Frank Morgan

Note added February 7, 2011. The announced counterexample was wrong. It does not satisfy the hypothesis of the conjecture. Geodesic circles about the origin in Enneper’s surface are unstable.

The proposed generalization of the Log-convex Density Conjecture to separate densities for volume and perimeter fell at the centennial congress of the Royal Spanish Mathematical Society.  The counterexample—Enneper’s Surface—emerged from discussions with participants in the Geometric Analysis session, notably Manuel Ritoré, Antonio Ros, Cesar Rosales, and Antonio Cañete.

Another way of stating the hypothesis for a single density for volume and perimeter is that the generalized Ricci curvature, equal to the Riemannian Ricci curvature (0 for Euclidean space) minus the Hessian of the log of the density be nonpositive. General separate perimeter and volume densities are equivalent to a conformal change of metric together with a density. The counterexample, Enneper’s Surface, is conformally the plane (with unit density) and intrinsically a surface of revolution. As a minimal surface, it has negative curvature. Since the curvature is negative at the origin and approaches 0 at infinity, small balls about the origin have larger perimeter than small balls near infinity, and the generalized conjecture fails.

Preliminary announcement February 4, 2011.