论文标题
沿着指标路径的霍奇laplacian和卷曲操作员的光谱简单性
On spectral simplicity of the Hodge Laplacian and Curl Operator along paths of metrics
论文作者
论文摘要
我们证明，在封闭式$ 3 $ manifolds上的卷发操作员，即霍德·拉普拉斯（Hodge laplacian）的平方根，在其伴侣光谱上，通常具有$ 1 $尺寸的特征宽面积，甚至沿着$ 1 $ $ $ - 参数的$ - 参数为$ \ mathcal {c}^c}^c}^c}^k $ riemann $ riemann $ ke n $ k n $ k n $ k k ege $ k k n $ k k n $ k k n $ k。我们进一步表明，尺寸$ 3 $的Hodge laplacian在仿制药$ 1 $ 1 $ - 参数的Riemannian Metrics家族中有两个可能的来源：要么来自正面的特征值，要么来自curl操作员交叉的curl eigenvalues，或者来自curl ocernvalues of Curl Operator Cross，或者是一个精确的和Ececexact eigenvalue cross。我们提供了这两种现象的例子。为了证明我们的结果，我们概括了一种teytel \ cite {teytel1999}的方法，使我们能够计算curl oterator和hodge laplacian具有一定的特征性多种多样性的Riemannian指标集的微薄编织。结果的一个结果是，虽然hodge laplacian频谱的简单性$ 3 $是微薄的codimension $ 1 $属性，相对于$ \ mathcal {c}^k $ topology，由enciso and peralta-salas证明，\ cite in \ cite {enciso2012;
We prove that the curl operator on closed oriented $3$-manifolds, i.e., the square root of the Hodge Laplacian on its coexact spectrum, generically has $1$-dimensional eigenspaces, even along $1$-parameter families of $\mathcal{C}^k$ Riemannian metrics, where $k\geq 2$. We show further that the Hodge Laplacian in dimension $3$ has two possible sources for nonsimple eigenspaces along generic $1$-parameter families of Riemannian metrics: either eigenvalues coming from positive and from negative eigenvalues of the curl operator cross, or an exact and a coexact eigenvalue cross. We provide examples for both of these phenomena. In order to prove our results, we generalize a method of Teytel \cite{Teytel1999}, allowing us to compute the meagre codimension of the set of Riemannian metrics for which the curl operator and the Hodge Laplacian have certain eigenvalue multiplicities. A consequence of our results is that while the simplicity of the spectrum of the Hodge Laplacian in dimension $3$ is a meagre codimension $1$ property with respect to the $\mathcal{C}^k$ topology as proven by Enciso and Peralta-Salas in \cite{Enciso2012}, it is not a meagre codimension $2$ property.
