No abstract

Motivated by the study of collapsing Calabi-Yau 3-folds with a Lefschetz K3 fibration, we construct a complete Calabi-Yau metric on 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 C 2 /Z 2 × C, and its Riemannian geometry has certain non-standard features near the singularity of the tangent cone, which are more typical of adiabatic limit problems. The proof uses an existence result in H-J. Hein's Ph.D. thesis to perturb an asymptotic approximate solution into an actual solution, and the main difficulty lies in correcting the slowly decaying error terms. 1 Motivations This work grows out of the attempt to model the collapsing behaviour of Calabi-Yau metrics on a K3 fibred compact Calabi-Yau manifold (i.e. Kähler Ricci-flat with parallel nonvanishing holomorphic volume form) over a Riemann surface, where the Kähler class has very small volume on the K3 fibres. Assuming the only singularities in the fibration are nodal, we wish to understand the metric near the critical points of the fibration, in the standard local model of the Lefschetz fibration f : C 3 → C with f = z 2 1 + z 2 2 + z 2 3. Knowledge of the model metric is often useful in gluing constructions. The prototype B Yang Li

We consider the Dirichlet boundary value problem for graphical maximal submanifolds inside Lorentzian type ambient spaces, and obtain general existence and uniqueness results which apply to any codimension.

We use the gluing method to give a refined description of the collapsing Calabi-Yau metrics on Calabi-Yau 3-folds admitting a Lefschetz K3 fibration.

We consider the Dirichlet boundary value problem for graphical maximal submanifolds inside Lorentzian-type ambient spaces and obtain general existence and uniqueness results that apply to any codimension.