The Calabi functional on a ruled surface

TL;DR

This paper investigates the behavior of the Calabi functional on a ruled surface over a genus two curve, analyzing minimising sequences and the Calabi flow under symmetry conditions.

Contribution

It characterizes minimising sequences of the Calabi functional on ruled surfaces and links the Calabi flow to these sequences under symmetry assumptions.

Findings

Minimising sequences can split the manifold into pieces.

Calabi flow starting from symmetric metrics produces minimising sequences.

The study provides insight into the Calabi functional's behavior on ruled surfaces.

Abstract

We study the Calabi functional on a ruled surface over a genus two curve. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising sequence.