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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A global invariant of conformal mappings in space


Author: James H. White
Journal: Proc. Amer. Math. Soc. 38 (1973), 162-164
MSC: Primary 53C45
DOI: https://doi.org/10.1090/S0002-9939-1973-0324603-1
MathSciNet review: 0324603
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Abstract: This paper shows that the total integral of the square of the mean curvature for a compact orientable surface in $ {E^3}$ is an invariant of a conformal space mapping. This result is then used to answer a problem raised by T. Willmore and B.-Y. Chen concerning embeddings of compact orientable surfaces, and in particular tori, for which this integral is a minimum.


References [Enhancements On Off] (What's this?)

  • [1] W. Blaschke, Vorlesungen über Differentialgeometrie. III, Springer, Berlin, 1929.
  • [2] B.-Y. Chen, On a variational problem of hypersurfaces (mimeograph).
  • [3] Katsuhiro Shiohama and Ryoichi Takagi, A characterization of a standard torus in 𝐸³, J. Differential Geometry 4 (1970), 477–485. MR 0276906
  • [4] T. J. Willmore, Note on embedded surfaces, An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 11B (1965), 493–496 (English, with Romanian and Russian summaries). MR 0202066

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0324603-1
Keywords: Conformal mappings in space, inversion, mean curvature, global invariant, anchor ring, torus
Article copyright: © Copyright 1973 American Mathematical Society