标题： 双曲自旋曲面的谱间隙界限 显示英文标题

标题： Bounds on spectral gaps of Hyperbolic spin surfaces

摘要： 我们描述了一种约束二维紧致可定向双曲自旋流形和轨道空间的拉普拉斯谱和狄拉克谱的方法。 关键要素是谱满足的无限族恒等式。 这些谱恒等式来源于1）自旋丛上函数的谱分解为$\mathrm{SL}(2,\mathbb{R})$的不可约表示，以及2）函数逐点乘法的结合性之间的一致性。 将半定规划方法应用于我们的恒等式，可以得到拉普拉斯谱间隙的严格上界，以及在前者条件下的狄拉克谱间隙的上界。 在几个例子中，我们的界几乎是最优的；基于塞尔伯格迹公式的一个数值算法表明，$[0;3,3,5]$轨道空间、一个具有符号$[1;3]$的特定曲面以及博尔扎曲面分别在亏格$0$、$1$和$2$时几乎达到这些界。 在关于自旋曲面上调和自旋子数量的额外假设下，我们得到了对拉普拉斯谱间隙更严格的限制。 特别是，这些限制适用于双椭圆曲面。 我们还确定了所有紧致可定向二维双曲自旋轨道空间所实现的拉普拉斯谱间隙的集合。 我们证明这个集合被$12.13798$上界限定；这个界限几乎被$[0;3,3,5]$陪集所饱和，其第一个非零拉普拉斯特征值是$\lambda^{(0)}_1\approx 12.13623$。 显示英文摘要

摘要： We describe a method for constraining Laplacian and Dirac spectra of two dimensional compact orientable hyperbolic spin manifolds and orbifolds. The key ingredient is an infinite family of identities satisfied by the spectra. These spectral identities follow from the consistency between 1) the spectral decomposition of functions on the spin bundle into irreducible representations of $\mathrm{SL}(2,\mathbb{R})$ and 2) associativity of pointwise multiplication of functions. Applying semidefinite programming methods to our identities produces rigorous upper bounds on the Laplacian spectral gap as well as on the Dirac spectral gap conditioned on the former. In several examples, our bounds are nearly sharp; a numerical algorithm based on the Selberg trace formula shows that the $[0;3,3,5]$ orbifold, a particular surface with signature $[1;3]$, and the Bolza surface nearly saturate the bounds at genus $0$, $1$ and $2$ respectively. Under additional assumptions on the number of harmonic spinors carried by the spin-surface, we obtain more restrictive bounds on the Laplacian spectral gap. In particular, these bounds apply to hyperelliptic surfaces. We also determine the set of Laplacian spectral gaps attained by all compact orientable two-dimensional hyperbolic spin orbifolds. We show that this set is upper bounded by $12.13798$; this bound is nearly saturated by the $[0;3,3,5]$ orbifold, whose first non-zero Laplacian eigenvalue is $\lambda^{(0)}_1\approx 12.13623$.