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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 $\Delta$ be the Laplacian with Dirichlet boundary condition. If $\{\lambda\sb n\}$ are the non-zero eigenvalues of $\Delta$, then the zeta function $\zeta (s,\Delta)=\sum\sb n\lambda\sb n\sp{-s}$ is holomorphic at $s=0$ and - $\zeta$ ’(0,$\Delta)$ is the functional determinant. This is a non-local spectral invariant. A metric g on M is said to be uniform if (i) $dM=\emptyset$ and the metric g has constant curvature or (ii) dM$\ne \emptyset$ 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

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