| Title:
|
On total curvature of immersions and minimal submanifolds of spheres (English) |
| Author:
|
Rotondaro, Giovanni |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
3 |
| Year:
|
1993 |
| Pages:
|
459-463 |
| . |
| Category:
|
math |
| . |
| Summary:
|
For closed immersed submanifolds of Euclidean spaces, we prove that $\int |\mu |^2\, dV\geq V/R^2$, where $\mu $ is the mean curvature field, $V$ the volume of the given submanifold and $R$ is the radius of the smallest sphere enclosing the submanifold. Moreover, we prove that the equality holds only for minimal submanifolds of this sphere. (English) |
| Keyword:
|
closed submanifold |
| Keyword:
|
total mean curvature |
| Keyword:
|
minimal submanifold |
| MSC:
|
53A05 |
| MSC:
|
53C40 |
| MSC:
|
53C42 |
| MSC:
|
53C45 |
| MSC:
|
58E12 |
| idZBL:
|
Zbl 0787.53049 |
| idMR:
|
MR1243078 |
| . |
| Date available:
|
2009-01-08T18:05:20Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118603 |
| . |
| Reference:
|
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| Reference:
|
[2] Chen B.-Y.: Total Mean Curvature and Submanifolds of Finite Type.World Scientific, Singapore, 1984. Zbl 0537.53049, MR 0749575 |
| Reference:
|
[3] Chern S.S., Hsiung C.C.: On the isometry of compact submanifolds in Euclidean space.Math. Ann. 149 (1962/63), 278-285. MR 0148011 |
| Reference:
|
[4] Kühnel W.: A lower bound for the $i$-th total absolute curvature of an immersion.Colloq. Math. 41 (1969), 253-255. MR 0591931 |
| Reference:
|
[5] Reilly R.: On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space.Comm. Math. Helv. 52 (1977), 525-533. Zbl 0382.53038, MR 0482597 |
| Reference:
|
[6] Spivak M.: A Comprehensive Introduction to Differential Geometry.Vol. I-V, Publish or Perish, Berkeley, 1970-1979. Zbl 0439.53005, MR 0532830 |
| Reference:
|
[7] Weiner J.L.: An inequality involving the length, curvature and torsions of a curve in Euclidean $n$-space.Pacific J. Math. 74 (1978), 531-534. Zbl 0377.53001, MR 0478025 |
| Reference:
|
[8] Willmore T.J.: Note on embedded surfaces.An. St. Univ. Iasi, s.I.a. Mat. 12B (1965), 493-496. Zbl 0171.20001, MR 0202066 |
| Reference:
|
[9] Willmore T.J.: Tight immersions and total absolute curvature.Bull London Math. Soc. 3 (1971), 129-151. Zbl 0217.19001, MR 0292003 |
| Reference:
|
[10] Willmore T.J.: Total Curvature in Riemannian Geometry.Ellis Horwood Limited, Chichester, 1982. Zbl 0501.53038, MR 0686105 |
| . |
