A sharp lower bound for the scalar curvature of certain steady gradient Ricci solitons (Q2838971)

A steady Ricci soliton is a Riemannian manifold \((M, g)\) which has a smooth vector field \(X\) such that \( L_X g + 2\mathrm{ Ric}(g) =0\), where \(L_X\) is the Lie derivative with respect to \(X\) and \(\mathrm{Ric}(g)\) denotes the Ricci tensor of \(g\). In the present paper the authors show a sharp lower bound for the scalar curvature of a steady gradient Ricci soliton in terms of the hyperbolic secant of the distance from a fixed point \(O \in M\) under the assumption that \(2 |\mathrm{Ric} (g) |^2 \leq\mathrm{Scal}(g)\). They prove that this condition is satisfied by every steady gradient Ricci soliton with nonnegative sectional curvature.