Communications in Mathematical Physics

, Volume 249, Issue 2, pp 273–303 | Cite as

Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model

  • Fanghua Lin
  • Yisong Yang


In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π3(S 2 )= Open image in new window . We prove in the full space situation that there exists an infinite subset Open image in new window of Open image in new window such that for any mOpen image in new window , the Faddeev energy, E, has a minimizer among the topological class Q=m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for Open image in new window = Open image in new window . As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality EC|Q|3/4, where C>0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.


Soliton Energy Minimizer Bounded Domain Theory Model Technical Result 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Fanghua Lin
    • 1
  • Yisong Yang
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsPolytechnic UniversityBrooklynUSA

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