\begin{theorem}

Any complete, conformal metric $g$ on $\mathbb S^n \setminus \mathbb S^{l}$ for $l\le \frac{n-2}{2}$ satisfying

\label{q1}

Q_g \equiv 1\; \text{or \ $0$},

and

\label{sp}

R_g \ge 0,

in $\mathbb S^n \setminus \mathbb S^{l}$ has to be symmetric with respect to rotations of $\mathbb S^n$ which leave $\mathbb S^l$ invariant.

\end{theorem}

This theorem is a corollary of the following

\begin{theorem}\label{thm1}

Let $g$ be a conformal, complete metric on $\Omega \subsetneqq \mathbb S^n$ such that \eqref{q1} and \eqref{sp} hold in $\Omega$. Then for any ball $B\subset \subset \Omega$ in the canonical metric $g_{\mathbb S^n}$, the mean curvature of its boundary $\partial B$ in metric $g$ with respect to its inner normal is nonnegative.

\end{theorem}