极小曲率子流形的积分不等式

Integral inequality of minimal curvature submanifolds

设φ:M^n→N■^(n+p)R^(n+p+1)是极小曲率闭子流形,N^(n+p)是欧氏空间R^(n+p+1)的超曲面,如果主曲率|λ|≥c(c>0),则有∫_M[np(c^2-2K)-S]Sd V≥0,其中K(x)为M中每一点处所有截面曲率的下确界.特别地,当对任意点x∈M^n,均有K≤0时,则∫_M[np(c^2-K)-S]Sd V≥0.此结论推广了Yau^([7])中常曲率空间极小子流形的情形.

Let M n is closed minimal curvature submanifolds of N^n+p , where N^n+p is hypersurface of R^n+p+1 . If the main curvature |λ|≥c(c>0), then we have ∫ M n [np(c^2-2K)-S]S d V≥0 where K is the function assigns to each point of M the infinimum of the sectional curratures of M at the point. in particular, if K(x)≤0 for any point x∈M n ,we have ∫ M n [np(c^2-K)-S]S d V≥0. Our conclusion promotes Yau [7]'s.