GENERIC PROPERTIES OF THE EIGENVALUE OF THE LAPLACIAN FOR COMPACT RIEMANNIAN MANIFOLDS

Introduction. In this paper, we discuss generic properties of the eigenvalues of the Laplacian for compact Riemannian manifolds without boundary. Throughout this paper, let M be an arbitrary fixed connected compact C°° manifold of dimension n without boundary, and ^t the set of all C°° Riemannian metrics on M. For g e^t, let Δg be the Laplacian (cf. (2.1)) of (M, g) acting on the space C°°(M) of all C°° real valued functions on M and 0 = λo(flf) < λΛflO ^ x2(g) ^ T°° the eigenvalues of the Laplacian Δg counted with their multiplicities. We regard each eigenvalue Xk(g), k = 0,1, 2, , as a function of g in ^. Let us consider the following problem: "Does each eigenvalue Xk(g) depend continuously on g in ^t with respect to the C°° topology*!" The continuous dependence of the eigenvalues of the Dirichlet problem upon variations of domains is well known (cf. [CH, p. 290]). Variations of coefficients of elliptic differential operators were dealt with by KodairaSpencer [KS] who gave a proof of the continuity of eigenvalues. In this paper, we give a simple proof of the above problem. To answer the above problem, in § 1, we introduce a complete distance p on ^t which gives the C°° topology. Then, in § 2, we assert that each χk(g)f k = 1, 2, , depends continuously on g e ^t with respect to the topology on Λ? induced by the distance p. More precisely, we have