On total curvature of immersions and minimal submanifolds of spheres
Commentationes Mathematicae Universitatis Carolinae (1993)
- Volume: 34, Issue: 3, page 459-463
- ISSN: 0010-2628
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topRotondaro, Giovanni. "On total curvature of immersions and minimal submanifolds of spheres." Commentationes Mathematicae Universitatis Carolinae 34.3 (1993): 459-463. <http://eudml.org/doc/247484>.
@article{Rotondaro1993,
abstract = {For closed immersed submanifolds of Euclidean spaces, we prove that $\int |\mu |^2\, dV\ge V/R^2$, 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.},
author = {Rotondaro, Giovanni},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {closed submanifold; total mean curvature; minimal submanifold; Willmore conjecture; total mean curvature},
language = {eng},
number = {3},
pages = {459-463},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On total curvature of immersions and minimal submanifolds of spheres},
url = {http://eudml.org/doc/247484},
volume = {34},
year = {1993},
}
TY - JOUR
AU - Rotondaro, Giovanni
TI - On total curvature of immersions and minimal submanifolds of spheres
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 3
SP - 459
EP - 463
AB - For closed immersed submanifolds of Euclidean spaces, we prove that $\int |\mu |^2\, dV\ge V/R^2$, 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.
LA - eng
KW - closed submanifold; total mean curvature; minimal submanifold; Willmore conjecture; total mean curvature
UR - http://eudml.org/doc/247484
ER -
References
top- Benson R., Euclidean Geometry and Convexity, Mc Graw-Hill, New York, 1966. Zbl0187.44103MR0209949
- Chen B.-Y., Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapore, 1984. Zbl0537.53049MR0749575
- Chern S.S., Hsiung C.C., On the isometry of compact submanifolds in Euclidean space, Math. Ann. 149 (1962/63), 278-285. (1962/63) MR0148011
- Kühnel W., A lower bound for the -th total absolute curvature of an immersion, Colloq. Math. 41 (1969), 253-255. (1969) MR0591931
- Reilly R., On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comm. Math. Helv. 52 (1977), 525-533. (1977) Zbl0382.53038MR0482597
- Spivak M., A Comprehensive Introduction to Differential Geometry, Vol. I-V, Publish or Perish, Berkeley, 1970-1979. Zbl0439.53005MR0532830
- Weiner J.L., An inequality involving the length, curvature and torsions of a curve in Euclidean -space, Pacific J. Math. 74 (1978), 531-534. (1978) Zbl0377.53001MR0478025
- Willmore T.J., Note on embedded surfaces, An. St. Univ. Iasi, s.I.a. Mat. 12B (1965), 493-496. (1965) Zbl0171.20001MR0202066
- Willmore T.J., Tight immersions and total absolute curvature, Bull London Math. Soc. 3 (1971), 129-151. (1971) Zbl0217.19001MR0292003
- Willmore T.J., Total Curvature in Riemannian Geometry, Ellis Horwood Limited, Chichester, 1982. Zbl0501.53038MR0686105
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