A new complete Calabi-Yau metric on ${\mathbb{C}}^3$

Motivated by the study of collapsing Calabi-Yau threefolds with a Lefschetz K3 fibration, we construct a complete Calabi-Yau metric on ${\mathbb{C}}^3$ with maximal volume growth, which in the appropriate scale is expected to model the collapsing metric near the nodal point. This new Calabi-Yau metric has singular tangent cone at infinity, and its Riemannian geometry has certain non-standard features near the singularity of the tangent cone $$\mathbb{C}$^2/$\mathbb{Z}$_2 \times $\mathbb{C}$$, which are more typical of adiabatic limit problems. The proof uses an existence result in H-J. Hein’s PhD thesis to perturb an asymptotic approximate solution into an actual solution, and the main difficulty lies in correcting the slowly decaying error terms.