Adiabatic limits of Ricci-flat Kähler metrics

ADIABATIC LIMITS OF RICCI - FLAT KÄHLER METRICS 3 from

We study adiabatic limits of Ricci-flat Kähler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Ampère equation, we show that the Ricci-flat metrics collapse (away from the singular fibers) to a metric on the base of the fibration. This metri...

Numerical Ricci - flat metrics on K 3

We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kähler parameters. We show that Kähler geometry can be exploited for significant gains in computational efficiency. As a proof of principle, we apply our methods to a one-parameter family of K3 surfaces constructed as blow-ups of the T /Z2 orbifold with ma...

Ricci-flat K Ahler Metrics on Canonical Bundles

We prove the existence of a (unique) S-invariant Ricci-flat Kähler metric on a neighbourhood of the zero section in the canonical bundle of a realanalytic Kähler manifold X, extending the metric on X. In the important paper [3], Calabi proved existence of Ricci-flat Kähler metrics on two classes of manifolds: a) cotangent bundles of projective spaces; b) canonical bundles of Kähler-Einstein man...

Numerical Ricci - flat metrics on K 3 Matthew

We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kähler parameters. We show that Kähler geometry can be exploited for significant gains in computational efficiency. As a proof of principle, we apply our methods to a one-parameter family of K3 surfaces constructed as blow-ups of the T /Z2 orbifold with ma...

Ricci Flow on Kähler-einstein Surfaces