| Title:
|
A half-space type property in the Euclidean sphere (English) |
| Author:
|
Velásquez, Marco Antonio Lázaro |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
58 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
49-63 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the notion of strong $r$-stability for the context of closed hypersurfaces $\Sigma ^n$ ($n\ge 3$) with constant $(r+1)$-th mean curvature $H_{r+1}$ immersed into the Euclidean sphere $\mathbb{S}^{n+1}$, where $r\in \lbrace 1,\ldots ,n-2\rbrace $. In this setting, under a suitable restriction on the $r$-th mean curvature $H_r$, we establish that there are no $r$-strongly stable closed hypersurfaces immersed in a certain region of $\mathbb{S}^{n+1}$, a region that is determined by a totally umbilical sphere of $\mathbb{S}^{n+1}$. We also provide a rigidity result for such hypersurfaces. (English) |
| Keyword:
|
Euclidean sphere |
| Keyword:
|
closed hypersurfaces |
| Keyword:
|
$(r+1)$-th mean curvature |
| Keyword:
|
strong $r$-stability |
| Keyword:
|
geodesic spheres |
| Keyword:
|
upper (lower) domain enclosed by a geodesic sphere |
| MSC:
|
53C21 |
| MSC:
|
53C42 |
| idZBL:
|
Zbl 07511507 |
| idMR:
|
MR4412967 |
| DOI:
|
10.5817/AM2022-1-49 |
| . |
| Date available:
|
2022-02-23T12:12:27Z |
| Last updated:
|
2022-06-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149446 |
| . |
| Reference:
|
[1] Alencar, H., do Carmo, M., Colares, A.G.: Stable hypersurfaces with constant scalar curvature.Math. Z. 213 (1993), 117–131. Zbl 0792.53057, 10.1007/BF03025712 |
| Reference:
|
[2] Alías, L.J., Barros, A., Brasil Jr., A.: A spectral characterization of the $H(r)$-torus by the first stability eigenvalue.Proc. Amer. Math. Soc. 133 (2005), 875–884. MR 2113939, 10.1090/S0002-9939-04-07559-8 |
| Reference:
|
[3] Alías, L.J., Brasil Jr., A., Perdomo, O.: On the stability index of hypersurfaceswith constant mean curvature in spheres.Proc. Amer. Math. Soc. 135 (2007), 3685–3693. MR 2336585, 10.1090/S0002-9939-07-08886-7 |
| Reference:
|
[4] Alías, L.J., Brasil Jr., A., Sousa Jr., L.: A characterization of Clifford tori with constant scalar curvatrue one by the first stability eigenvalue.Bull. Braz. Math. Soc. 35 (2004), 165–175. MR 2081021, 10.1007/s00574-004-0009-8 |
| Reference:
|
[5] Aquino, C.P., de Lima, H.: On the rigidity of constant mean curvature complete vertical graphs in warped products.Differential Geom. Appl. 29 (2011), 590–596. MR 2811668, 10.1016/j.difgeo.2011.04.039 |
| Reference:
|
[6] Aquino, C.P., de Lima, H.F., dos Santos, Fábio R., Velásquez, Marco A.L.: On the first stability eigenvalue of hypersurfaces in the Euclidean and hyperbolic spaces.Quaest. Math. 40 (2017), 605–616. MR 3691472, 10.2989/16073606.2017.1305463 |
| Reference:
|
[7] Barbosa, J.L.M., Colares, A.G.: Stability of hypersurfaces with constant $r$-mean curvature.Ann. Global Anal. Geom. 15 (1997), 277–297. Zbl 0891.53044, 10.1023/A:1006514303828 |
| Reference:
|
[8] Barbosa, J.L.M., do Carmo, M.: Stability of hypersurfaces with constant mean curvature.Math. Z. 185 (1984), 339–353. Zbl 0513.53002, 10.1007/BF01215045 |
| Reference:
|
[9] Barbosa, J.L.M., do Carmo, M., Eschenburg, J.: Stability of hypersurfaces with constant mean curvature in Riemannian manifolds.Math. Z. 197 (1988), 1123–138. |
| Reference:
|
[10] Barros, A., Sousa, P.: Compact graphs over a sphere of constant second order mean curvature.Proc. Amer. Math. Soc. 137 (2009), 3105–3114. Zbl 1189.53058, MR 2506469, 10.1090/S0002-9939-09-09862-1 |
| Reference:
|
[11] Chavel, I.: Eigenvalues in Riemannian Geometry.Academic Press, Inc., 1984. |
| Reference:
|
[12] Cheng, Q.: First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature.Proc. Amer. Math. Soc. 136 (2008), 3309–3318. MR 2407097, 10.1090/S0002-9939-08-09304-0 |
| Reference:
|
[13] Cheng, S.Y., Yau, S.T.: Hypersurfaces with constant scalar curvature.Math. Ann. 225 (1977), 195–204. Zbl 0349.53041, 10.1007/BF01425237 |
| Reference:
|
[14] de Lima, H.F., Velásquez, Marco A.L.: A new characterization of $r$-stable hypersurfaces in space forms.Arch. Math. (Brno) 47 (2011), 119–131. MR 2813538 |
| Reference:
|
[15] Montiel, S.: Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds.Indiana Univ. Math. J. 48 (1999), 711–748. 10.1512/iumj.1999.48.1562 |
| Reference:
|
[16] Reilly, R.C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms.J. Differential Geom. 8 (1973), 465–477. Zbl 0277.53030, 10.4310/jdg/1214431802 |
| . |
