论文标题
第一个卷曲特征值的等级问题
Isoperimetric problem for the first curl eigenvalue
论文作者
论文摘要
我们考虑了一个等速度问题，涉及抽象环境流形上最小和最大的负卷曲特征值，重点是标准模型空间。我们特别表明，在欧几里得和双曲线设置中，假设存在最佳结构域，最佳域上的相应特征值必须很简单。这概括了Enciso和Peralta-Salas的最新结果，他们在欧几里得空间中表现出具有连接边界的轴对称最佳结构域的简单性。然后，我们概括了Enciso和Peralta-Salas的另一个最新结果，即，在欧几里得空间中，任何旋转对称的最佳域的点具有连接的边界，它们最接近对称轴必须与边界相连，与双曲线设置，并在欧近体案例中加强连接的边界。最后，我们展示了如何使用与同一问题相关的第二个变异不平等，以将杀死贝特拉米字段的存在与环境空间的几何形状联系起来。
We consider an isoperimetric problem involving the smallest positive and largest negative curl eigenvalues on abstract ambient manifolds, with a focus on the standard model spaces. We in particular show that the corresponding eigenvalues on optimal domains, assuming optimal domains exist, must be simple in the Euclidean and hyperbolic setting. This generalises a recent result by Enciso and Peralta-Salas who showed the simplicity for axisymmetric optimal domains with connected boundary in Euclidean space. We then generalise another recent result by Enciso and Peralta-Salas, namely that the points of any rotationally symmetric optimal domain with connected boundary in Euclidean space which are closest to the symmetry axis must disconnect the boundary, to the hyperbolic setting, as well as strengthen it in the Euclidean case by getting rid of the connected boundary assumption. Lastly, we show how a second variation inequality related to the isoperimetric problem may be used in order to relate the existence of Killing-Beltrami fields to the geometry of the ambient space.
