A dual monotonicity formula for harmonic mappings

The well-known monotonicity formula for harmonic maps says that the scaled energy functional over a ball of radius r is a non-decreasing function of r. The proof uses the fact that the energy functional is critical under any compactly supported variation on the domain of the map. In this article, we will instead use the fact that the energy is critical under variations of the map on the image of the map. By choosing the variational vector field suitably it will be shown that a scaled energy considered as an integral functional over a ball of radius r where r is the distance from a point on the image manifold, is monotonically non-decreasing. The formula takes a stronger form when the image is one dimensional.Received: 29 June 2002, Accepted: 30 September 2002, Published online: 14 February 2003Supported in part by NSF DMS0071862