A Conformal Mean Value Theorem for Solutions of the Ultrahyperbolic Equation

TL;DR

This paper extends Asgeirsson's mean value theorem for solutions of the ultrahyperbolic equation by identifying conjugate conic sections and establishing their invariance under conformal transformations in pseudo-Euclidean space.

Contribution

It generalizes the mean value property to pairs of conjugate conics, including hyperbolae, parabolae, and lines, under conformal invariance in four-dimensional pseudo-Euclidean space.

Findings

Mean value theorem extended to conjugate hyperbolae and parabolae.

Established a correspondence between conjugate conics and rulings of doubly ruled surfaces.

Proved invariance of the ultrahyperbolic equation under conformal maps.

Abstract

Asgeirsson's theorem establishes a mean value property for solutions of the ultrahyperbolic equation. 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, the invariance of the four dimensional ultrahyperbolic equation under conformal maps of the pseudo-Euclidean space of signature 2+2 is used to get the most general version of the mean value theorem. The name non-degenerate conjugate conics is used for the most general pairs of curves over which solutions of the ultrahyperbolic equation enjoy the mean value property. These are proven to be pairs of conic sections, so that, in addition to conjugate circles which were known to exist, the picture is completed by finding mean value theorems over conjugate hyperbolae, conjugate parabolae, and line-empty pairs. In addition, Fritz John established a link between conjugate circles and the two rulings of a hyperboloid of revolution. The line incidence property of doubly ruled surfaces is used to prove a one-to-one correspondence between non-degenerate conjugate conics and pairs of rulings of doubly ruled surfaces in Euclidean 3-space.