Spectral Problems on Riemannian Manifolds
Spectral problems on Riemannian manifolds Spectral problems on Riemannian manifolds Pierre Bérard Université Joseph Fourier - Grenoble geometrias géométries IMPA, April 13-15,2009 1/53 Spectral problems on Riemannian manifolds Introduction to the spectrum Introduction to the spectrum Let (M, g) be a compact Riemannian manifold (possibly with boundary). We consider the Laplacian on M, acting on functions, ∆g (f ) = δg (df ), where δg is the divergence operator on 1-forms. The divergence of a 1-form ω is given by n n δ (ω) = X(Dg ω)(E ) = X E · ω(E ) − ω(Dg E ), g Ej j j j Ej j j=1 j=1 n where {Ej }j=1 a local orthonormal frame. 2/53 Spectral problems on Riemannian manifolds Introduction to the spectrum n In a local coordinate system {xj }j=1, the Laplacian is given by 1 n ∂ ∂f ∆f = − X (v (x)g ij (x) ), v (x) ∂x g ∂x g i,j=1 i j ij −1 where g (x) is the inverse matrix gij (x) , the ∂ ∂ gij (x) = g( , ) are the coefficients of the Riemannian metric ∂xi ∂xj 1/2 in the local coordinates, and vg (x) = Det(gij (x)) . In local coordinates, the Riemannian measure dvg on (M, g) is given by dvg = vg (x) dx1 ... dxn. 3/53 Spectral problems on Riemannian manifolds Introduction to the spectrum We are interested in the eigenvalue problem for the Laplacian on (M, g), i.e. in finding the pairs (λ, u), where λ is a (real) number and u a non-zero function, such that ∆u = λu and, when M has a boundary ∂M, u|∂M = 0 (Dirichlet eigenvalue problem).
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