Parallel Mean Curvature Submanifolds Gap Theorem And Rigidity Theorem

In the present thesis, we mainly study the L^n/2 curvature gap and the geometric rigidity problems of the complete submanifold with parallel mean curvature .In the first part of this paper, we mainly study the L^n/2-pinching problem for the complete submanifold with parallel mean curvature in the Euclidean space or a sphere, and get the following results:Let Mⁿ(n ≥ 3) be an n-dimensional complete submanifold with parallel mean curvature in Euclidean space R^n+p. Denote by H and S the mean curvature and the squared length of the second fundamental form of M respectively. If ∫_M(S — nH²)^n/2dM < C{n), where C{n) is an explicit positive constant depending only on n, then 5 = nH², i.e., Mⁿ is a totally umbilical submanifold. In particular, if H = 0, then M = Rⁿ;if H ≠ 0, then M — Sⁿ(1/H). It improves the gap theorems due to L. Ni and H. W. Xu.More general, we obtain the following:Let Mⁿ(n ≥ 3) be an n-dimensional complete submanifold with parallel mean curvature in F^n+p(c), where F^n+p(c) is an (n + p)-dimensional complete simply connected space form with non-negative constant curvature c. Denote by H and S the mean curvature and the squared length of the second fundamental form of M respectively. If ∫_M(S - nH²)^n/2dM < C(n), where C(n) is an explicit positive constant depending only on n, then S = nH², i.e., Mⁿ is a totally umbilical submanifold. In particular, if c + H² = 0, then M = Rⁿ;if c + H² ≠0, then M = Sⁿ( ).In the second part, we study the case that the ambient space is the standard hyperbolic space H^n+p(—1), and have the following:Let Mⁿ(n ≥ 3) be an n-dimensional complete submanifold with parallel mean curvature in H^n+p(—1). Denote by H and S the mean curvature satisfying H > 1 and the squared length of the second fundamental form of M respectively. By using a direct method, we prove that if ∫_M(S - nH²)^n/2dM < C'(n, H), where C'(n, H) is an explicit positive constant depending on n and H , then 5 ≡ nH², i.e., Mⁿ is a totally umbilical submanifold Sⁿ( ).In the third part, we study the geometric rigidity problem for complete submanifolds in pinched Riemannian manifold, and prove the following result:Let Mⁿ be an n(> 3)-dimensional complete submanifold with parallel mean curvature in a complete and simple connected (n + p)-dimensional Riemannian manifold N^n+p. Let K_n be the sectional curvature of N satisfying c := inf K_N ≤ 0, d := smpK_N ≥ 0and c + H2 > 0, then there exist constants Ti(n,p,H)(< 0), T2(n,p, H)(> 0), here T2(n,p,H) + T2(n,p,H) ^ 0 such that if K^ is pinched in [ri(n,p,H), T2(n,p, H)\ andnH2 + Ai(n,p)(d -c) + A2(n,p)[n(n - \)< ao(n, H) - B1(n,p)(d - c) - B2(n,p)[n(n -then A^n+P is isometric to the Euclidean space Rn+P. Moreover, if sup^fS < o;o(n, H), then M is congruent to Sn(l/H). Here the constant ao(n, H) = r￡z^, and the other constants Ti(n,p,H), T2(n,p,H), Ai(n,p), A2(n,p), B\(n,p), B2(n,p) will be given in the paper.