Pacific Journal of Mathematics

Soap bubbles in ${\bf R}^2$ and in surfaces.

Frank Morgan

Article information

Pacific J. Math., Volume 165, Number 2 (1994), 347-361.

First available in Project Euclid: 8 December 2004

Permanent link to this document

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] 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]


Morgan, Frank. Soap bubbles in ${\bf R}^2$ and in surfaces. Pacific J. Math. 165 (1994), no. 2, 347--361.

Export citation


  • [All] W.K. Allard, On the first variation of a varifold, Annals of Math., 95 (1972), 417-491.
  • [Aim] F.J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs AMS, 4:165 (1976).
  • [B] M.H. Bleicher, Isoperimetric division into a finite number of cells in the plane, Studia Scien. Math. Hung., 22 (1987), 123-137.
  • [CHK] C.Cox, L. Harrison, M. Hutchings, S. Kim, J. Light, A. Mauer and M. Tilton, The shortest enclosure of three connected areas in M2, Real Anal. Exch., to appear.
  • [Fe] H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.
  • [Fo] J. Foisy, M. Alfaro, J. Brock, N. Hodges and J. Zimba, The standard double soap bubble in W2 uniquely minimizes perimeter, Pacific J. Math., 159 (1993), 47-59.
  • [HM] J. Hass and F. Morgan, Geodesies and soap bubbles in surfaces, preprint (1991).
  • [Ho] H. Howards, Soap bubbles on surfaces, Honors thesis, Williams College, 1992.
  • [Ml] F. Morgan, Compound soap bubbles, shortest networks, and minimal sur- faces, AMS video, 1992.
  • [M2] F. Morgan, Geometric Measure Theory:a Beginner's Guide, Academic Press, second edition, 1995.
  • [M3] F. Morgan, Mathematicians,including undergraduates, look at soap bub- bles, Amer. Math. Monthly, 101 (1994), 343-351.
  • [M4] F. Morgan, Riemannian Geometry: a Beginner's Guide, A.K. Peters, Welles-
  • [M5] F. Morgan, Soap bubbles and soap films, in Joseph Malkevitch and Donald McCarthy, ed, Mathematical Vistas: New and Recent Publications in Mathematics from the New York Academy of Sciences, 607 (1990), 98- 106.
  • [M6] F. Morgan, (M,,)-mnimalcurve regularity, Proc. AMS, 120 (1994), 677-686.
  • [MS] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure and Applied Math., 42 (1989), 577-685.
  • [P] H. Poincare, Sur les lignes geodesiques des surfaces convexes, Trans. AMS, 6 (1905), 237-274.
  • [Tl] J.E. Taylor, The structure of singularities in soap-bubble-like and soap- film-like minimal surfaces, Ann. Math., 103 (1976), 489-539.
  • [T2] J.E. Taylor, The structure of singularities in solutions to ellipsoidal vari- ational problems with constraints in 1R3, Ann. Math., 103 (1976), 541- 546.