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Conformal metrics and Ricci tensors on the sphere

Authors: Romildo Pina and Keti Tenenblat
Journal: Proc. Amer. Math. Soc. 132 (2004), 3715-3724
MSC (2000): Primary 53C21, 53C50, 53C80
Published electronically: July 22, 2004
MathSciNet review: 2084096
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Abstract: We consider tensors $T=fg$ on the unit sphere $S^n$, where $n\geq 3$, $g$ is the standard metric and $f$ is a differentiable function on $S^n$. For such tensors, we consider the problems of existence of a Riemannian metric $\bar{g}$, conformal to $g$, such that $\mbox{Ric }\bar{g} = T$, and the existence of such a metric that satisfies $\mbox{Ric }\bar{g} - {\bar{K}}\bar{g}/2 = T$, where $\bar{K}$ is the scalar curvature of $\bar{g}$. We find the restrictions on the Ricci candidate for solvability, and we construct the solutions $\bar{g}$ when they exist. We show that these metrics are unique up to homothety, and we characterize those defined on the whole sphere. As a consequence of these results, we determine the tensors $T$ that are rotationally symmetric. Moreover, we obtain the well-known result that a tensor $T=\alpha g$, $\alpha>0 $, has no solution $\bar{g}$ on $S^n$if $\alpha\neq n-1$ and only metrics homothetic to $g$ admit $(n-1)g$ as a Ricci tensor. We also show that if $\alpha\neq -(n-1)(n-2)/2$, then equation $\mbox{Ric }\bar{g} - \displaystyle {\bar{K}}\bar{g}/2 = \alpha g $ has no solution $\bar{g}$, conformal to $g$ on $S^n$, and only metrics homothetic to $g$ are solutions to this equation when $\alpha= -(n-1)(n-2)/2$. Infinitely many $C^\infty$ solutions, globally defined on $S^n$, are obtained for the equation

\begin{displaymath}- \varphi \Delta_g \varphi +\displaystyle\frac{n}{2} \vert\na... ...displaystyle\frac{n}{2}\left( \lambda + \varphi^2 \right) = 0, \end{displaymath}

where $\lambda\in R$. The geometric interpretation of these solutions is given in terms of existence of complete metrics, globally defined on $R^n$ and conformal to the Euclidean metric, for certain bounded scalar curvature functions that vanish at infinity.

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Additional Information

Romildo Pina
Affiliation: IME, Universidade Federal de Goi\a’as, 74001-970 Goi\a^{a}nia, GO, Brazil

Keti Tenenblat
Affiliation: Departamento de Matem\a’atica, Universidade de Bras\a’ılia, 70910-900, Bras\a’ılia, DF, Brazil

Keywords: Ricci tensor, conformal metric, scalar curvature
Received by editor(s): May 30, 2002
Received by editor(s) in revised form: May 14, 2003
Published electronically: July 22, 2004
Additional Notes: The first author was partially supported by FUNAPE/UFG and PROCAD
The second author was partially supported by CNPq, PRONEX and PROCAD
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2004 American Mathematical Society