$L^2$ representations of the Second Variation and Łojasiewicz-Simon gradient estimates for a decomposition of the Möbius energy

Abstract

Knot energies, one of which is the Möbius energy, are constructed to measure the well-proportioned of the knot. The best-proportioned knot in the given knot class may be determined by the gradient flow of the energy. Indeed, Blatt showed the global existence and convergence of the gradient flow of the Möbius energy near stationary points. The Łojasiewicz inequality played an important role in proving this result. The inequality derived from $L^2$ representation of the first and second variations. On the other hand, Ishizeki and Nagasawa showed that the Möbius energy can be decomposed into three parts that are Möbius invariant. Each part has an $L^2$ representation of the first variation. In this paper, we discuss the $L^2$ representation of the second variation for each decomposed part of the Möbius energy, and derive explicit formulas for it. As a consequence of this and Chill's findings the Łojasiewicz inequality is derived.