I am interested in the first non zero eigenvalue of the Laplace-Beltrami operator in a 2D compact manifold, and if there is a geometric characterization of its value.

I am interested in the case when you fix the volume of the manifold to some value (say $Vol = 1$), and let the other modes of the metric fluctuate. The average curvature of the manifold is imposed by the Gauss-Bonet theorem, but can let the curvature to fluctuate from one point to another.

My intuition says that the first non zero eigenvalue should approach 0 in the limit when the "fluctuations of the curvature" grow, but I can not give a precise meaning to this statement.

so the question is: Is there any characterization of the first eigenvalue(s) of the Laplace-Beltrami operator in a 2D compact riemann manifold as functions of the curvature or its powers (i.g. $\int R^2 \sqrt{g} d^2x$).

So let me be more specific. Imagine a manifold topologically equivalent to a Torus. The metric can be written as

$ds^2=f(x_1,x_2)(dx^2_1+dx^2_2)$

and the scalar curvature

$R=−2\Delta \log(f)$

For the case $f=cte$ we have a flat torus. Now expanding f as a fourier series we will have some regions of the torus with poritive curvature and some regions with negative curvature (this invalid some known theorems, that need wither the curvature to be always positive or always negative). Gauss bonnet says that:

$\int R=0$

So the concrete question is: Is there a characterisation of the first eigenvalue of the Laplace operator in terms of $\int R^2$?