Arkiv för Matematik

Volume 55 (2017)

Number 1

A four-dimensional Neumann ovaloid

Pages: 185 – 198

DOI: https://dx.doi.org/10.4310/ARKIV.2017.v55.n1.a9

Authors

Lavi Karp (Department of Mathematics, ORT Braude College, Karmiel, Israel)

Erik Lundberg (Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Fl., U.S.A)

Abstract

What is the shape of a uniformly massive object that generates a gravitational potential equivalent to that of two equal point-masses? If the weight of each point-mass is sufficiently small compared to the distance between the points then the answer is a pair of balls of equal radius, one centered at each of the two points, but otherwise it is a certain domain of revolution about the axis passing through the two points. The existence and uniqueness of such a domain is known, but an explicit parameterization is known only in the plane where the region is referred to as a Neumann oval. We construct a four-dimensional “Neumann ovaloid”, solving explicitly this inverse potential problem.

Keywords

quadrature domain, Schwarz function, Neumann oval, inverse potential problem, elliptic integral

2010 Mathematics Subject Classification

Primary 30C20, 31A35. Secondary 35R35.

Received 16 October 2016

Accepted 8 February 2017

Published 26 September 2017