论文标题
扁平托里（Flat Tori
Flat tori with large Laplacian eigenvalues in dimensions up to eight
论文作者
论文摘要
我们考虑了最大化$ k $ -th laplacian特征值的优化问题，$λ_{k} $，固定音量的flat $ d $ d $ d $二维摩托车。对于$ k = 1 $，此问题等同于最密集的晶格球形包装问题。对于较大的$ k $，这相当于找到NP硬的问题，即找到最长$ k $的$ d $数（双）晶格 - 最短的晶格矢量。由于广泛的计算，对于$ d \ leq 8 $，我们获得了一系列flat tori，$ t_ {k，d} $，每个卷。对于每个（有限）$ k $，$ k $ -th特征值超过了（$ k \ to \ ismptotic $渐近）Weyl's Law的值，其因素在1.54到2.01之间，具体取决于维度。平稳性条件被得出并以$ t_ {k，d} $进行数值验证，我们将托里的变性描述为$ k \ to \ infty $。
We consider the optimization problem of maximizing the $k$-th Laplacian eigenvalue, $λ_{k}$, over flat $d$-dimensional tori of fixed volume. For $k=1$, this problem is equivalent to the densest lattice sphere packing problem. For larger $k$, this is equivalent to the NP-hard problem of finding the $d$-dimensional (dual) lattice with longest $k$-th shortest lattice vector. As a result of extensive computations, for $d \leq 8$, we obtain a sequence of flat tori, $T_{k,d}$, each of volume one, such that the $k$-th Laplacian eigenvalue of $T_{k,d}$ is very large; for each (finite) $k$ the $k$-th eigenvalue exceeds the value in (the $k\to \infty$ asymptotic) Weyl's law by a factor between 1.54 and 2.01, depending on the dimension. Stationarity conditions are derived and numerically verified for $T_{k,d}$ and we describe the degeneration of the tori as $k \to \infty$.
