Ricci flow on quasiprojective manifolds II

TL;DR

This paper investigates how the Kähler-Ricci flow affects different types of asymptotic geometries on non-compact complex manifolds, demonstrating preservation of asymptotics and analyzing long-term behavior and limits.

Contribution

It extends previous work by analyzing three new types of spatial asymptotics and their preservation under the Kähler-Ricci flow on quasiprojective manifolds.

Findings

Asymptotics are preserved under the flow for cylindrical, bulging, and conical cases.

Long-time existence of the flow is established in these settings.

Parabolic blowdown limits and divisor roles are analyzed.

Abstract

We study the Ricci flow on complete Kaehler metrics that live on the complement of a divisor in a compact complex manifold. In earlier work, we considered finite-volume metrics which, at spatial infinity, are transversely hyperbolic. In the present paper we consider three different types of spatial asymptotics: cylindrical, bulging and conical. We show that in each case, the asymptotics are preserved by the Kaehler-Ricci flow. We address long-time existence, parabolic blowdown limits and the role of the Kaehler-Ricci flow on the divisor.