scholarly journals Supersymmetric rigidity of asymptotically locally hyperbolic manifolds

2014 ◽  
Vol 25 (03) ◽  
pp. 1450020 ◽  
Author(s):  
Oussama Hijazi ◽  
Sebastián Montiel

Let (M, g) be an Asymptotically Locally Hyperbolic (ALH) manifold which is the interior of a conformally compact manifold and (∂M, [γ]) its conformal infinity. Suppose that the Ricci tensor of (M, g) dominates that of the hyperbolic space and that its scalar curvature satisfies a certain decay condition at infinity. If the Yamabe invariant of (∂M, [γ]) is non-negative, we prove that there exists a conformal metric on M with non-negative scalar curvature and whose boundary ∂M has either positive or zero constant inner mean curvature. In the spin case, we make use of a previous estimate obtained by X. Zhang and the authors for the Dirac operator of the induced metric on ∂M. As a consequence, we generalize and simplify the proof of the result by Andersson and Dahl in [Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom.16 (1998) 1–27] about the rigidity of the hyperbolic space when the prescribed conformal infinity ∂M is a round sphere. We also provide non-existence results for conformally compact ALH spin metrics when ∂M is conformal to a Riemannian manifold with special holonomy.

2013 ◽  
Vol 24 (03) ◽  
pp. 1350020 ◽  
Author(s):  
PAK TUNG HO

In this paper, we consider the problem of prescribing pseudo-Hermitian scalar curvature on a compact strictly pseudoconvex CR manifold M. Using geometric flow, we prove that for any negative smooth function f we can prescribe the pseudo-Hermitian scalar curvature to be f, provided that dim M = 3 and the CR Yamabe invariant of M is negative. On the other hand, we establish some uniqueness and non-uniqueness results on prescribing pseudo-Hermitian scalar curvature.


2015 ◽  
Vol 26 (02) ◽  
pp. 1550014 ◽  
Author(s):  
Uğur Dursun ◽  
Rüya Yeğin

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.


2015 ◽  
Vol 365 (3-4) ◽  
pp. 1257-1277 ◽  
Author(s):  
Jie Qing ◽  
Wei Yuan

2014 ◽  
Vol 57 (3) ◽  
pp. 653-663 ◽  
Author(s):  
CÍCERO P. AQUINO ◽  
HENRIQUE F. DE LIMA ◽  
MARCO ANTONIO L. VELÁSQUEZ

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.


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