Conformal metrics and Ricci tensors on the sphere

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 \[ - \varphi \Delta _g \varphi +\displaystyle \frac {n}{2} |\nabla _g \varphi |^2 - \displaystyle \frac {n}{2}\left ( \lambda + \varphi ^2 \right ) = 0, \] 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.