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Invariant conformal metrics on $ \mathbb{S}^n$


Author: José M. Espinar
Journal: Trans. Amer. Math. Soc. 363 (2011), 5649-5661
MSC (2000): Primary 53A10; Secondary 49Q05, 53C42
DOI: https://doi.org/10.1090/S0002-9947-2011-05123-8
Published electronically: April 29, 2011
MathSciNet review: 2817403
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Abstract: In this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constraints on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on the sphere which are invariant by a $ k-$parameter subgroup of conformal diffeomorphisms of the sphere, giving a bound on its maximum dimension.

Moreover, we classify conformal metrics on the sphere whose eigenvalues of the Shouten tensor are all constant (we call them isoparametric conformal metrics), and we use a classification result for radial conformal metrics which are solutions of some $ \sigma _k -$Yamabe type problem for obtaining existence of rotational spheres and Delaunay-type hypersurfaces for some classes of Weingarten hypersurfaces in $ \mathbb{H} ^{n+1}$.


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Additional Information

José M. Espinar
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
Email: jespinar@ugr.es

DOI: https://doi.org/10.1090/S0002-9947-2011-05123-8
Keywords: Conformal metric, $\sigma_{k}$ curvature, radial solution, Schouten tensor, hyperbolic Gauss map, rotational hypersurfaces, Weingarten hypersurfaces
Received by editor(s): November 17, 2008
Received by editor(s) in revised form: May 22, 2009
Published electronically: April 29, 2011
Additional Notes: The author was partially supported by Spanish MEC-FEDER Grant MTM2007-65249, and Regional J. Andalucía Grants P06-FQM-01642 and FQM325.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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