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Co-area, liquid crystals, and minimal surfaces. (English) Zbl 0645.58015
Partial differential equations, Proc. Symp., Tianjin/China 1986, Lect. Notes Math. 1306, 1-22 (1988).
[For the entire collection see Zbl 0631.00004.]
Authors’ abstract: “Oriented n area minimizing surfaces (integral currents) in \(M^{m+n}\) can be approximated by level sets (slices) of nearly m-energy minimizing mappings \(M^{m+n}\to S\) m with essential but controlled discontinuities. This gives new perspective on multiplicity, regularity, and computation questions in least area surface theory.”
The main general theorem tells that the n-area of such an area minimizing surface in a given homology class can be obtained as infima of various m- energies. The paper avoids technicalities and is pleasant to read also for non-experts. It explains the basic ideas and sketches the proofs for the general theorems and some more concrete special cases. Also the motivation from the geometry of liquid crystals is discussed.
Reviewer: P.Mattila

MSC:
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
49Q20 Variational problems in a geometric measure-theoretic setting