No Access

Variational formulae and estimates of O’Hara’s knot energies

Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama-shi, Saitama 338-8570, Japan

E-mail Address: s.kawakami.649@ms.saiatama-u.ac.jp

Search for more papers by this author

and

Takeyuki Nagasawa

Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama-shi, Saitama 338-8570, Japan

E-mail Address: tnagasaw@rimath.saitama-u.ac.jp

Search for more papers by this author

https://doi.org/10.1142/S0218216520500170Cited by:2 (Source: Crossref)

Abstract

O’Hara’s energies, introduced by Jun O’Hara, were proposed to answer the question what the canonical shape in a given knot type is, and were configured so that the less the energy value of a knot is, the “better” its shape is. The existence and regularity of minimizers has been well studied. In this paper, we calculate the first and second variational formulae of the $(α, p)$ $(α, p)$ -O’Hara energies and show absolute integrability, uniform boundedness, and continuity properties. Although several authors have already considered the variational formulae of the $(α, 1)$ $(α, 1)$ -O’Hara energies, their techniques do not seem to be applicable to the case $p > 1$ $p > 1$ . We obtain the variational formulae in a novel manner by extracting a certain function from the energy density. All of the $(α, p)$ $(α, p)$ -energies are made from this function, and by analyzing it, we obtain not only the variational formulae but also the estimates in several function spaces.

Keywords:

AMSC: 53A04, 49Q10

References

1. A. Abrams, J. Cantarella, J. H. G. Fu, M. Ghomi and R. Howard, Circles minimize most knot energies, Topology 42(2) (2003) 381–394. Crossref, Web of Science, Google Scholar
2. S. Blatt, The gradient flow of the Möbius energy near local minimizers, Calc. Var. Partial Differential Equations 43(3–4) (2012) 403–439. Crossref, Web of Science, Google Scholar
3. S. Blatt, Boundedness and regularizing effects of O’Hara’s knot energies, J. Knot Theory Ramifications 21 (2012) 1250010, 9 pp. Link, Web of Science, Google Scholar
4. S. Blatt, The gradient flow of O’Hara’s knot energies, Math. Ann. 370(3–4) (2018) 993–1061. Crossref, Web of Science, Google Scholar
5. S. Blatt and P. Reiter, Stationary points of O’Hara’s knot energies, Manuscripta Math. 140(1–2) (2013) 29–50. Crossref, Web of Science, Google Scholar
6. S. Blatt, P. Reiter and A. Schikorra, Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth, Trans. Amer. Math. Soc. 368(9) (2016) 6391–6438. Crossref, Web of Science, Google Scholar
7. S. Blatt, P. Reiter and A. Schikorra, On O’Hara knot energies I: Regularity for critical knots, preprint (2019) arXiv:1905.06064. Google Scholar
8. S. Blatt and N. Vorderobermeier, On the analyticity of critical points of the Möbius energy, Calc. Var. Partial Differential Equations 58(1) (2019) 28 pp. Crossref, Web of Science, Google Scholar
9. M. H. Freedman, Z.-X. He and Z. Wang, Möbius energy of knots and unknots, Ann. of Math. 139 (1994) 1–50. Crossref, Web of Science, Google Scholar
10. Z.-X. He, The Euler–Lagrange equation and heat flow for the Möbius energy, Comm. Pure Appl. Math. 53 (4) (2000) 399–431. Crossref, Web of Science, Google Scholar
11. A. Ishizeki and T. Nagasawa, A decomposition theorem of the Möbius energy I: Decomposition and Möbius invariance, Kodai. Math. J. 37(3) (2014) 737–754. Crossref, Web of Science, Google Scholar
12. A. Ishizeki and T. Nagasawa, A decomposition theorem of the Möbius energy II: Variational formulae and estimates, Math. Ann. 363(1–2) (2015) 617–635. Crossref, Web of Science, Google Scholar
13. A. Ishizeki and T. Nagasawa, Decomposition of generalized O’Hara’s energies, preprint (2019), arXiv:1904.06812. Google Scholar
14. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems (Birkhäuser Verlag, Basel, 1995). Crossref, Google Scholar
15. J. O’Hara, Energy of a knot, Topology 30(2) (1991) 241–247. Crossref, Web of Science, Google Scholar
16. J. O’Hara, Family of energy functionals of knots, Topology Appl. 48(2) (1992) 147–161. Crossref, Web of Science, Google Scholar
17. J. O’Hara, Energy functionals of knots II, Topology Appl. 56(1) (1994) 45–61. Crossref, Web of Science, Google Scholar
18. P. Reiter, Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family $E^{(α)}$ $E^{(α)}$ , $α \in [2, 3)$ $α \in [2, 3)$ , Math. Nachr. 285(7) (2012) 889–913. Crossref, Web of Science, Google Scholar