Almgren, F.; Browder, W.; Lieb, E. H. 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 Cited in 27 Documents MSC: 58E12 Variational problems concerning minimal surfaces (problems in two independent variables) 49Q20 Variational problems in a geometric measure-theoretic setting Keywords:area minimizing integral current; co-area formula; area minimizing surfaces; m-energy minimizing mappings; liquid crystals Citations:Zbl 0631.00004 PDFBibTeX XML