Index estimates for harmonic Gauss maps

Let $\mathrm{\Sigma }$ denote a closed surface with constant mean curvature in ${\mathbb{G}}^3$, a 3-dimensional Lie group equipped with a bi-invariant metric. For such surfaces, there is a harmonic Gauss map which maps values to the unit sphere within the Lie algebra of $\mathbb{G}$. We prove that the energy index of the Gauss map of $\mathrm{\Sigma }$ is bounded below by its topological genus. We also obtain index estimates in the case of complete non compact surfaces.