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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. $\int R^2 \sqrt{g} d^2x$)

Thanks.