标题： 短测地线和双曲曲面上特征值的重数 显示英文标题

标题： Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces

摘要： 在本文中，我们根据短闭测地线的数量和亏格$g$来获得闭双曲曲面上拉普拉斯特征值重数的上界。 例如，我们证明如果短闭测地线的数量在$g$中是次线性的，那么第一个特征值的重数也是在$g$中次线性的。 这在1980年代中期Colin de Verdière提出的猜想上取得了新的进展。 显示英文摘要

摘要： In this paper, we obtain upper bounds on the multiplicity of Laplacian eigenvalues for closed hyperbolic surfaces in terms of the number of short closed geodesics and the genus $g$. For example, we show that if the number of short closed geodesics is sublinear in $g$, then the multiplicity of the first eigenvalue is also sublinear in $g$. This makes new progress on a conjecture by Colin de Verdi\`ere in the mid 1980s.