A family of asymptotically hyperbolic manifolds with arbitrary energy-momentum vectors

TL;DR

This paper constructs a family of asymptotically hyperbolic manifolds with arbitrary energy-momentum vectors, challenging the positive energy-momentum theorem by removing the completeness condition.

Contribution

It introduces non-radial solutions to the Yamabe equation that produce new asymptotically hyperbolic metrics with prescribed energy-momentum vectors.

Findings

Family of metrics with arbitrary energy-momentum vectors constructed

Counter-examples to positive energy-momentum theorem without completeness

Metrics have scalar curvature greater than -n(n-1)

Abstract

A family of non-radial solutions of the Yamabe equation, with reference the hyperbolic space, is constructed using power series. As a result, we obtain a family of asymptotically hyperbolic metrics, with spherical conformal infinity, with scalar curvature greater than -n(n - 1), but which are a priori not complete. Moreover, any vector of R^n+1 is performed by an energy-momentun vector of one suitable metric of this family. They can in particular provide counter-examples to the positive energy-momentum theorem when one removes the completeness assumption.