Conelike soap films spanning tetrahedra

Huff, Robert

doi:10.1090/S0002-9947-2010-04899-8

Conelike soap films spanning tetrahedra
HTML articles powered by AMS MathViewer

by Robert Huff PDF

Trans. Amer. Math. Soc. 362 (2010), 5063-5081 Request permission

Abstract:

In this paper we provide the first examples of non-flat soap films proven to span tetrahedra. These are members of a continuous two parameter family of soap films with tetrahedral boundaries. Of particular interest is a two parameter subfamily where each spanning soap film has the property that two minimal surfaces meet along an edge of the boundary at an angle greater than $120^\circ$.

References

Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165, viii+199. MR 420406, DOI 10.1090/memo/0165
C. Caratheodory, Theory of functions of a complex variable. Vol. 2, Chelsea Publishing Co., New York, 1954. Translated by F. Steinhardt. MR 0064861
Manfredo P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. Translated from the Portuguese. MR 0394451
David Hoffman and Hermann Karcher, Complete embedded minimal surfaces of finite total curvature, Geometry, V, Encyclopaedia Math. Sci., vol. 90, Springer, Berlin, 1997, pp. 5–93. MR 1490038, DOI 10.1007/978-3-662-03484-2_{2}
Robert Huff and John McCuan, Scherk-type capillary graphs, J. Math. Fluid Mech. 8 (2006), no. 1, 99–119. MR 2205153, DOI 10.1007/s00021-004-0140-8
Robert Huff, Soap films and Kelvin’s curved, truncated octahedron, J. Geom. Anal. 15 (2005), no. 3, 425–443. MR 2190240, DOI 10.1007/BF02930980
Robert Huff, Soap films spanning rectangular prisms, Geom. Dedicata 123 (2006), 223–238. MR 2299736, DOI 10.1007/s10711-007-9128-x
Gary Lawlor and Frank Morgan, Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms, Pacific J. Math. 166 (1994), no. 1, 55–83. MR 1306034
Tibor Radó, On the problem of Plateau. Subharmonic functions, Springer-Verlag, New York-Heidelberg, 1971. Reprint. MR 0344979
Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489–539. MR 428181, DOI 10.2307/1970949