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Extremals of determinants of Laplacians. (English) Zbl 0653.53022
Let M be a compact Riemann surface with smooth boundary dM. Let Δ be the Laplacian with Dirichlet boundary condition. If {λ n } are the non-zero eigenvalues of Δ, then the zeta function ζ(s,Δ)= n λ n -s is holomorphic at s=0 and - ζ ’(0,Δ) is the functional determinant. This is a non-local spectral invariant. A metric g on M is said to be uniform if (i) dM= and the metric g has constant curvature or (ii) dM and g is flat. The authors show that the uniform metric minimizes ths inducing normal variation. Then structures are preserved by invariant, isometric infinitesimal variations.
Reviewer: Y.Muto

53C20Global Riemannian geometry, including pinching
58J50Spectral problems; spectral geometry; scattering theory