On the Existence of a Closed Convex Surface Realizing a Given Riemannian Metric.

1 This problem corresponds to Douglas' generalization of Plateau's original problem, cf. the papers by Douglas, Journ. Math. Phys., 15, 55 ff (Feb., 1936) and 106 ff (June, 1936) and Courant, these PROCEEDINGS, June, 1936, 369 ff and Ann. Math., 38, 679 ff (1937). For the case of boundary planes, cf. also the doctoral thesis by E. Ritter, soon to appear. 2 It is noteworthy that the Lemma and the following Theorem subsist if instead of harmonic functions general monotonic functions of u and v are admitted. ' Incidentally, at least a subsequence of the 9, satisfying this last condition can always be found. 4 A knowledge of the first two sections of the paper in the Ann. Math. (see footnote 1) is required for the following. ' Cf. the similar situation with the boundary value problem of elliptic partial linear differential equations. See Courant-Hilbert, Meth. math. Physik, Vol. II, Chapter 7.