# Fundamental Gap Estimate for Hyperbolic Space

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Guofang Wei, University of California, Santa Barbara

Online Talk

The fundamental (or mass) gap refers to the difference between the first two eigenvalues of the Laplacian or more generally for Schr\"{o}dinger operators. It is a very interesting quantity both in mathematics and physics as the eigenvalues are possible allowed energy values in quantum physics.  In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture that difference of first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter D in the Euclidean space is greater than or equal to $3\pi^2/D^2$. In several joint works with X. Dai, Z. He, S. Seto, L. Wang (in various subsets) the estimate is generalized, showing the same lower bound holds for convex domains in the unit sphere. In sharp contrast, in recent joint work with T. Bourni, J. Clutterbuck, X. Nguyen, A. Stancu and V. Wheeler (a group of women mathematicians), we prove that there is no lower bound at all for the fundamental gap of convex domains in hyperbolic space in terms of the diameter. Very recently, jointed with X. Nguyen,  A. Stancu,  we show that even  for horoconvex (which is much stronger than convex) domains in the hyperbolic space, the product of their fundamental gap with the square of their diameter has no positive lower bound.