We investigate topological and differentiable structures of submanifolds under extrinsic restrictions. We first obtain a topological sphere theorem for compact submanifolds in a Riemannian manifold. Secondly, we prove an optimal differentiable sphere theorem for 4-dimensional complete submanifolds in a space form, which provides a partial solution of the smooth Poincaré conjecture. Finally, we prove some new differentiable sphere theorems for n-dimensional submanifolds in a Riemannian manifold.

In this note, we first prove that the solution of mean curvature flow on a finite time interval [0, T ) can be extended over time T if the space-time integration of the norm of the second fundamental form is finite. Secondly, we prove that the solution of certain mean curvature flow on a finite time interval [0, T ) can be extended over time T if the space-time integration of the mean curvature is finite. Moreover, weshow that these conditions are optimal in some sense.

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a pinching condition in a hyperbolic space form to a round point in finite time.2000 Mathematics Subject Classification. 53C44, 53C40.

Dedicated to Professor Katsuhiro Shiohama on the occasion of his 70th birthday.Let M be an n-dimensional complete locally conformally flat Riemannian manifold with constant scalar curvature R and n ≥ 3. We first prove that if R = 0 and the L n/2 norm of the Ricci curvature tensor of M is pinched in [0, C 1 (n)), then M is isometric to a complete flat Riemannian manifold, which improves Pigola, Rigoli, and Setti's pinching theorem. Next, we prove that if n ≥ 6, R = 0, and the L n/2 norm of the trace-free Ricci curvature tensor of M is pinched in [0, C 2 (n)), then M is isometric to a space form. Finally, we prove an L n trace-free Ricci curvature pinching theorem for complete locally conformally flat Riemannian manifolds with constant nonzero scalar curvature. Here C 1 (n) and C 2 (n) are explicit positive constants depending only on n.

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