Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-03T01:45:05.344Z Has data issue: false hasContentIssue false
  • English
  • Français

Tangential properties of Fréchet surfaces

Published online by Cambridge University Press:  24 October 2008

H. G. Eggleston
H. G. Eggleston
Affiliation:
7 Hauxton Road, Trumpington, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

The first two papers of Reifenberg ((4), (5)) under the general heading ‘Parametric Surfaces’ contain a detailed and profound study of the tangential properties of these surfaces. Since their publication one of the fundamental problems in the subject, that of obtaining a convenient representation for the surface, has been solved by Cesari (1). In this paper we obtain direct proofs of Reifenberg's results from Cesari's theorem. Whereas Reifenberg had to contend with both topological and real-variable problems combined the effect of Cesari's theorem is to remove the topological difficulties and to leave a straightforward real variable problem. The definition of approximate tangential plane used here is not the same as either of the two employed by Reifenberg, but the differences between it and one of Reifenberg's definitions ((5), definition 2) are not very important.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 2 , April 1958 , pp. 187 - 196
Copyright
Copyright © Cambridge Philosophical Society 1958

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Cesari, L.Surface area (Princeton, 1956).Google Scholar
(2)Federer, H.Measure and area. Bull. Amer. Math. Soc. 58 (1952), 306–78.CrossRefGoogle Scholar
(3)Morrey, C. B.An analytic characterization of surfaces of finite Lebesgue area. Amer. J. Math. 57 (1935), 692–702 and 58 (1936), 312–22.CrossRefGoogle Scholar
(4)Reifenberg, E. R.Parametric surfaces. I. Proc. Camb. Phil. Soc. 47 (1951), 687–98.CrossRefGoogle Scholar
(5)Reifenberg, E. R.Parametric surfaces. II. Proc. Camb. Phil. Soc. 48 (1952), 46–69.CrossRefGoogle Scholar
(6)Youngs, J. W.The representation problem for Fréchet surfaces. Memoirs Amer. Math. Soc. 8 (1951).Google Scholar