Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Min-max theory and the energy of links

Authors: Ian Agol, Fernando C. Marques and André Neves
Journal: J. Amer. Math. Soc. 29 (2016), 561-578
MSC (2010): Primary 57M25; Secondary 53C42, 49Q20
Published electronically: June 4, 2015
MathSciNet review: 3454383
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Freedman, He, and Wang conjectured in 1994 that the Möbius energy should be minimized, among the class of all nontrivial links in Euclidean space, by the stereographic projection of the standard Hopf link. We prove this conjecture using the min-max theory of minimal surfaces.

References [Enhancements On Off] (What's this?)

  • [1] Frederick Justin Almgren Jr., The homotopy groups of the integral cycle groups, Topology 1 (1962), 257-299. MR 0146835 (26 #4355)
  • [2] Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418 (2002e:53053)
  • [3] Michael H. Freedman, Zheng-Xu He, and Zhenghan Wang, Möbius energy of knots and unknots, Ann. of Math. (2) 139 (1994), no. 1, 1-50. MR 1259363 (94j:58038),
  • [4] Zheng-Xu He, On the minimizers of the Möbius cross energy of links, Experiment. Math. 11 (2002), no. 2, 244-248. MR 1959266 (2003k:58016)
  • [5] Denise Kim and Rob Kusner, Torus knots extremizing the Möbius energy, Experiment. Math. 2 (1993), no. 1, 1-9. MR 1246479 (94j:58039)
  • [6] Fernando C. Marques and André Neves, Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014), no. 2, 683-782. MR 3152944,
  • [7] Frank Morgan, Geometric measure theory, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginner's guide. MR 1775760 (2001j:49001)
  • [8] Jun O'Hara, Energy of a knot, Topology 30 (1991), no. 2, 241-247. MR 1098918 (92c:58017),
  • [9] Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417 (87a:49001)
  • [10] T. J. Willmore, Note on embedded surfaces, An. Şti. Univ. ``Al. I. Cuza'' Iaşi Secţ. I a Mat. (N.S.) 11B (1965), 493-496 (English, with Romanian and Russian summaries). MR 0202066 (34 #1940)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 57M25, 53C42, 49Q20

Retrieve articles in all journals with MSC (2010): 57M25, 53C42, 49Q20

Additional Information

Ian Agol
Affiliation: Department of Mathematics, University of California, 970 Evans Hall #3840, Berkeley, California 94720-3840

Fernando C. Marques
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544

André Neves
Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2RH, United Kingdom

Received by editor(s): September 19, 2012
Received by editor(s) in revised form: April 10, 2015
Published electronically: June 4, 2015
Additional Notes: The first author was supported by DMS-0806027 and DMS-1105738.
The second author was partly supported by CNPq-Brazil, FAPERJ, and Math-Amsud.
The third author was partly supported by Marie Curie IRG Grant, ERC Start Grant, and Leverhulme Award.
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society