On the Problem of Plateau / Subharmonic Functions

A convex function f may be called sublinear in the following sense; if a linear function l is ::=: j at the boundary points of an interval, then l:> j in the interior of that interval also. If we replace the terms interval and linear junction by the terms domain and harmonic function, we obtain a statement which expresses the characteristic property of subharmonic functions of two or more variables. This ge­ neralization, formulated and developed by F. RIEsz, immediately at­ tracted the attention of many mathematicians, both on account of its intrinsic interest and on account of the wide range of its applications. If f (z) is an analytic function of the complex variable z = x + i y. then If (z) I is subharmonic. The potential of a negative mass-distribu­ tion is subharmonic. In differential geometry, surfaces of negative curvature and minimal surfaces can be characterized in terms of sub­ harmonic functions. The idea of a subharmonic function leads to significant applications and interpretations in the fields just referred to, and· conversely, every one of these fields is an apparently in­ exhaustible source of new theorems on subharmonic functions, either by analogy or by direct implication.

Inhalt

I. Curves and surfaces.- II. Minimal surfaces in the small.- III Minimal surfaces in the large.- IV. The non-parametric problem.- V. The problem of Plateau in the parametric form.- VI. The simultaneous problem in the parametric form. Generalizations.- I. Definition and preliminary discussion.- II. Integral means.- III. Criterions and constructions.- IV. Examples.- V. Harmonic majorants.- VI. Representation in terms of potentials.- VII. Analogies between harmonic and subharmonic functions.- References.