Geometric flows and K\"ahler reduction

TL;DR

This paper explores how different geometric flows of K"ahler metrics can be derived from static equations on higher-dimensional manifolds through K"ahler reduction, unifying several important flows in complex geometry.

Contribution

It identifies static equations that generate key K"ahler flows via reduction, providing a unified geometric framework for these flows.

Findings

Derived the geodesic equation for Mabuchi's metric from static equations.

Re-derived the V-soliton equation of La Nave-Tian within this framework.

Unified various K"ahler flows as reductions of static equations on higher-dimensional manifolds.

Abstract

We investigate how to obtain various flows of K\"ahler metrics on a fixed manifold as variations of K\"ahler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that induce the geodesic equation for the Mabuchi's metric, the Calabi flow, the pseudo-Calabi flow of Chen-Zheng and the K\"ahler-Ricci flow. In the latter case we re-derive the V-soliton equation of La Nave-Tian.