A Global Rigidity Theorem For Submanifolds With Parallel Mean Curvature In Space Forms

Yau and Itoh proved rigidity theorems for compact minimal submanifold in the sphere under sectional curvature pinching condition.J.R.Gu and H.W.Xu improved the pinching constants in Yau and Itoh’s theorems and generalized the result to submanif’olds with parallel mean curvature in space f’orms.In this paper,we investigate the geometric rigidity of submanif’olds with parallel mean curvature in the space form under integral pinching condition of sectional curvature.Theoreml.Let Mn(n≥3)be a compact and oriented minimal submanifold in the unit sphere Sn+p.For positive constants q,λ satisfying q≥1.2,α(n,p)<λ≤1,there exists a positive cofnstant C(n,p,λ,q)depending only on n,p,λ,q,such that if K(λ,q)<C(n,p,λ,q),then M is totally geodesic.Here α(n,p)is a positive constant depending only on n,p.K(λ,q)=∫M(max(λ-Kmin,0})qdM and Kmin(x)is the minimum of the sectional curvature of M at x.Theorem2.Let Mn(n≥3)be a compact and oriented submanifold with paral-lel mean curvature(the mean curvature H≠0)in the space form Fn+p(c)with constant curvature c.For positive constants q,λ satisfying q≥n/2,β(n,p,c,H)<λ≤c+H2,there exists a positive constant C(n,p,λ,q,c,H)depending only on n,p,λ,q,c and H,such that if c+H2>0 and K(λ,q)<C(n,p,λ,q,c,H),then M is totally umbilical.Here β(n,p,c,H)is a positive constant depending only on n,p,c,H.