Extremal curves in (2+1)-dimensional Yang-Mills theory

We examine the structure of the potential energy of (2 + 1)-dimensional Yan g-Mills theory on a torus with gauge group SU(2). We use a standard definit ion of distance on the space of gauge orbits. The curves of extremal potent ial energy in orbit space satisfy a certain partial differential equation. We argue that the energy spectrum is gapped because the extremal curves are of finite length. Though classical gluon waves satisfy our differential eq uation, they are not extremal curves. We construct examples of extremal cur ves and find how the length of these curves depends on the dimensions of th e torus. The intersections with the Gribov horizon are determined explicitl y. The results are discussed in the context of Feynman's ideas about the or igin of the mass gap. (C) 2000 Elsevier Science B.V. All rights reserved.