Sphere Willmore Hypersurface Overall Pinching Theorem

Let M be an n-dimensional(n≥3)compact minimal submanifold in n+p-dimensional unit sphere S^n+p.H denotes the mean curvature of M,S denotesthe square of the length of the second fundamental form.Then we have:If S≤n/2-1/p, Then either S≡0,and M is totally geodesic,or S≡n/2-1/p. Inthe latter case.1.either, p=1and M is a Clifford torus C_m,n-m in Sⁿ⁺¹.2.or n=2, p=2 and M is the Veronese surface in S⁴(1).After that,[2] [4] [6]improved the pinching condition. Recently, H.Z.Li, ob-tained the following results when M is a Willmore submanifolds (see [3] [5] [9])Theorem: Let M be an n-dimensional(n≥2) compact Willmore subman-ifold in n+p-dimensional unit sphere S^n+p. then we have integral from Mρⁿ(n/2-1/P-ρ²)dv≤0In particular,if 0≤ρ²≤n/2-1/pthen eitherρ²=0 and M is totally umbilical, orρ²≡n/2-1 p. In the lattercase,either p=1 and M is a Willmore torus W_m,n-m defined by the followingequation: W_m,n-m=S^m n-m/n^1/2×S^m-m m/n^1/2,1≤m≤n-1. or n=2,p=2 and M is the Veronese surface.Obviously, all the above results have pointwise condition for S or S-nH² Itseems to be interesting to study the L_q-pinching theorem. It is first initiated byC.Li.Shen[8], latter H. Wang[16], J. M. Lin C. Y. xia[7], H. W. Xu[9] improvedthe theorems.In this thesis, when M is Willmore compact hypersurface we proved.Theorem. Let M(n≥3)be n-dimensional compact Willmore hypersur-face in unit sphere Sⁿ⁺¹. H Denotes the mean curvature of M and S denotes the square of the length of the second fundamental form.if‖ρ²‖n/2＜n(n-2)³/C²(n)[n³(n-1)²+(n-2)³(1+H₀²)]Where H₀=(?) H.Thenρ²=0, ie M is totally umbilical hypersurface inSⁿ⁺¹.