G_2-structures on Einstein solvmanifolds

We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-K"ahler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $\phi$ such that the induced metric $g_{\phi }$ is Einstein, unless $g_{\phi }$ is flat. We give an example of 7-dimensional solvmanifold admitting a left-invariant calibrated $G_2$-structure $\phi$ such that $g_{\phi }$ is Ricci-soliton. Moreover, we show that a 7-dimensional (non-flat) Einstein solvmanifold $(S,g)$ cannot admit any left-invariant cocalibrated $G_2$-structure $\phi$ such that the induced metric $g_{\phi }=g$.