| Title:
|
A remark on almost umbilical hypersurfaces (English) |
| Author:
|
Roth, Julien |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
49 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
1-7 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this article, we prove new stability results for almost-Einstein hypersurfaces of the Euclidean space, based on previous eigenvalue pinching results. Then, we deduce some comparable results for almost umbilical hypersurfaces. (English) |
| Keyword:
|
hypersurfaces |
| Keyword:
|
rigidity |
| Keyword:
|
pinching |
| Keyword:
|
Ricci curvature |
| Keyword:
|
umbilicity tensor |
| Keyword:
|
higher order mean curvatures |
| Keyword:
|
$\theta $-quasi-isometry |
| MSC:
|
53A07 |
| MSC:
|
53A10 |
| MSC:
|
53C20 |
| MSC:
|
53C24 |
| idZBL:
|
Zbl 06321142 |
| idMR:
|
MR3073010 |
| DOI:
|
10.5817/AM2013-1-1 |
| . |
| Date available:
|
2013-05-28T13:23:47Z |
| Last updated:
|
2014-07-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143292 |
| . |
| Reference:
|
[1] Aubry, E.: Finiteness of $\pi _1$ and geometric inequalities in almost positive Ricci curvature.Ann. Sci. École Norm. Sup. (4) 40 (4) (2007), 6757–695. Zbl 1141.53034, MR 2191529 |
| Reference:
|
[2] Colbois, B., Grosjean, J. F.: A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space.Comment. Math. Helv. 82 (2007), 175–195. Zbl 1112.53003, MR 2296061, 10.4171/CMH/88 |
| Reference:
|
[3] Fialkow, A.: Hypersurfaces of a space of constant curvature.Ann. of Math. (2) 39 (1938), 762–783. Zbl 0020.06601, MR 1503435, 10.2307/1968462 |
| Reference:
|
[4] Grosjean, J. F.: Upper bounds for the first eigenvalue of the Laplacian on compact manifolds.Pacific J. Math. 206 (1) (2002), 93–111. MR 1924820, 10.2140/pjm.2002.206.93 |
| Reference:
|
[5] Grosjean, J. F., Roth, J.: Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces.Math. Z. 271 (1) (2012), 469–488. Zbl 1243.53071, MR 2917153, 10.1007/s00209-011-0872-0 |
| Reference:
|
[6] Hoffman, D., Spruck, D.: Sobolev and isoperimetric inequalities for Riemannian submanifolds.Comm. Pure Appl. Math. 27 (1974), 715–727. Zbl 0295.53025, MR 0365424, 10.1002/cpa.3160270601 |
| Reference:
|
[7] Reilly, R.C.: On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space.Comment. Math. Helv. 52 (1977), 525–533. Zbl 0382.53038, MR 0482597, 10.1007/BF02567385 |
| Reference:
|
[8] Roth, J.: Pinching of the first eigenvalue of the Laplacian and almost-Einstein hypersurfaces of Euclidean space.Ann. Global Anal. Geom. 33 (3) (2008), 293–306. MR 2390836, 10.1007/s10455-007-9086-4 |
| Reference:
|
[9] Shiohama, K., Xu, H.: Rigidity and sphere theorems for submanifolds.Kyushu J. Math. 48 (2) (1994), 291–306. Zbl 0826.53045, MR 1294532, 10.2206/kyushujm.48.291 |
| Reference:
|
[10] Shiohama, K., Xu, H.: Rigidity and sphere theorems for submanifolds II.Kyushu J. Math. 54 (1) (2000), 103–109. Zbl 1005.53028, MR 1762795, 10.2206/kyushujm.54.103 |
| Reference:
|
[11] Thomas, T. Y.: On closed space of constant mean curvature.Amer. J. Math. 58 (1936), 702–704. MR 1507192, 10.2307/2371240 |
| Reference:
|
[12] Vlachos, T.: Almost-Einstein hypersurfaces in the Euclidean space.Illinois J. Math. 53 (4) (2009), 1221–1235. Zbl 1213.53072, MR 2741186 |
| . |
