A gap theorem for Ricci-flat 4-manifolds

الأصل · EN

Let (M,g) be a compact Ricci-flat 4-manifold. For p ∈ M let Kmax(p) (respectively Kmin(p)) denote the maximum (respectively the minimum) of sectional curvatures at p. We prove that if Kmax (p) ≤ -c Kmin(p) for all p ∈ M, for some constant c with 0 ≤ c < 2+√ 6/4, then (M,g) is flat. We prove a similar result for compact Ricci-flat Kähler surfaces. Let (M,g) be such a surface and for p ∈ M let Hmax(p) (respectively Hmin(p)) denote the maximum (respectively the minimum) of holomorphic sectional curvatures at p. If Hmax (p) ≤ -c Hmin(p) for all p ∈ M, for some constant c with 0 ≤ c < 1+√ 3/2, then (M,g) is flat.