Celebratio Mathematica

article M. H. Freedman and Z.-X. He: “Factoring the logarithmic spiral,” Invent. Math. 92 : 1 (1988), pp. 129–138. MR 0931207 Zbl 0622.30011

article M. H. Freedman and Z.-X. He: “A remark on inherent differentiability,” Proc. Amer. Math. Soc. 104 : 4 (1988), pp. 1305–1310. MR 0937012 Zbl 0689.57021

article M. H. Freedman and Z.-X. He: “Links of tori and the energy of incompressible flows,” Topology 30 : 2 (1991), pp. 283–287. MR 1098922 Zbl 0731.57003

The moduli of curve families have been a useful bridge between analysis and geometric-topological argument. Caratheódory exploited the notion to construct his theory of prime ends. In dimensions three and higher this connection has been extensively developed by Fred Gehring (for example, see [Gehring 1971; 1986]). We continue in this tradition by showing that the naturally defined “conformal moduli” for a disjoint collection of solid tori in \( \mathbb{R}^3 \) cannot all be greater than the constant \( 125\pi/48 \) if the tori are linked in any essential manner. It is natural to conjecture that the optimal lower bound is \( (\sqrt{2}-1)/2\pi \), the modulus of the “solid” Clifford torus.

As an application, we use the topology of linking flow lines to estimate a lower bound on the energy of certain incompressible flows. Roughly, one thinks that an invariant solid torus of spinning fluid may give up energy by elongating like a soda straw, but that this should be prevented if several such tori are linked. To make this precise, an inequality relating modulus and a variant of energy is derived.

article M. H. Freedman and Z.-X. He: “Divergence-free fields: Energy and asymptotic crossing number,” Ann. of Math. (2) 134 : 1 (1991), pp. 189–229. MR 1114611 Zbl 0746.57011

incollection M. H. Freedman and Z.-X. He: “Research announcement on the ‘energy’ of knots,” pp. 219–222 in Topological aspects of the dynamics of fluids and plasmas (University of California at Santa Barbara, August–December 1991). Edited by H. K. Moffatt, G. M. Zaslavasky, P. Comte, and M. Tabor. NATO ASI Series E: Applied Sciences 218. Kluwer Academic Publishers (Dordrecht), 1992. MR 1232232 Zbl 0788.53004

Zheng-Xu He	Related
Pierre Comte	Related
Michael Tabor	Related
Henry Keith Moffatt	Related
George M. Zaslavsky	Related

article S. Bryson, M. H. Freedman, Z.-X. He, and Z. Wang: “Möbius invariance of knot energy,” Bull. Amer. Math. Soc. (N.S.) 28 : 1 (1993), pp. 99–103. MR 1168514 Zbl 0776.57003 ArXiv math/9301212

Zhenghan Wang	Related
Zheng-Xu He	Related
Steve Bryson	Related

article M. H. Freedman, Z.-X. He, and Z. Wang: “Möbius energy of knots and unknots,” Ann. of Math. (2) 139 : 1 (1994), pp. 1–50. MR 1259363 Zbl 0817.57011

Zhenghan Wang	Related
Zheng-Xu He	Related

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Michael H. Freedman

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