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Consider a configuration in Euclidean 3-space consisting of a surface T and of a rectifiable Jordan arc r = { j = j ( r ) ; 0 g r g l } having its end points on T, but no other point in common with T. Denote by P the semidisc in the (w, ü)-plane P = {w, v; u+v <1 , fl>0}, by d'P and d"P its boundary portions {u, v; u+v=l, v>0} and {u,v; 1 < « < 1 , » = 0) , respectively, and by P the domain PUd'P. A surface 5 = {f = f(w, v); (u, v)ÇzP'} is said to be bounded by the above configuration, or chain (I\ T), if its position vector f(w, v) = {x(uf v), y(u, v), z(u, v)} satisfies the following conditions: (0 t(u,v)ec»(P'). (ii) i(u, v) maps the arc d'P onto the open arc ( r ) = { ^ = j(r); 0 < r < 1} monotonically in such a way that lim j(cos 0, sin 6) = $(0), Urn j(cos 6, sin 6) = g(l).

Surfaces of least area with partially free boundary on a manifold satisfying the chord-arc condition