scholarly journals Hypersurfaces in hyperbolic space associated with the conformal scalar curvature equation Δu+kun+2n−2=0

2009 ◽  
Vol 27 (2) ◽  
pp. 279-295 ◽  
Author(s):  
Walterson Ferreira ◽  
Pedro Roitman
Keyword(s):  
Scalar Curvature ◽  
Hyperbolic Space ◽  
Curvature Equation

Hyperbolic submanifolds with finite type hyperbolic Gauss map

2015 ◽  
Vol 26 (02) ◽  
pp. 1550014 ◽  
Author(s):  
Uğur Dursun ◽  
Rüya Yeğin
Keyword(s):  
Scalar Curvature ◽  
Finite Type ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Gauss Map ◽  
Hyperbolic Spaces ◽  
Nonzero Constant ◽  
Hyperbolic Gauss Map

We study submanifolds of hyperbolic spaces with finite type hyperbolic Gauss map. First, we classify the hyperbolic submanifolds with 1-type hyperbolic Gauss map. Then we prove that a non-totally umbilical hypersurface Mn with nonzero constant mean curvature in a hyperbolic space [Formula: see text] has 2-type hyperbolic Gauss map if and only if M has constant scalar curvature. We also classify surfaces with constant mean curvature in the hyperbolic space [Formula: see text] having 2-type hyperbolic Gauss map. Moreover we show that a horohypersphere in [Formula: see text] has biharmonic hyperbolic Gauss map.

On positive entire solutions of the elliptic equation Δu + K(x)up = 0 and its applications to Riemannian geometry

1996 ◽  
Vol 126 (2) ◽  
pp. 225-237 ◽  
Author(s):  
Changfeng Gui
Keyword(s):  
Elliptic Equation ◽  
Asymptotic Behaviour ◽  
Scalar Curvature ◽  
Riemannian Geometry ◽  
Semilinear Elliptic Equation ◽  
Entire Space ◽  
Entire Solutions ◽  
Curvature Equation ◽  
Semilinear Elliptic ◽  
Special Case

We study the existence and asymptotic behaviour of positive solutions of a semilinear elliptic equation in entire space. A special case of this equation is the scalar curvature equation which arises in Riemannian geometry.

scholarly journals LINEAR WEINGARTEN HYPERSURFACES WITH BOUNDED MEAN CURVATURE IN THE HYPERBOLIC SPACE

2014 ◽  
Vol 57 (3) ◽  
pp. 653-663 ◽  
Author(s):  
CÍCERO P. AQUINO ◽  
HENRIQUE F. DE LIMA ◽  
MARCO ANTONIO L. VELÁSQUEZ
Keyword(s):  
Scalar Curvature ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Maximum Principles ◽  
Rigidity Result ◽  
The Second Fundamental Form ◽  
Linear Weingarten Hypersurfaces ◽  
Compact Case ◽  
Suitable Restriction

AbstractWe apply appropriate maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space. By supposing a suitable restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder, when its scalar curvature is positive, or to a spherical cylinder, when its scalar curvature is negative. Related to the compact case, we also establish a rigidity result.

scholarly journals Multiplicity results for the scalar curvature equation

2015 ◽  
Vol 259 (8) ◽  
pp. 4327-4355 ◽  
Author(s):  
Isabel Flores ◽  
Matteo Franca
Keyword(s):  
Scalar Curvature ◽  
Multiplicity Results ◽  
Curvature Equation
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