Invariant conformal metrics on 𝕊ⁿ

Espinar, José

doi:10.1090/S0002-9947-2011-05123-8

Invariant conformal metrics on $\mathbb {S}^n$
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by José M. Espinar

Trans. Amer. Math. Soc. 363 (2011), 5649-5661

DOI: https://doi.org/10.1090/S0002-9947-2011-05123-8

Published electronically: April 29, 2011

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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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