Summary of Gradient Flows on Nonpositively Curved Metric Spaces and Harmonic Maps

Gradient flows for energy functionals have been studied extensively in the past. Well known examples are the heat flow or the mean curvature flow. To make sense of the term gradient an inner product structure is assumed. One works on a Hilbert space, or on the tangent space to a manifold, for example. However, it is possible to do without an inner product. The domain of the energy functionals considered herein is assumed to be a nonpositively curved metric space (L,D). A complete metric space (L,D) is called a nonpositively curved (NPC) space if it satisfies the following two conditions (see for example [9, 12]). (a) L is a length space. Distance realizing curves are called geodesics. (b) For any three points v, u0, u1 and choices of connecting geodesics γv,u0 , γu0,u1 , γu1,v the following comparison principle holds. Let ut be the point on γu0,u1 which is a fraction t of the distance from u0 to u1 . The NPC hypothesis is the following inequality for 0 ≤ t ≤ 1 : D(v, ut) ≤ (1− t)D(v, u0) + tD(v, u1)− t(1− t)D(u0, u1) . (1)