LINEAR WEINGARTEN HYPERSURFACES IN RIEMANNIAN SPACE FORMS

Title & Authors
LINEAR WEINGARTEN HYPERSURFACES IN RIEMANNIAN SPACE FORMS
Chao, Xiaoli; Wang, Peijun;

Abstract
In this note, we generalize the weak maximum principle in [4] to the case of complete linear Weingarten hypersurface in Riemannian space form $\small{\mathbb{M}^{n+1}(c)}$ (c = 1, 0,-1), and apply it to estimate the norm of the total umbilicity tensor. Furthermore, we will study the linear Weingarten hypersurface in $\small{\mathbb{S}^{n+1}(1)}$ with the aid of this weak maximum principle and extend the rigidity results in Li, Suh, Wei [13] and Shu [15] to the case of complete hypersurface.
Keywords
linear Weingarten hypersurface;maximum principle;space form;Clifford torus;circular cylinder;hyperbolic cylinder;
Language
English
Cited by
1.
On the Gauss map of Weingarten hypersurfaces in hyperbolic spaces, Bulletin of the Brazilian Mathematical Society, New Series, 2016, 47, 4, 1051
2.
Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds, Mathematische Nachrichten, 2016, 289, 11-12, 1309
3.
Linear Weingarten submanifolds in unit sphere, Archiv der Mathematik, 2016, 106, 6, 581
References
1.
H. Alencar and M. do Carmo, Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1223-1229.

2.
L. J. Alias and S. C. Garcia-Martinez, On the scalar curvature of constant mean curvature hypersurfaces in space forms, J. Math. Anal. Appl. 363 (2010), no. 2, 579-587.

3.
L. J. Alias and S. C. Garcia-Martinez, An estimate for the scalar curvature of constant mean curvature hypersurfaces in space forms, Geom. Dedicata 156 (2012), 31-47.

4.
L. J. Alias, S. C. Garcia-Martinez, and M. Rigoli, A maximum principle for hypersurfaces with constant scalar curvature and applications, Ann. Global Anal. Geom. 41 (2012), no. 3, 307-320.

5.
C. Aquino, H. de Lima, and M. Velasquez, A new characterization of complete linear Weingarten hypersurfaces in real space forms, Pacific J. Math. 261 (2013), no. 1, 33-43.

6.
A. Brasil Jr., A. G. Colares, and O. Palmas, Complete hypersurfaces with constant scalar curvature in spheres, Monatsh. Math. 161 (2010), no. 4, 369-380.

7.
A. Caminha, On hypersurfaces into Riemannian spaces of constant sectional curvature, Kodai Math. J. 29 (2006), no. 2, 185-210.

8.
X. L. Chao, On complete spacelike submanifolds in semi-Riemannian space forms with parallel normalized mean curvature vector, Kodai Math. J. 34 (2011), no. 1, 42-54.

9.
H. Chen and X. F. Wang, Stability and eigenvalue estimates of linear Weingarten hypersurfaces in a sphere, J. Math. Anal. Appl. 397 (2013), no. 2, 658-670.

10.
S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195-204.

11.
A. V. Corro, W. Ferreira, and K. Tenenblat, Ribaucour transformations for constant mean curvature and linear Weingarten surfaces, Pacific J. Math. 212 (2003), no. 2, 265-297.

12.
H. Z. Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann. 305 (1996), no. 4, 665-672.

13.
H. Z. Li, Y. J. Suh, and G. X. Wei, Linear Weingarten hypersurfaces in a unit sphere, Bull. Korean Math. Soc. 46 (2009), no. 2, 321-329.

14.
M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207-213.

15.
S. C. Shu, Linear Weingarten hypersurfaces in a real space form, Glasg. Math. J. 52 (2010), no. 3, 635-648.

16.
D. Yang, Linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz space, Bull. Korean Math. Soc. 49 (2012), no. 2, 271-284.

17.
Q. Zhang, Scalar curvature of hypersurfaces with constant mean curvature in spheres, Glasg. Math. J. 54 (2012), no. 1, 67-75.