## 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

### MSC:

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

Zbl 0653.53021
Full Text:

### References:

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