黎曼流形中常平均曲率超曲面的稳定性

STABILITY OF HYPERSURFACES OF CONSTANT MEAN CURVATURE IN RIEMANNIAN MANIFOLDS

通过对曲面的Gauss像的讨论证明了:如果X(M)的Gauss像位于一个开半球内,则x(M)是稳定的.接着又讨论了n=2时的情况,通过对共形变换Gauss曲率的估计得出:a>0时,设(?)M是一个单连通区域,如果(?)满足条件:C,则(?)是稳定的.a<0时,如果(?)满足:H^2≥-a时,integral from n=(?)(1/2(s+2a)dM<2π);H^2<-a时,integral from n=(?)((2/3)a+(2/3)H^2-K)dM<2π,则■是稳定的.

Let Nn+1(a) be a (n+1)-dimensional space form with constant sectional curvature a and x2 M→Nn+1(a) a immersion hypersurface with constant mean curvature in Nn+1(a). If the image of Gauss map lies in an open semisphere, then x(M) is stable.when n=1, let DCM be a simply-connected domain with compact closure in M. Denote the square of the length of the second fundamental form by s, the Gauss curvature of x(M)by k.If a>0 and 1/2(s+2a)d M<2π or if a<0, H<-a and (2/3a+2/3H2-K) d M<2π,then D is stable.