A nonuniform extension of the Möbius energy

Abstract

In this paper, we discuss a nonuniform extension of the Möbius energy for curves of nonuniform thickness. After defining the nonuniform energy and stating its basic properties, we show that its C1 minimizers in R3 must be convex planar curves. As this energy is not Möbius invariant, this requires a novel approach. We then describe two variations of the nonuniform energy: the equitotal variation, which allows the free reallocation of weight along a curve, and the translatory variation, which only allows the translation of a weight profile along a curve. We prove that equitotal variation has no absolute minimizer when the curve has a point of zero curvature. We also provide conditions that ensure invariance of the energy of a weighted curve under translatory variation—a standard parametrization of torus knots with sinusoidal weight distributions are such examples.