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 Ω1 and Ω2,simultaneously(?)Ω1=M =(?)Ω2.In general,this paper discusses the first eigenvalue estimation of Δ which is Laplace operator in area Ω1 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 Ω1 and Ω2,and(?)Ω1= M=(?)Ω2,let λ1 is the first eigenvalue of Laplace operator in Ω1,then Robin first eigenvalue λ1≥(n + 1)K.theorem 1.8 using Ricci identity in sphere proved that,the first Dirichlet eigen-value λ1≥ 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.