The First Eigenvalue Estimation Of Convex Domain In Riemann Manifold With Positive Ricci Curvature

For Riemann manifold N with positive Ricci curvature,there exist a hypersurface M which can divide N into two connected areas ?1 and ?2,simultaneously(?)?1=M =(?)?2.In general,this paper discusses the first eigenvalue estimation of ? which is Laplace operator in area ?1 when M is a cnvex hypersurface.There is the main result of this paper:This paper gives a new proof of lemma 1.5,the proving method is same as the method in[13],and Robin eigenvalue also can be proved,that is,N is n + 1-dimensional Riemann manifold,Ric(N)is Ricci curvature of N,and Ric(N)? nK.M is closed(compact no-boundary)orientable connected smooth embedded hypersurface in N,M divides N into two connected areas ?1 and ?2,and(?)?1= M=(?)?2,let ?1 is the first eigenvalue of Laplace operator in ?1,then Robin first eigenvalue ?1?(n + 1)K.theorem 1.8 using Ricci identity in sphere proved that,the first Dirichlet eigen-value ?1? 2(n + 1)in n + 1-dimensional sphere Sn+1(1).This paper improves the proof method,first calculateing in Riemann manifold,then replacing Ricci curvature with spherical curvature,finally we can get the same consequence.