Extremals of determinants of Laplacians. (English) Zbl 0653.53022

Osgood, B.; Phillips, R.; Sarnak, P.

J. Funct. Anal. 80, No. 1, 148-211 (1988).

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 this 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

Keywords:

Riemann surface; Laplacian; uniform metric; infinitesimal variations

Citations:

Zbl 0653.53021

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