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 \(\{\lambda_ n\}\) are the non-zero eigenvalues of \(\Delta\), then the zeta function \(\zeta (s,\Delta)=\sum_ n\lambda_ n^{-s}\) 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\(\neq \emptyset\) 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


53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds


Zbl 0653.53021
Full Text: DOI


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