Moduli space, heights and isospectral sets of plane domains*
Moduli space, heights and isospectral sets of plane domains*
复制标题
平面域的模空间、高度和等谱集*
DOI：
10.2307/1971449
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发表时间：
1989
影响因子：
4.9
通讯作者：
P. Sarnak
中科院分区：
文献类型：
--
作者：
B. Osgood;R. Phillips;P. Sarnak
设Q是具有光滑边界的有限连通度n的平面域，并选择同一类型的固定域2。则在2上存在一个平坦度量g，使得q与29g等距。在下面的内容中，我们不区分等角域。29的谱是指Laplace-Beltrami算子Ag在29上的Dirichlet边界条件下的谱。高度h(Eg)=-log Det Ag是一个光谱不变量，在本文中起着中心作用。在2上与给定度量g共形的所有适当正规化的平坦度量中，有一个唯一的平坦度量的高度是最小的。这个度量的特征是d2具有常测地曲率；我们称这样的度量为一致的，用u表示。所有这样的度量的集合用u(2)表示。因此，我们可以(用模空间#(2))来确定E上的共形结构。对于n？3，我们对Mu(2)引入了一种特殊的参数化法，利用它，我们证明了当u逼近Fu(2)的边界时，h(U)-00。利用它和拉普拉斯的热不变量，我们证明了平面区域的任何等谱集在C‘拓扑中都是紧的。当n=1和2时，类似的结果也成立。
Let Q be a plane domain of finite connectivity n with smooth boundary and choose a fixed domain 2 of the same type. Then there exists a flat metric g on 2 such that Q is isometric with 29g. In what follows we do not distinguish between isometric domains. By the spectrum of 29 we mean the spectrum of the Laplace-Beltrami operator Ag on 29 with Dirichlet boundary conditions. The height h(Eg) = - log det Ag is a spectral invariant and plays a central role in this paper. Among all suitably normalized flat metrics on 2 conformal to a given metric g there is a unique flat metric for which the height is a minimum. This metric is characterized by the fact that d 2 has constant geodesic curvature; we call such a metric uniform and denote it by u. The set of all such metrics is denoted by u(2). We can therefore identify ( with the moduli space #(2) of conformal structures on E. For n ? 3 we introduce a special parametrization for Mu(2) by means of which we show that h(u) - oo as u approaches the boundary of fu(2). Using this along with the heat invariants for the Laplacian we then show that any isospectral set of plane domains is compact in the C' topology. Similar results hold for n = 1 and 2.