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The Most General Form of the Problem of Plateau

1939; Johns Hopkins University Press; Volume: 61; Issue: 3 Linguagem: Inglês

10.2307/2371315

1080-6377

Jesse Douglas,

Computational Geometry and Mesh Generation

S 5 [8 ], formula (7.17); [9 1, formula (2); for the case h = 0, [2], formula (2).6 The dot denotes the scalar product of vectors.The exponent two (formulas 14, 15) will denote the scalar product of a vector by itself, i.e., the sum of the squares of its com- ponents.7 Except that we interpret a as a derivative, rather than a differential, as is more cus- tomary.8 Formulas ( 41) and (33) of [2], reproduced as (34) and (37) of the foregoing note in the present issue of these PROCEEDINGS.This is for a general topological form of the Riemann surface B. With increasing complexity of It, the appropriate identity involved successively: algebraic functions, trigonometric functions, elliptic functions, 0-functions.See, respectively [4], p. 243; [7], formula (5.4); [5], formula (7.1); [2], formulas (41), (33).In [7], the functions actually appearing are hyperbolic, due to the rotation through a right angle of the parallel strip representing the Riemann surface R.Given in [3], and in preceding abstracts in Bull.Amer.Math.Soc., 36, 50 (1930). 10We suppose that the contact of the tangents t, t to A at Q, Q, respectively, is ordi- nary two-point contact.Higher contact can always be avoided by a preliminary bi- rational transformation.11 Or its conformally equivalent orthogonal projections It = Rt, or itv (see art. 2).The notation is supposed arranged so that the semi-surface R contains the point Q.12 We may again remark that this one-to-one character breaks down in the vicinity of the points of contact Q, Q of the tangents t, t.Otherwise, it extends beyond the real branches to the rest of the Riemann surfaces involved, with deletion of the stated neigh- borhoods.