标题： 三维公司在六维中：不可约的、共谱的、非等距的平坦环面 显示英文标题

标题： Three's company in six dimensions: irreducible, isospectral, non-isometric flat tori

摘要： 1964年，约翰·米尔诺利用维特构造的两个格子，首次给出了两个不全局等距的平坦环面的例子，它们的外形式拉普拉斯算子具有相同的特征值序列。 上述平坦环面是十六维的。 一个是可约的，另一个是不可约的。 在随后的几年里，人们已经证明在四维及更高维度中存在非等距但具有共同拉普拉斯谱的平坦环面对。 在三维及更低维度中，亚历山大·希曼在1994年证明了任何等谱的平坦环面实际上都是等距的，因此四维是这种环面对存在的最低维度。 使用这样一个四维的环面对，可以很容易地构造出一个八维的环三元组。 然而，在四维、五维、六维和七维中，相互非等距但具有共同拉普拉斯谱的环三元组一直未被研究人员发现——直到现在。 我们在此呈现第一个这样的例子。 显示英文摘要

摘要： In 1964, John Milnor, using a construction of two lattices by Witt, produced the first example of two flat tori that are not globally isometric and whose Laplacians for exterior forms have the same sequence of eigenvalues. The aforementioned flat tori are sixteen-dimensional. One is reducible while the second is irreducible. In the ensuing years, pairs of non-isometric flat tori that share a common Laplace spectrum have been shown to exist in dimensions four and higher. In dimensions three and lower, Alexander Schiemann proved in 1994 that any flat tori that are isospectral are in fact isometric, so four is the lowest dimension in which such pairs exist. Using a four-dimensional such pair, one can easily construct an eight-dimensional such triplet. However, triplets of mutually non-isometric flat tori that share a common Laplace spectrum in dimensions 4, 5, 6, and 7 have eluded researchers - until now. We present here the first example.