The first eigenvalue of the Laplacian and the ground flow of a compact surface

Abstract

We present some results whose central theme is that the phenomenon of the first eigenvalue of the Laplacian and the ground flow of the compact surface (bitorus). Our main tool is a method for studying how the hyperbolic metric on a Riemann surface behaves under deformation of the surface. With this model, we show that there are variation of the first eigenvalue of the Laplacian and the ground flow with the Fenchel-Nielsen coordinates, that characterize the surface.