A global invariant of conformal mappings in space
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- by James H. White PDF
- Proc. Amer. Math. Soc. 38 (1973), 162-164 Request permission
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
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W. Blaschke, Vorlesungen über Differentialgeometrie. III, Springer, Berlin, 1929.
B.-Y. Chen, On a variational problem of hypersurfaces (mimeograph).
- Katsuhiro Shiohama and Ryoichi Takagi, A characterization of a standard torus in $E^{3}$, J. Differential Geometry 4 (1970), 477–485. MR 276906
- 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 202066
Additional Information
- © Copyright 1973 American Mathematical Society
- 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