The nonexistence of quasi-einstein metrics

We study complete Riemannian manifolds satisfying the equation by studying the associated PDE Δ_ff+ mμ exp(2f/m) = 0 for μ < 0. By developing a gradient estimate for f, we show there are no nonconstant solutions. We then apply this to show that there are no nontrivial Ricci flat warped products with fibers that have nonpositive Einstein constant. We also show that for nontrivial steady gradient Ricci solitons, the quantity R + |∇f|² is a positive constant.