Continuity of extremal transitions and flops for calabi–yau manifolds

In this paper, we study the behavior of Ricci-flat Kähler metrics on Calabi–Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa’s conjecture: Ricci–flat Calabi–Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi–Yau manifolds and a compact metric space in the Gromov–Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci–flat Kähler metrics on Calabi–Yau manifolds along a smoothing is established, which can be of independent interest.