Freedman–He–Wang conjecture for Möbius cross energy of links

Let (γ1,γ2)(\gamma_1,\gamma_2) be a nontrivial 22-component link in R3\mathbb R^3, and let its Möbius cross energy be

E(γ1,γ2)=S1×S1γ1(s)γ2(t)γ1(s)γ2(t)2dsdt.E(\gamma_1,\gamma_2)=\int_{S^1\times S^1}\frac{|\gamma_1'(s)||\gamma_2'(t)|}{|\gamma_1(s)-\gamma_2(t)|^2}\,ds\,dt.

Freedman–He–Wang conjecture. The Möbius energy is minimized, among the class of all nontrivial links in R3\mathbb R^3, by the stereographic projection of the standard Hopf link in S3S^3. This is a conformally invariant variational problem for links, motivated by the lower bound in terms of linking number. The source presents the minimizing-link assertion as an open conjecture.

Sources & referencesView supporting material

Primary source

André Neves, “New applications of Min-max Theory”, arXiv:1409.7537 (2014).

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