Conformal metrics of the ball with constant ${\sigma }_k$-curvature and constant boundary mean curvature

On the upper hemisphere, we use the Obata-Escobar argument to classify conformal metrics with constant ${\sigma }_k$ curvature and constant boundary mean curvature in all types of cones including positive and negative cones. This extends a result of Escobar in \cite{Es} for $k=1$.