Solution of the Problem of Plateau*

ion being made, in case the representation g is improper of the first kind, of the values of 0, at most denumerably infinite in number, where gi(8) is discontinuous. (19.9) rests on the fact (whose proof is trivial) that if fiim)(l) tends uniformly to the continuous f((t) when m—>, then if tm—H as m—>«>, we have lim/i(m)(í„) =/i(i) for m—>°o. The assertion is now easily proved that if (19.10) xí = SRFi(w) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 304 JESSE DOUGLAS [January are the harmonic functions determined by g,(0), then (19.11) ¿F/2(w)=0, i=l so that the surface (19.10) is minimal. For consider (18.2) without the factor w: tmy 1 r 2e*6 (19.12) p\> (w) = —--,«(-) (B)d8. 2t Jc (e" — w)2 Since all the polygons r(m) are contained in a finite region of space, the functions g>(m)(0) are uniformly bounded; and if w is any fixed point interior to the unit circle, the denominator {e^—w)2 remains superior in absolute value to a fixed positive quantity when ew describes C. Therefore the integrand in (19.12) remains uniformly bounded during the limit process (19.9); consequently the limit of the integral is equal to the integral of the limit: (19.13) lim FTM'(w) =F¡(w). It is evident that in case g is improper of the first kind this result is not affected by the circumstance that the points of discontinuity of g,(0) are not considered in the limit relation (19.9), since these points, being at most denumerably infinite in number, form a set of zero measure. The result (19.11) now follows from (19.13) and the subsistence of (19.5) for every m. 20. The minimal surface is bounded by T. To show that the minimal surface whose existence is proved in the preceding section is bounded by T, we must prove that the representation (19.8) of T is proper. That it cannot be improper of the second kind is proved in §18, which, being based on the relation (19.11), applies here with full validity. We cannot however apply the argument of §17 to prove that (19.8) cannot be improper of the first kind. For although we would still have for a g of this kind A (g) = 4°o, it would not be true in the case of a general Jordan contour that A (g) sometimes takes finite values. We therefore use the following argument, based on the relation (19.11), to obtain the desired result. Suppose that under g the point P of C corresponds to the arc Q'Q" of T. Since T is a Jordan curve, Q' and Q" are distinct: and if fl< denote the coordinates of Q', Z>< of Q", the distance Q'Q" or / with (20.1) I2 = ¿(i* ad2 is not equal to zero. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1931] THE PROBLEM OF PLATEAU 305 There is no loss of generality in supposing P to be at w = 1, for this may be achieved by a rotation of the unit circle, which changes nothing essential.