an:04052621
Zbl 0645.58015
Almgren, F.; Browder, W.; Lieb, E. H.
Co-area, liquid crystals, and minimal surfaces
EN
Partial differential equations, Proc. Symp., Tianjin/China 1986, Lect. Notes Math. 1306, 1-22 (1988).
1988
a
58E12 49Q20
area minimizing integral current; co-area formula; area minimizing surfaces; m-energy minimizing mappings; liquid crystals
[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.
P.Mattila
Zbl 0631.00004