By E. Brian Davies, Yuri Safarov

This quantity brings jointly lectures from a convention on spectral conception and geometry held less than the auspices of the foreign Centre for Mathematical Sciences in Edinburgh. The contributions via global specialists contain extended models of a number of the unique lectures. jointly, they survey the center fabric and transcend to arrive deeper effects. For graduate scholars and specialists alike, this e-book may be a hugely resource.

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Dirichlet eigenvalue problem. For M connected with compact closure and smooth boundary, find all real numbers A for which there exists a nontrivial solution (j) G C2{M) n C°(M) to A(j> + X(j) = 0, (f) | <9M = 0. The Laplacian on Riemannian manifolds 35 Neumann eigenvalue problem. 2. Basic facts for elementary eigenvalue problems The desired numbers A in each of the eigenvalue problems are referred to as eigenvalues of A, and the vector space of solutions of the eigenvalue problem with given A — it is a linear problem in all the above instances — its eigenspace E\.

R dr / 00 sinh n — 1 Jo n-1 o 2 r A00 2 «_! i 1 / 2 r r°°, ,x2 «-i i 1/2 < < / 0 sinh rdr> • < / ( 0 ) sinh rdr> , n - l u o J [Jo ) 52 I. Chavel which implies (n - I) 2 /^^sinh- 1 rdr. 4 f°° (f)A\2 • Jo Now integrate over Sn~l. Then = L '«.. r d |grad|2sinhn ~l r dr (d^sinh"- 1 ^ (n - If 4 7 d /i _i / n which implies the theorem, by Rayleigh's principle. Additional 02sir\hn~lr dr ^0 qed references Berger-Gauduchon-Mazet [6], Berard [5], Courant-Hilbert [11], Stein-Weiss [19]. §3. Spectra of noncompact Riemannian manifolds Let M denote an arbitrary Riemannian manifold, and A some self-adjoint extension of A|C£°(Af).

We therefore have: (V<)2y = sinh"tU(t) = sinhtU{t) Basic Riemannian geometry 21 whence Take an inner product with U to get Ktf MJ) = -\ and so conclude that {Dn,g) has constant curvature — 1. The same argument (that is, differentiate the image under expm of a family of straight lines through the origin) computes Jacobi fields in normal coordinates: Theorem. 6 Conjugate points and the Cartan-Hadamard theorem Let £ G Mp and let 7 = 7^ : 1$ —> R. We say that q = j{t\) is conjugate to p along 7 if there is a non-zero Jacobi field Y with In view of the theorem just stated, this happens exactly when (dexpp)tl^ is singular.