Langevin, R.; Levitt, G.; Rosenberg, H. Complete minimal surfaces with long line boundaries. (English) Zbl 0637.53007 Duke Math. J. 55, 985-995 (1987). The authors prove Bernstein-type theorems for complete minimal surfaces bounded by lines in \({\mathbb{R}}^ 3.\) For example, if M is a complete minimal surface whose interior is a graph over a square J in the (x,y) plane, and if the boundary of M is composed of the four vertical lines over the vertices of J, then M is Scherk’s surface. Reviewer: F.Gackstatter Cited in 4 Documents MSC: 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature Keywords:Bernstein-type theorems; minimal surfaces PDF BibTeX XML Cite \textit{R. Langevin} et al., Duke Math. J. 55, 985--995 (1987; Zbl 0637.53007) Full Text: DOI OpenURL References: [1] M. Beeson, The behavior of a minimal surface in a corner , Arch. Ration. Mech. Anal. 65 (1977), no. 4, 379-393. · Zbl 0371.53003 [2] G. Darboux, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal, Vol. 1 , Chelsea Publishing Co., Bronx, New York, 1972. · Zbl 0257.53001 [3] H. Jenkins and J. Serrin, Variational problems of minimal surface type. II. Boundary value problems for the minimal surface equation , Arch. Rational Mech. Anal. 21 (1966), 321-342. · Zbl 0171.08301 [4] R. Osserman, Global properties of minimal surfaces in \(E^{3}\) and \(E^{n}\) , Ann. of Math. (2) 80 (1964), 340-364. JSTOR: · Zbl 0134.38502 [5] B. Riemann, Ueber die Fläche vom kleinsten Inhalt bei gegebner Begrenzung , Œuvres complètes, Mémoires de la Société Royale de Goettingen, vol. 13, 1867, p. 305. [6] H. Rosenberg, Deformations of complete minimal surfaces , Trans. Amer. Math. Soc. 295 (1986), no. 2, 475-489. JSTOR: · Zbl 0598.53004 [7] J. Serret, Sur la moindre surface comprise entre des lignes droites données, non situeés dans le même plan , Compt. Rend. Acad. Sc. 40 (1855), 1078-1083. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.