A uniqueness theorem for properly embedded minimal surfaces bounded by straight lines (Q2707065)

The Plateau problem of determining minimal surfaces spanned by a given boundary has occupied the attention of geometers for quite some time. The results on minimal surfaces bounded by a given polygon with sides possibly of infinite length belong to the classical minimal surface theory of the nineteenth century inaugurated through the works of Schwarz, Weierstrass and Riemann. One of the fundamental questions about the Plateau problem is that of uniqueness, or more broadly, to determine how many distinct minimal surfaces can be spanned by a given boundary. NEWLINENEWLINENEWLINEIn 1966 \textit{H. Jenkins} and \textit{J. Serrin} proved an existence and uniqueness theorem for minimal graphs bounded by straight lines [Arch. Rat. Mech. Anal. 21, 321-342 (1966; Zbl 0171.08301)]. Recently, \textit{F. J. Lopez} and \textit{F. Martin} constructed a deformation of some particular Jenkins-Serrin graphs which consists of properly embedded minimal discs bounded by straight lines, contained in a wedge of a slab [Minimal surfaces in a wedge of a slab, Commun. Anal. Geom. 9, 683-723 (2001)]. NEWLINENEWLINENEWLINEThe purpose of the paper under review is to obtain a uniqueness theorem for the minimal surfaces in the family of examples exhibited in the work of the authors just mentioned. In the problems considered, the generalized Plateau's problem as formulated in the paper may have a unique solution, exactly two solutions or no solution at all.