| J.Simon, H.B.Lawson, Chern-do Coma-Kobayashi proved the following theorem: Let Mn be a n-dimensional compact manifold minimally immersed in a n+p-dimensional space Sn+1, moreover S≤n/(2-1/p). Then S (?) 0 and hence M is congruent to Sn(1);orS (?)n/(2-1/p), moreover M is either the Veronese surface in S4, or the Clifford minimalHypersurface in Sn+1.Several geometers improved above results in [1], [6], [8]and [12].J. R. Gu and H. W. Xu studied the geometric rigidity problem for complete submanifolds in pinched Riemannian manifold, and proved the following results:Let Mn be an n-dimensional complete submanifold with parallel mean curvature in a complete and simple connected n+p-dimensional Riemannian manifold Nn+p.Let KN be the sectional curvature of N satisfying c := inf KN ≤ 0,d := supKN ≥ 0,denote by H and S the mean curvature and the squared length of the second fundamental form of M respectively then there exist constant τ1(n,p,H)(≤ 0), τ2(n,p,H)(≥ 0), here τ12(n,p,H) + τ22(n,p,H) ≠0, such that if KN ∈ [τ1(n,p,H),τ2(n,p,H)], and if, c + H2 > 0nH2 + A1(n,p)(d - c) + A2(n,p)[(n(n - 1)-1H3]1/2(d - c)1/4 ≤ S≤ C0(n,p,H) - B2(n,p)(d - c) - B2(n,p)[n(n - l)-1H3]1/2(d-c)1/4, then Nn+P is isometric to the Euclidean space Rn+p.Moreover,then M is congruent toSn(1/H),the Veronese surface in S4(1/H) or the generalized cylinderSn-1((n- 1)/nH) × R1 in Rn+1.here the constant C0(n,p,H) is given byand τ1(n,p,H), τ2(n,p,H), A1(n,p),A2(n,p), B1(n,p). B2(n,p) are explicit positive constants.H. W. Xu and F. Wang proved a global pinching theorem for submanifolds with parallel mean curvature in a pinched Riemannian manifold, thus improve the above Ln/2 - pinching theorem. Then under weaker geometric condition we prove a global theorem for compact minimal submanifolds in a pinched locally symmetric manifold.In this thesis,we prove the followingThoerem3.4. let Mn be an compact submanifold with parallel mean curvature in a complete simply connected n + p-dimensional Riemannian manifold Nn+P with sectional curvature 5\ < Kk < 82 {8182 < 0. Let the relative mean curvature H of the composition of isometric immersions M" —? Nn+P ^ Rl, satisfy H < Hq. if\\S-nH2 \\n2> (<52 - 6i)a(n,p)vol{M),then TV is isometric to i?"+p,and M is congruent to Sn(jj). Here C{n,p,8\,52,H,Hq) is a positive number depending only on n,p, 81,82, H, Hq\ a(n,p) is a positive number depending only on n,p.Corollary. Let Mn(n > 3) be an oriented closed submanifold with parallel mean curvature in Rn+P,ii \\ S — nH2 ||n < C(n), where C(n) is a positive number depending only on n, then M is a totally umbilical sphere. |
