be an oriented compact submanifold with parallel mean curvature in a complete and simply connected (n + p)-dimensional Riemannian manifold Nn+P with sectional curvature — 1 < Ks < 6o(n,p,H), H be the mean curvature of M, with H > 1. and S be the square of the length of the second fundamental form . Let the relative mean curvature H of the composition of isometric immersionsMn ₊ Nn+p ?_, R\\S-nH2\\n/22 > (1 + 6)a(n,p)vol(M), then Nn+P is isometric to Hn+P{-1), and Mn is congruent to Sn{l/^H2 - 1). where5 :— supKfif, vol(M) is the volume of M, 6o(n,p, H) is a impositive number depending on n,p, a(n,p) is a positive number depending on n,p, C(n,p,5,H,Ho) is a positive number depending on n,p, 5, H, Hq.In particular, when 6 = — 1, we can getCorollary. Let Mn(n > 3) is an oriented compact submanifold with parallel mean curvature in a Hn+P(-1), If fM(S - nH2)% < C(n), where C(n) is a positive number depending only on n, then M is a totally umbilical sphere.