This issue is proffesor H. Hopf. He drew our attention to the connection between differential geometry and potential theory which is revealed by relations (1.3) and (1.4). For example, the function u(x,y) is subharmonic in a certain (x,у)-parameter region if and only if in the corresponding domain on M. This fact had already been used by E. F. Beckenbach and T. Rado in their proof of the isoperimetric inequality on surfaces of negative curvature. Analogously, и is super-harmonic if and only if . Furthermore, (1.4) discloses an even deeper connection: The surface integral of К, considered as a set function, is essentially the measure associated with и. Consequently, results of differential geometry in the large involving the curvatura integra, such as those due to S. Cohn- Vossen, F. Fiala, Ch. Blanc and F. Fiala, have a potentialtheoretical meaning. It is therefore natural to apply functiontheoretical methods to this field in the hope that not only other (and eventually simpler) proofs of known results will be found, but also theorems which are new in both their differential geometrical and potential-theoretical aspects.When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/35035
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