论文标题
网络，nijenhuis操作员和天上的pdes
Webs, Nijenhuis operators, and heavenly PDEs
论文作者
论文摘要
1989年，梅森（Mason）和纽曼（Newman）证明，满足爱因斯坦真空方程式的自动双重指标（在复杂的情况下或中性签名中）与依赖于向量域的通勤参数$ x_1（λ）（λ），x_2（λ）$之间的通勤参数成对。早些时候（1975年），普兰斯基（Plebański）展示了此类矢量场的实例，具体取决于满足所谓的I或IIPlebańskiHeavenly Pdes的四个变量的一个函数。导致梅森 - 新闻媒介领域的其他PDE在文献中也是众所周知的：Husain-Park（1992---94），Schief（1996）。在本文中，我们在网络理论的背景下讨论了这些问题，即从尼尼豪斯运营商的角度来理解的歧管上的叶子收集理论。特别是我们展示了如何将这种理论应用于基于4D的nijenhuis运算符的不同形式的新``天堂''PDE，它们与其前任类似。在所有情况下，都与Konopelchenko-Schief-Szereszewski一起观察到的与PDE的无海洋分散系统和相应的Veronese网络有关。我们还讨论了``heavelny''pdes的一些较高维度的概括，以及在4D案例中存在相关真空爱因斯坦指标的存在。
In 1989 Mason and Newman proved that there is a 1-1-correspondence between self-dual metrics satisfying Einstein vacuum equation (in complex case or in neutral signature) and pairs of commuting parameter depending vector fields $X_1(λ),X_2(λ)$ which are divergence free with respect to some volume form. Earlier (in 1975) Plebański showed instances of such vector fields depending of one function of four variables satisfying the so-called I or II Plebański heavenly PDEs. Other PDEs leading to Mason--Newman vector fields are also known in the literature: Husain--Park (1992--94), Schief (1996). In this paper we discuss these matters in the context of the web theory, i.e. theory of collections of foliations on a manifold, understood from the point of view of Nijenhuis operators. In particular we show how to apply this theory for constructing new ``heavenly'' PDEs based on different normal forms of Nijenhuis operators in 4D, which are integrable similarly to their predecessors. Relation with the Hirota dispersionless systems of PDEs and the corresponding Veronese webs, which was recently observed by Konopelchenko--Schief--Szereszewski, is established in all the cases. We also discuss some higher dimensional generalizations of the ``heavelny'' PDEs and the existence of related vacuum Einstein metrics in 4D-case.
