A gap theorem for Ricci-flat 4-manifolds

Let (M,g)(M,g) be a compact Ricci-flat 4-manifold. For p∈Mp∈M let Kmax(p)Kmax(p) (respectively Kmin(p)Kmin(p)) denote the maximum (respectively the minimum) of sectional curvatures at p. We prove that ifKmax(p)≤−cKmin(p)Kmax(p)≤−cKmin(p) for all p∈Mp∈M, for some constant c   with 0≤c<2+64, then (M,g)(M,g) is flat.We prove a similar result for compact Ricci-flat Kähler surfaces. Let (M,g)(M,g) be such a surface and for p∈Mp∈M let Hmax(p)Hmax(p) (respectively Hmin(p)Hmin(p)) denote the maximum (respectively the minimum) of holomorphic sectional curvatures at p. IfHmax(p)≤−cHmin(p)Hmax(p)≤−cHmin(p) for all p∈Mp∈M, for some constant c   with 0≤c<1+32, then (M,g)(M,g) is flat.