论文标题
Asgeirsson的平均值定理的扩展，用于尺寸四的Ultra-Hyperbolic方程解决方案
An Extension of Asgeirsson's Mean Value Theorem for Solutions of the ultra-hyperbolic Equation in Dimension Four
论文作者
论文摘要
1937年，阿斯吉尔森（Asgeirsson）建立了一个平均价值属性，用于以$ 2N $变量为$ 2N $变量的一般超纤维方程的解决方案。在四个变量的情况下，它指出，在某些成对的共轭圆圈上的解决方案的积分是相同的。在本文中，我们将此结果扩展到非脱位结合锥体，其中包括结合圆圈的原始情况，并添加了新的结合双曲线。 该结果的更广泛的背景是弗里茨·约翰（Fritz John）1938年对超纤维方程的分析的几何化。该方程的解决方案是从欧几里得3空间中函数的线积分中的函数函数的兼容性出现的。在定向线的空间上引入了规范中性的Kaehler指标，阐明了这种关系，并扩大了范式以允许新见解。 特别是，事实证明，超纤维方程的解决方案在任何一对曲线上具有平均值属性，而这些曲线是在保形映射下作为约翰共轭圆圈的图像所产生的。这些曲线随后显示为结合锥体，其中包括圆圈和双曲线。 约翰与1页的双曲线裁决一起确定了共轭圈。结合双曲线是用1片双曲线或双曲线抛物面的两块裁决确定的。
In 1937 Asgeirsson established a mean value property for solutions of the general ultra-hyperbolic equation in $2n$ variables. In the case of four variables, it states that the integrals of a solution over certain pairs of conjugate circles are the same. In this paper we extend this result to non-degenerate conjugate conics, which include the original case of conjugate circles and adds the new case of conjugate hyperbolae. The broader context of this result is the geometrization of Fritz John's 1938 analysis of the ultra-hyperbolic equation. Solutions of the equation arise as the compatibility for functions on line space to come from line integrals of functions in Euclidean 3-space. The introduction of the canonical neutral Kaehler metric on the space of oriented lines clarifies the relationship and broadens the paradigm to allow new insights. In particular, it is proven that a solution of the ultra-hyperbolic equation has the mean value property over any pair of curves that arise as the image of John's conjugate circles under a conformal map. These pairs of curves are then shown to be conjugate conics, which include circles and hyperbolae. John identified conjugate circles with the two rulings of a hyperboloid of 1-sheet. Conjugate hyperbolae are identified with the two rulings of either a piece of a hyperboloid of 1-sheet or a hyperbolic paraboloid.
