Degenerations of exotic Calabi-Yau metrics through Atiyah's flop

Abstract: We construct new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the conifold $$\mathcal{Z} = {z_1² + z_2² + z_3² + z_4² = 0} \subset \mathbb{C}<sup>4$$. These metrics have tangent cone $$\mathbb{C} \times (\mathbb{C}² / \mathbb{Z}_2)$$ at infinity and are parametrised by their Kähler class. As the Kähler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a Calabi-Yau metric on $\mathcal{Z}$ with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point and tangent cone at infinity $\mathbb{C} \times (\mathbb{C}<sup>2/\mathbb{Z}_2)$, thereby providing a new metric realisation of the Atiyah flop.