Pacific Journal of Mathematics

Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms.

Gary Lawlor and Frank Morgan

Article information

Pacific J. Math., Volume 166, Number 1 (1994), 55-83.

First available in Project Euclid: 8 December 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58E12: Applications to minimal surfaces (problems in two independent variables) [See also 49Q05]
Secondary: 49Q05: Minimal surfaces [See also 53A10, 58E12]


Lawlor, Gary; Morgan, Frank. Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms. Pacific J. Math. 166 (1994), no. 1, 55--83.

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  • [Al] Manuel Alfaro, Mark Conger, Kenneth Hodges, Adam Levy, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, and Karen von Haam, Segments can meet in fours in energy-minimizing networks, J. of Undergraduate Math., 22 (1990), 9-20 .
  • [A2] Manuel Alfaro, Mark Conger, Kenneth Hodges, Adam Levy, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, and Karen von Haam, The structure of singularities in -minimizing networks in R2, Pacific J. Math., 149 (1991), 201-210.
  • [A3] Manuel Alfaro, Existence of shortest directed networks in R2, Pacific J. Math., to appear.
  • [A] F. J. Almgren, Jr., Existence and regularity almost everywhere of solu- tions to elliptic variational problems with constraints, Memoirs Amer. Math. Soc, 165 (1976), 1-199.
  • [AT] F. J. Almgren, Jr. and Jean Taylor, The geometry of soap films and soap bubbles, Sci. Amer., 235 (1976), 82-93.
  • [B] Leonard M. Blumenthal, Theory and Applications of Distance Geometry, 2nd ed., Chelsea, New York, 1970
  • [Bl] Kenneth Brakke, Minimal cones on hypercubes, J. Geom. Anal., 1 (1991), 329-338.
  • [B2] Kenneth Brakke, Soap films and covering spaces, to appear in J. Geom. Anal, (also Minnesota Geometry Center research report GCG54).
  • [B3] Kenneth Brakke, The Surface Evolver, Experimental Mathematics, 1 (1992), 141-165.
  • [CG] G. D. Chakerian and M. A. Ghandehari, The Fermat problem in Minkowski spaces, Geom. Dedicata, 17 (1985), 227-238.
  • [Coc] E. J. Cockayne, On the Steiner problem, Canad. Math. Bull., 10 (1967), 431-450.
  • [Con] Mark Conger, Energy-minimizing networks in Rn, Honors thesis, Williams College, 1989, expanded 1989.
  • [CR] R. Courant and H. Robbins, What is Mathematics?, Oxford Univ. Press, New York, 1941.
  • [F] Federer, H., Geometric Measure Theory, New York, Springer-Verlag, 1969.
  • [FLM] Z. Fiiredi, J. C. Lagarias, and F. Morgan, Singularities of minimal sur- faces and networks and related problems in Minkowski space, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol.
  • [G] Branko Griinbaum, Strictly antipodal sets, Israel J. Math., 1 (1963), 5-10.
  • [H] M. Hanan, On Steiner's problem with rectilinear distance, J. SIAM Appl. Math., 14 (1966) 255-265.
  • [K] Robert B. Kusner, in Richard K. Guy, An olla-podrida of open problems, often oddly posed, Amer. Math. Monthly, 90 (1983), 196-199.
  • [LM] Gary Lawlor and Frank Morgan, Minimizing cones and networks: im- miscible fluids, norms, and calibrations, Extended abstracts of lectures presented at the AMS special session, "Computing Optimal Geometries", AMS Selected Lectures in Math. (Jean E Taylor, ed.), 1991, 27.
  • [L] Adam Levy, Energy-minimizing networks meet only in threes, J. of Un- dergraduate Math., 22 (1990), 53-59.
  • [Ml] Frank Morgan, Area-minimizing surfaces, faces of Grassmannians, and calibrations, Amer. Math. Monthly, 95 (1988), 813-822.
  • [M2] Frank Morgan, Calibrations and new singularities inarea-minimizing surfaces: a survey, in Henri Berestycki, Jean-Michel Coron, and Ivar Ekeland, ed., Variational Methods, Prog. Nonlin. Diff. Eqns., Vol. 4 (Proc. conf., Paris, 1988), Birkhauser, Boston, 1990, pp. 329-342.
  • [M3] Frank Morgan, The cone over the Clifford iorus in R4 is^-minimizing, Math. Ann., 289 (1991), 341-354; Announced in A sharp counterexample on the regularity of ^-minimizinghypersurf aces, Bull. Amer.Math. Soc, 22 (1990), 295-299.
  • [M4] Frank Morgan, Geometric Measure Theory: a Beginner's Guide, Aca- demic Press, New York, 1988; Second edition, 1995.
  • [M5] Frank Morgan, Minimal surfaces, crystals, and norms on Rn, Proc. 7th Annual Symp. on Computational Geom., June, 1991, N. Conway, NH.
  • [M6] Frank Morgan, Minimal surfaces, crystals, shortest networks, and un- dergraduate research, Math. Intel., 14 (1992), 37-44.
  • [M7] Frank Morgan, Riemannian Geometry: a Beginner's Guide, Jones and Bartlett, 1992
  • [M8] Frank Morgan, Size-minimizing rectifiable currents, Inventiones math., 96 (1989), 333-348.
  • [M9] Frank Morgan, Compound soap bubbles, shortest networks, and minimal surfaces, video of AMS-MAA address, San Francisco, 1991.
  • [M10] Frank Morgan, Soap bubbles and soap films, in Joseph Malkevitch and Donald McCarthy, ed., Mathematical Vistas: New and Recent Publica- tions in Mathematics from the New York Academy of Sciences, vol. 607, 1990.
  • [Mil] Frank Morgan, Strict calibrations, constant mean curvature, and triple junctions, preprint (1994).
  • [P] CM. Petty, Equilateral sets in Minkowski spaces, Proc. Amer. Math. Soc, 29 (1971), 369-374.
  • [S] I. J. Schoenberg, Remarks to Maurice FrecheVs articleuSur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vec- toriellement sur espace de Hubert", Ann. of Math., 36 (1935), 724-732.
  • [Tl] Jean Taylor, The structure of singularities in soap-bubble-like and soap- film-like minimal surfaces, Ann. Math., 103 (1976), 489-539.
  • [T2] Jean Taylor, The structure of singularities in solutions to ellipsoidal vari- ational problems with constraints in R3, Ann. Math., 103 (1976), 541- 546.
  • [Wl] Brian White, Existence of least-area mappings of N-dimensional do- mains, Annals Math., 118 (1983), 179-185.
  • [W2] Brian White, Regularity of the singular sets in immiscible fluid interfaces and solutions to other Plateau-type problems, Proc. Center Math. Anal., Australian Natl. Univ., 10 (1985), 244-249.