complete hypersurfaces
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2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ning Zhang

In this paper, applying the weak maximum principle, we obtain the uniqueness results for the hypersurfaces under suitable geometric restrictions on the weighted mean curvature immersed in a weighted Riemannian warped product I × ρ M f n whose fiber M has f -parabolic universal covering. Furthermore, applications to the weighted hyperbolic space are given. In particular, we also study the special case when the ambient space is weighted product space and provide some results by Bochner’s formula. As a consequence of this parametric study, we also establish Bernstein-type properties of the entire graphs in weighted Riemannian warped products.



2019 ◽  
Vol 27 (6) ◽  
pp. 1251-1279
Author(s):  
Manfredo do Carmo ◽  
Maria Fernanda Elbert


2018 ◽  
Vol 153 (2) ◽  
pp. 149-162
Author(s):  
Jogli G. Araújo ◽  
Márcio Batista ◽  
Henrique F. de Lima






2016 ◽  
Vol 103 (1) ◽  
pp. 45-58
Author(s):  
C. P. AQUINO ◽  
M. BATISTA ◽  
H. F. DE LIMA

In this paper, we establish new characterization results concerning totally umbilical hypersurfaces of the hyperbolic space$\mathbb{H}^{n+1}$, under suitable constraints on the behavior of the Lorentzian Gauss map of complete hypersurfaces having some constant higher order mean curvature. Furthermore, working with different warped product models for$\mathbb{H}^{n+1}$and supposing that certain natural inequalities involving two consecutive higher order mean curvature functions are satisfied, we study the rigidity and the nonexistence of complete hypersurfaces immersed in$\mathbb{H}^{n+1}$.



2016 ◽  
Vol 27 (08) ◽  
pp. 1650063
Author(s):  
Feng Jiang Li ◽  
Jian Bo Fang

Let [Formula: see text] be an umbilical free hypersurface in the unit sphere [Formula: see text]. Four basic invariants of [Formula: see text], under the Möbius transformation group of [Formula: see text] are the Möbius metric [Formula: see text], the Möbius second fundamental form [Formula: see text], the Blaschke tensor [Formula: see text] and the Möbius form [Formula: see text]. In this paper, we study complete hypersurfaces with constant normalized Möbius scalar curvature [Formula: see text] and vanishing Möbius form [Formula: see text]. By computing the Laplacian of the funtion [Formula: see text], where the trace-free Blaschke tensor [Formula: see text], and applying the well known generalized maximum principle of Omori–Yau, we obtain the following result: [Formula: see text] must be either Möbius equivalent to a minimal hypersurface with constant Möbius scalar curvature, when [Formula: see text]; [Formula: see text] in [Formula: see text], when [Formula: see text]; the pre-image of the stereographic projection [Formula: see text] of the circular cylinder [Formula: see text] in [Formula: see text], when [Formula: see text]; or the pre-image of the projection [Formula: see text] of the hypersurface [Formula: see text] in [Formula: see text], when [Formula: see text].



2016 ◽  
Vol 32 (6) ◽  
pp. 715-724
Author(s):  
Jian Bo Fang ◽  
Feng Jiang Li


2016 ◽  
Vol 54 (1) ◽  
pp. 1-11
Author(s):  
Hilário Alencar ◽  
Gregório Silva Neto


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