论文标题
Milnor的同一圆锥形和谐波图
Milnor's isospectral tori and harmonic maps
论文作者
论文摘要
一个众所周知的问题询问Riemannian歧管$（M，G）$上的Laplacian的频谱是否确定了Riemannian Metric $ g $ to Isometry。一个类似的问题是，从给定的Riemannian歧管$（σ，h）$到$ m $的所有谐波图的能源是否确定目标空间上的Riemannian指标。在扁平托里之间的谐波图的情况下，我们考虑了这个问题。特别是，我们表明，米尔诺（Milnor）发现的两个同一，非等法$ 16 $二维的扁平摩托车无法通过谐波地图的能量谱从$ d $ d $ d $二维的flat tori（$ d \ leq 3 $）区分，但可以以某些Flat tori为$ d \ egeq 4 $区分。这与Siegel Theta系列的属性有关，以$ D $ $ D $相关的$ 16 $维晶格有关。
A well-known question asks whether the spectrum of the Laplacian on a Riemannian manifold $(M,g)$ determines the Riemannian metric $g$ up to isometry. A similar question is whether the energy spectrum of all harmonic maps from a given Riemannian manifold $(Σ,h)$ to $M$ determines the Riemannian metric on the target space. We consider this question in the case of harmonic maps between flat tori. In particular, we show that the two isospectral, non-isometric $16$-dimensional flat tori found by Milnor cannot be distinguished by the energy spectrum of harmonic maps from $d$-dimensional flat tori for $d\leq 3$, but can be distinguished by certain flat tori for $d\geq 4$. This is related to a property of the Siegel theta series in degree $d$ associated to the $16$-dimensional lattices in Milnor's example.
