An Extension of Asgeirsson's Mean Value Theorem for Solutions of the ultra-hyperbolic Equation in Dimension Four

TL;DR

This paper extends Asgeirsson's mean value theorem for solutions of the ultra-hyperbolic equation in four variables from conjugate circles to conjugate conics, including hyperbolae, using geometric and conformal mapping techniques.

Contribution

It generalizes the mean value property to conjugate conics, broadening the geometric understanding of solutions to the ultra-hyperbolic equation in four dimensions.

Findings

Extended mean value property to conjugate hyperbolae

Connected solutions to line integrals in Euclidean 3-space

Linked conjugate conics with rulings of hyperboloids and paraboloids

Abstract

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.