Eguchi-Hanson空间中超曲面族的几何和解析性质
Geometric and analytic properties of families of hypersurfaces in Eguchi-Hanson space
摘要：
我们研究了江口-汉森空间中的超曲面族的几何，这些超曲面族是$S^2$中的曲线上的复线丛，并且是三维的、非紧的黎曼流形，它们在闭曲线的Hopf环面中是叶状的。它们是负曲的渐近平坦空间，我们计算了它们的完全三维曲率张量和第二基本型，并给出了它们的测地流的一些结果.本文证明了对任何$p \geq 1$和任意曲线，在这些超曲面上不存在$Ł^p$ -调和函数，并确定了在闭曲线情形下Laplace算子的本质谱和Dirac算子的平方的下确界.对于圆，我们还计算的$Ł^2$ -内核的狄拉克算子的意义上的谱理论，并表明，它是无限维的。我们进一步考虑这些空间上的Einstein Dirac系统，并在它们上构造T-Killing旋量的显式例子。
英文摘要：
We study the geometry of families of hypersurfaces in Eguchi-Hanson space that arise as complex line bundles over curves in $S^2$ and are three-dimensional, non-compact Riemannian manifolds, which are foliated in Hopf tori for closed curves. They are negatively curved, asymptotically flat spaces, and we compute the complete three-dimensional curvature tensor as well as the second fundamental form, giving also some results concerning their geodesic flow. We show the non-existence of $Ł^p$-harmonic functions on these hypersurfaces for every $p \geq 1$ and arbitrary curves, and determine the infima of the essential spectra of the Laplace and of the square of the Dirac operator in the case of closed curves. For circles we also compute the $Ł^2$-kernel of the Dirac operator in the sense of spectral theory and show that it is infinite dimensional. We consider further the Einstein Dirac system on these spaces and construct explicit examples of T-Killing spinors on them.
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