Invariant conformal metrics on S^n

TL;DR

This paper explores conformal metrics on the sphere, establishing conditions for radiality, invariance under subgroups, and classifying those with constant eigenvalues, linking to geometric structures in hyperbolic space.

Contribution

It introduces new criteria for radiality, bounds invariance group dimensions, and classifies isoparametric conformal metrics on the sphere.

Findings

Conditions for conformal metrics to be radial.

Bound on the dimension of symmetry groups.

Classification of isoparametric conformal metrics.

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 constrain 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 \emph{isoparametric conformal metrics}), and we use a classification result for radial conformal metrics which are solution of some $\sigma _k -$Yamabe type problem for obtaining existence of rotational spheres and Delaunay-type hypersurfaces for some classes of Weingarten hypersurfaces in $\h ^{n+1}$.