Frank Morgan

The proof of the regularity of an area-minimizing surface with a given smooth boundary has had a long and interesting history. Following work of Douglas, Rado, and Osserman, Gulliver [1973, G] proved that a least-area map of a disc into R³ with prescribed boundary is a smooth immersion on the interior. The map need not be an embedding and cannot be if the boundary curve is knotted. The possibility of boundary branch points remains open today. Contemporaneous with results in codimension 1 by De Giorgi [DeG] and the seminal paper of Federer and Fleming [FF], in a measure-theoretic tour de force Reifenberg [1960, R] had proved the existence and almost-everywhere regularity of m-dimensional least-area sets in Rⁿ, with given boundary in the strong sense of algebraic geometry. Almgren [1968, Alm1] generalized Reifenberg’s results from area to some more general integrands, in particular extending the results from Rⁿ to other smooth ambient manifolds. Using Almgren’s new theory of varifolds, Allard [1972, A] extended almost-everywhere regularity to sets of weakly bounded mean curvature. Everywhere interior regularity for area-minimizing hypersurfaces was proved through ambient dimension 7 by Fleming, Almgren, and Simons, with small singular sets possible in higher dimensions (see [M1, Chapts. 8 and 10]). Boundary regularity was proved in R³ by Allard [1969, Alm2] for unoriented surfaces and in Rⁿ by Hardt and Simon [1979, HS] for oriented hypersurfaces. All of these results generalize immediately to volume constraints, to smooth ambient manifolds, and to manifolds with density ([M1, §8.5], [M2]). For more general soap-film-like and soap-bubble-like surfaces, with codimension-1 singularities and multiple volume constraints, existence and almost-everywhere regularity was proved by Almgren [Alm], with a classification of the two types of interior singularities in R³ by Taylor [T] after Plateau and Lamarle.

[A] William K. Allard, On boundary regularity for Plateau’s problem, Bull. Amer. Math Soc. 75 (1969), 522–523.

[Alm1] F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math 87 (1968), 321–391.

[Alm2] F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. No. 165 (1976). Because the regularity theory herein treats other integrands and hence specifically applies to general ambient manifolds, it is often cited instead of simpler, more appropriate papers, which unfortunately do not explicitly state that their results apply in general smooth ambients; see [M1, §8.5].

[DeG] E. De Giorgi, Frontiere Orientate di Misura Minima (Sem. Mat. Scuola Norm. Sup. Pisa, 1960-61), Editrice Tecnico Scientifica, Pisa, 1961.

[FF] Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. Math. 72 (1960), 458-520.

[G]  Robert D. Gulliver II, Regularity of minimizing surfaces of prescribed mean curvature, Ann. of Math. 97 (1973), 275–305.

[HS] Robert Hardt and Leon Simon, Boundary regularity and embedded solutions for the oriented Plateau problem, Ann. of Math. 110 (1979), 439-486.

[M1] Frank Morgan, Geometric Measure Theory: a Beginner’s Guide, Academic Press, 4th ed., 2009.

[M2] Frank Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc. 355 (2003) 5041-5052.

[R] E. R. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Bull. Amer. Math. Soc. 66 (1960), 312–313.

[T] Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. 103  (1976), 489–539.