A dual-cubic proof of the four-point hyperbolic Atiyah conjecture

Abstract

The hyperbolic Atiyah conjecture assigns a binary form of degree n-1 to each point of an n-point configuration in hyperbolic three-space and asserts that the resulting forms are linearly independent. We give a complete geometric proof for four points. Starting from Malkoun's dual-cubic reduction, we exclude repeated roots by extremal arguments and use a function that is monotone along hyperbolic geodesics to show that any hypothetical dependence forces the configuration into one totallygeodesic plane. Direct arguments for collinear, concave, and convex configurations then complete the proof. The planar arguments give a self-contained treatment of previously established cases. We also identify an incorrect circular-domain assertion in Malkoun's published proof and give an explicit counterexample to that assertion.