论文标题
通过校准在3个维度的单位矢量场的体积上
On the volume of a unit vector field in 3 dimensions via calibrations
论文作者
论文摘要
我们给出了一个新的证明，证明了$ \ mathbb {s}^3（r）$上的最小音量向量字段是Hopf vector字段。这种证明再次依赖于校准理论，在这里是从自然差异形式源给出的系统观点出现的。我们的结果尤其适用于所有$ r $。 给出了$ t^1m $的相关校准的分类，用于每个定向的3个manifold $ m $恒定的截面曲率，继续研究\ textit {forthit {常规} riemannian几何形状的基本差分系统。显示这种差异系统的应用也是本文的目的之一。 我们推断出空间形式的地球流量矢量场的新属性，该特性与任何维度的椭圆形和双曲线几何形状的最小体积问题的解决方案相互作用。对于三维中的双曲线情况，解决方案 - 未知 - 最取决于域的域的同源类别和向量场的边界值。这是一个值得注意的示例来说明的，具有讽刺意味的是，它仅适用于曲率$ -1 $。
We give a new proof of the well-known result that the minimal volume vector fields on $\mathbb{S}^3(r)$ are the Hopf vector fields. Such proof relies again on calibration theory, arising here from a systematic point of view given by a natural source of differential forms. Our results serve in particular for all $r$. A classification of relevant calibrations on $T^1M$ for every oriented 3-manifold $M$ of constant sectional curvature is given, continuing the study of the \textit{usual} fundamental differential system of Riemannian geometry. Showing applications of this differential system is also one of the purposes of this article. We deduce new properties of the geodesic flow vector field of space forms, which interacts with the solutions of the minimal volume problem both in elliptic and hyperbolic geometry, in any dimension. The solution -- unknown -- for the hyperbolic case in 3-dimensions being most dependent on the homology class of the domain and boundary values of the vector fields. This is illustrated with a noteworthy example which ironically works just for curvature $-1$.
