Curvature estimates for 4-dimensional complete gradient expanding Ricci solitons

Abstract

Abstract 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 {\mathrm{{Rm}}} and its covariant derivative ∇ ⁡ Rm {\nabla\mathrm{{Rm}}} can be bounded by the scalar curvature R by | Rm | ≤ C a ⁢ R a {|\mathrm{{Rm}}|\leq C_{a}R^{a}} and | ∇ ⁡ Rm | ≤ C a ⁢ R a {|\nabla\mathrm{{Rm}}|\leq C_{a}R^{a}} (on M ∖ K {M\setminus K} ), for any 0 ≤ a < 1 {0\leq a<1} and some constant C a > 0 {C_{a}>0} . Moreover, if the scalar curvature has at most polynomial decay at infinity, then | Rm | ≤ C ⁢ R {|\mathrm{{Rm}}|\leq CR} (on M ∖ K {M\setminus K} ). As an application, it follows that if a 4-dimensional complete gradient expanding Ricci soliton ( M 4 , g , f ) {(M^{4},g,f)} has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, hence admits C 1 , α {C^{1,\alpha}} asymptotic cones at infinity ( 0 < α < 1 ) {(0<\alpha<1)} according to Chen and Deruelle (2015).[21].