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Decomposition of generalized O’Hara’s energies. (English) Zbl 1481.53004

This paper considers a generalization of a special case of O’Hara’s knot energy, and shows some results similar to those for the Möbius energy, mainly containing the decomposition formula, cosine formula and the first and second variational formulas. The main result is Theorem 1. The formulation of this theorem is a little complicated, namely the conditions (A.1) to (A.5) are not convenient to use directly. Then the authors give several more complicated conditions to implying (A.1) to (A.5), and show that a special class of O’Hara’s energy satisfies these conditions.
Section 1, the introduction, is well written, especially the historical background part. Besides, the whole paper is well organized. Section 4 is somewhat hard to read, and the result of this section seems not to be used in the following part.

MSC:

53A04 Curves in Euclidean and related spaces
58J70 Invariance and symmetry properties for PDEs on manifolds
49Q10 Optimization of shapes other than minimal surfaces
57K10 Knot theory
53A31 Differential geometry of submanifolds of Möbius space
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References:

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