Extremal K\"ahler metrics and energy functionals on projective bundles

TL;DR

This paper establishes the equivalence between extremal Kähler metrics and the properness of the modified K energy on projective bundles, exploring energy functional relations and providing a counterexample.

Contribution

It proves the equivalence of extremal Kähler metrics and properness of the modified K energy on projective bundles, and analyzes related energy bounds and extremal polynomials.

Findings

Equivalence of extremal Kähler metrics and properness of modified K energy

Relation between lower boundedness of K energy and extremal polynomials

Counterexample with bounded but not proper modified K energy

Abstract

In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum of the Calabi energy and the extremal polynomials. In particular, we give an example where the modified K energy is bounded from below but not proper.