拟常曲率Riemann流形的极小超曲面

本文主要考察QC流形的浸入极小超曲面M.建立了类似于〔2〕,〔3〕的“4次式”和“6次式”的积分不等式,并利用这些积分式,作出了关于M的第二基本形式长度平方S的值域估计.

Minimal Hypersurfaces of a Quasi-Constant Curvature Riemannian Manifold

Suppose that M is an n-dimensional compact oriented manifold which is minimally immersed in a quasi-constant curvature Riemannian manifold QC of dimension n+1, the generator of whose is . Let h be the second funda-mental form of this immersion, and we denote by S the square of the length of h. Let K be the scalar curvature of QC, and T be the Ricci principal curvature corresponding to . In the present paper we establish the following main results. Therem If is a vector field in the tangent bundie T(M), then (i) the following integral inequalties hold: (iii) If S=const. and hold everywhere on M, then M is totally geodesic, or (iv) H M has nonnegative sectional curvature> and if hold everywhere on M, then we have same conclusion as in (iii).