INSTABILITY OF THE CONSERVATIVE PROPERTY UNDER QUASI-ISOMETRIES

The paper shows by example the existence of a pair of quasi-isometrically equivalent complete Riemannian metrics g, g approximately on a manifold N so that (N, g) is conservative while (N, g approximately) is not. This contrasts with known sufficient criteria for conservativeness which are stable under such changes in metric. In doing so we also give an example of a complete manifold with no nonconstant bounded solutions to Laplace's equation but with nonconstant bounded solutions to the heat equation.