The aim of this thesis is to describe the collapsing Calabi-Yau (CY) metrics
on a CY 3-fold with a Lefschetz K3 fibration, from both the a priori estimate
perspective and the gluing perspective. Collapsing CY metric is a well studied
subject, but most of the previous works concentrate on the behaviour away from
the singular fibres, and the full description of the metric was only available in a
very small number of cases, mostly relying on very favourable gluing ansatz.
From the nonlinear perspective, the essential realisation is that by restricting
the type of singularities, and under some conjecture in pluripotential theory,
then a small neighbourhood of the singular fibre has a local noncollapsing
bound, which enables us to understand the pointed Gromov-Hausdor limit of
the singular fibre in the scale where the fibre volume is 1.
This gives strong heuristic evidence that there is a much finer scale near the
nodal points in the fibration, where the scaled limit is a CY metric on C3 with
maximal volume growth and singular tangent cone at infinity. This model CY
metric is rigorously constructed using a noncompact version of Yau's solution
to the Calabi conjecture.
The model metric enables a gluing description of the collapsing CY metrics.
The diffi culty of the gluing lies in the coarse nature of the gluing ansatz, and
the fact that the metric has many types of characteristic behaviours at different
scales. We overcome this by developing a sharp linear theory, using some earlier
ideas of Gabor Sz eklyhidi.