\(L^2\) representations of the second variation and Łojasiewicz-Simon gradient estimates for a decomposition of the Möbius energy (Q2178048)

Knot energies, one of which is the Möbius energy, are constructed to measure how well proportioned knots. The best-proportioned knot in a 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 Lojasiewicz inequality played an important role in proving this result. The inequality is derived from \(L^2\) representation of the first and second variations. On the other hand, Ishizeki and Nagasawa showed that the Möbius energy can decomposed into three parts that are Möbius invariant. Each part has an \(L^2\) representation of the first variation. In this paper, the author discusses 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 Lojasiewicz inequality is derived.