论文标题
卷曲旋转在三个manifold上
Curl through spin on three-manifold
论文作者
论文摘要
在过去的几十年中，许多数学家在紧凑型（或没有空边界）上研究了{\ em curl oterator}三个manifolds，主要是其频谱的行为和某些ISO \ -pe \ -ri \ -ri \ -Me \ -Me \ -Me \ -me \ - 有问题的问题。在本文中，我们揭示了该卷曲操作员与狄拉克运算符之间的（意外？）关系，与歧管上的任何旋转$^c $结构相对应。然后，我们将椭圆度（卷发不是）和$ D $ d $ specutrum所知的许多事实，以恢复几乎立即证明上述卷曲的一些结果，并为我获得了其他结果。 {例如，我们会发现卷曲的特征值（删除零频谱零）始终为固定常数，由dirac的特征频谱降至固定常数，而dirac的特征值和平等表征了三个球的表征}。另外，我们还表明，{\ em {\ em不存在与卷发相关的等速度问题的均值$ l^2 $ - 溶解}，就像cantarella，de turck，de turck，gluck y teytel \ citite \ cite {cdtgt}一样，是通过这些不知名的属性来实现这些不知名的解决方案（是否可以为这些不知名的解决方案提供）（是否可以为这些属性提供了一种，是否构成了这种不可思议的解决方案，是否有这种属性（是否有这种疾病），以实现这些疾病（是否可以构成强制性的解决方案）。 {\ em Helicity。}
In the last decades, many mathematicians have studied the {\em curl operator} on compact (both with or without empty boundary) three-manifolds, mainly the behaviour of its spectrum and some iso\-pe\-ri\-me\-tric problems associated with it. In this paper, we reveal an (unexpected?) relation between this curl operator and the Dirac operator correspondingto any of the spin$^c$ structures on the manifold. Then, we make the ellipticity of $D$ (curl is not) and the many facts already known about the spectrum of $D$ to recuperate with almost immediate proofs some results above curl and obtain others unknown for me. {\em For example, we will find that the eigenvalues of curl, removing the point spectrum zero, are always, up to a fixed constant, lower bounded by those of the Dirac and the equality characterize the round three-sphere}. Also, we also show that {\em there do not exist mean-convex $L^2$-solutions for the isoperimetric problem associated to curl}, as Cantarella, de Turck, Gluck y Teytel \cite{CdTGT} had conjectured, while other authors proved properties for these unknown solutions (adding always whether optimal domains exist) perhaps by thinking in the case of the successful maximization of the {\em helicity.}
