On the asymptotic behavior of expanding gradient Ricci solitons

Let (M, g, f) be an n-dimensional expanding gradient Ricci soliton with faster-than-quadratic-decay curvature, i.e., ${\lim_{{\rm dist}(O,x)\rightarrow\infty} |{\rm Sect}(x)|\cdot {\rm dist}(O,x)^2=0}$ . When M is simply connected at infinity and n??? 3, we show that its tangent cone at infinity must be a manifold and is isometric to ${\mathbb{R}^n}$ . Here, we also assume that M has only one end for the simplicity of the statement. A crucial step to gain the regularity of the tangent cone at infinity is to prove that the injectivity radius grows linearly. This can be achieved by combining the curvature assumption and a lower bound estimate of volume ratio of all geodesic balls, which is attained as Theorem 3. On the other hand, we also study the asymptotic volume ratio of non-steady gradient Ricci solitons under other weaker conditions.