American Institute of Mathematical Sciences

Advanced
September  2010, 9(5): 1463-1471. doi: 10.3934/cpaa.2010.9.1463

Regularity theory for the Möbius energy

Philipp Reiter 1,

1. 

Institut für Mathematik, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany

Received  August 2009 Revised  December 2009 Published  May 2010

The Möbius energy, defined 1991 by O'Hara, is the most prominent example of a knot energy. In this text we will focus on the regularity of local minimizers (within a prescribed knot class) whose arc-length parametrization was shown to be $C^{1,1}$ by Freedman, He, and Wang. Later on, He improved this result to $C^\infty$ regularity. In this text we will briefly outline the main ideas of these two steps which require completely different approaches involving techniques from geometry and analysis. Moreover we explain how to rigorously derive the first variation of the Möbius energy and fix a gap in He's treatise.
Keywords: Knot energy, Möbius energy, regularity., bilinear Fourier multiplier.
Mathematics Subject Classification: Primary: 57M25, 58E10; Secondary: 53A04, 42A4.
Citation: Philipp Reiter. Regularity theory for the Möbius energy. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1463-1471. doi: 10.3934/cpaa.2010.9.1463
[1]

Konovenko Nadiia, Lychagin Valentin. Möbius invariants in image recognition. Journal of Geometric Mechanics, 2017, 9 (2) : 191-206. doi: 10.3934/jgm.2017008
[2]

Xiuting Li. The energy conservation for weak solutions to the relativistic Nordström-Vlasov system. Evolution Equations & Control Theory, 2016, 5 (1) : 135-145. doi: 10.3934/eect.2016.5.135
[3]

Petr Kůrka. Minimality in iterative systems of Möbius transformations. Conference Publications, 2011, 2011 (Special) : 903-912. doi: 10.3934/proc.2011.2011.903
[4]

Petr Kůrka. Iterative systems of real Möbius transformations. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 567-574. doi: 10.3934/dcds.2009.25.567
[5]

Wen Huang, Jianya Liu, Ke Wang. Möbius disjointness for skew products on a circle and a nilmanifold. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3531-3553. doi: 10.3934/dcds.2021006
[6]

Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133
[7]

Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499
[8]

Rich Stankewitz, Hiroki Sumi. Backward iteration algorithms for Julia sets of Möbius semigroups. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6475-6485. doi: 10.3934/dcds.2016079
[9]

Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003
[10]

Livio Flaminio, Giovanni Forni. Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows. Electronic Research Announcements, 2019, 26: 16-23. doi: 10.3934/era.2019.26.002
[11]

Wen Huang, Zhiren Wang, Guohua Zhang. Möbius disjointness for topological models of ergodic systems with discrete spectrum. Journal of Modern Dynamics, 2019, 14: 277-290. doi: 10.3934/jmd.2019010
[12]

Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021076
[13]

Jorge Cortés. Energy conserving nonholonomic integrators. Conference Publications, 2003, 2003 (Special) : 189-199. doi: 10.3934/proc.2003.2003.189
[14]

Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2063-2077. doi: 10.3934/dcds.2012.32.2063
[15]

Luisa Berchialla, Luigi Galgani, Antonio Giorgilli. Localization of energy in FPU chains. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 855-866. doi: 10.3934/dcds.2004.11.855
[16]

Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230
[17]

Patricia Bauman, Andrea C. Rubiano. Energy-minimizing nematic elastomers. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 259-282. doi: 10.3934/dcdss.2015.8.259
[18]

Antonin Chambolle, Francesco Doveri. Minimizing movements of the Mumford and Shah energy. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 153-174. doi: 10.3934/dcds.1997.3.153
[19]

Gisèle Ruiz Goldstein, Jerome A. Goldstein, Fabiana Travessini De Cezaro. Equipartition of energy for nonautonomous wave equations. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 75-85. doi: 10.3934/dcdss.2017004
[20]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy