Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set $K$). More precisely, we prove that the norm of the curvature tensor $Rm$ and its covariant derivative $\mathrm{\nabla }Rm$ can be bounded by the scalar curvature $R$ by $\mathrm{\mid }Rm\mathrm{\mid }\le C_a{R}^a$ and $\mathrm{\mid }\mathrm{\nabla }Rm\mathrm{\mid }\le C_a{R}^a$ (on $M\mathrm{\backslash }K$), for any $0\le a<1$ and some constant $C_a>0$. Moreover, if the scalar curvature has at most polynomial decay at infinity, then $|Rm| \le C R$(on$M\backslash K$). As an application, it follows that that if a 4-dimensional complete gradient expanding Ricci soliton $(M^4,g,f)$ has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, and $C^{1,\alpha }$ asymptotic cones at infinity ($0<\alpha <1$) according to Chen-Deruelle [20].