Small eigenvalues on degenerating hyperbolic Riemann surfaces

Mathematics Research Report MRR95-016

Small eigenvalues on degenerating hyperbolic Riemann surfaces

Joseph F. Grotowski, Jonathan Huntley and Jay Jorgenson

Abstract: Consider a semi-stable family of compact, connected algebraic curves M_t which degenerates to a stable, noded curve M₀. The uniformization theorem allows us to endow each curve in the family, as well as the limit curve, with its natural complete hyperbolic metric, so that we are considering a degenerating family of compact hyperbolic Riemann surfaces. Assume that M₀ has k components and n nodes, so there are n families of geodesics whose lengths approach zero through degeneration and k-1 families of eigenvalues of the Laplacian which approach zero through degeneration. A problem which has received considerable attention is to compare the rate at which the eigenvalues and the geodesics approach zero. In this paper, we will use results from complex algebraic geometry and from heat kernel analysis to obtain a precise relation involving the small eigenvalues, the small geodsics, and the period matrix of the underlying complex curve M_t. Our method leads naturally to a conjecture in the general setting of an arbitrary degenerating family of hyperbolic Riemann surfaces of finite volume.

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