Area measure and Radó’s lower area

Abstract:

The theory of Geöcze area for two-dimensional surfaces in threedimensional space had been essentially completed by the mid 1950’s. The only hypothesis needed for all theorems in this case is the finiteness of the area. See [2] for an account of this theory. In the early 1960’s, H. Federer established, in his paper [6], fundamental facts concerning his integral geometric area for higher dimensional area theory by employing the theory of normal and integral currents. These facts employ not only the finiteness of area as a basic hypothesis but certain other hypotheses as well. The extensions of Geöcze type area to higher dimensions also employ not only the finiteness of area but certain added hypotheses. These hypotheses are of such a nature as to allow the use of the theory of quasi-additivity [3], [11]. The present paper concerns these added hypotheses which play such an important part of higher-dimensional area theory of today. It is shown that Radó’s lower area is the best Geöcze type area to describe these added hypotheses. That is, it is shown that the quasi-additivity hypotheses of Geöcze area in [11] imply the quasi-additivity hypotheses of lower area. Second, it is shown that the quasi-additivity hypotheses for lower area imply that the surface has the essential cylindrical property defined by J. Breckenridge in [5]. This essential cylindrical property is proved to be equivalent to the existence of area measures on the middle space of the mapping representing the surface. Finally, it is shown that the essential cylindrical property of a surface is equivalent to the quasi-additivity condition for lower area. Thus, an intrinsic property of the surface characterizes the quasi-additivity condition for the lower area of a surface.