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Title:
New upper bounds on the spectral gap of hyperbolic manifolds
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Abstract:
I will describe a method for constraining Laplacian spectra of hyperbolic surfaces and d-manifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of representation theory of SO(1,d) and linear programming, the method yields rigorous upper bounds on the spectral gap. In several examples, the bound is nearly sharp. The bounds also allow us to determine the set of spectral gaps attained by all hyperbolic 2-orbifolds. The ideas were inspired by recent developments in the conformal bootstrap. The linear program is similar to the Cohn+Elkies linear program for bounding sphere packing density. Based on https://arxiv.org/abs/2111.12716 with P. Kravchuk and S. Pal and work in progress with the same collaborators and J. Bonifacio.
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