Variation of Laplace spectra of compact "nearly" hyperbolic surfaces

We use the real analyticity of the Ricci flow with respect to time proved by B. Kotschwar to extend a result of P. Buser, namely, we prove that the Laplace spectra of negatively curved compact orientable surfaces having the same genus gamma >= 2, the same area and the same curvature bounds vary in a " controlled way", of which we give a quantitative estimate in our main theorem. The basic technical tool is a variational formula that provides the derivative of an eigenvalue branch under the normalized Ricci flow. In a related manner, we also observe how the above- mentioned real analyticity result can lead to unexpected conclusions concerning the spectral properties of generic metrics on a compact surface of genus gamma >= 2. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.