Double CMC surfaces in Euclidean and hyperbolic spaces

We establish a rigidity theorem for $2$-manifolds immersed in Euclidean and hyperbolic spaces, which have constant mean curvatures with respect to both induced metrics. Using the relation between the two mean curvatures, we prove that the two immersions are both totally umbilical. Moreover, we classify totally umbilical surfaces in $\mathbb R^3$ and $\mathbb H^3$ with respect to the value of the mean curvatures.