# Calabi type functionals for coupled K\"ahler-Einstein metrics

## TL;DR

This paper introduces new functionals to measure deviations from coupled K"ahler-Einstein metrics, providing inequalities and Hessian formulas that relate to algebraic invariants and obstructions.

## Contribution

It defines coupled Ricci-Calabi and H-functionals, establishes inequalities linking them to algebraic invariants, and derives Hessian formulas with applications to existence obstructions.

## Key findings

- Inequalities estimating functionals via algebraic invariants
- Hessian formulas at critical points of the functionals
- Obstruction theorem for coupled K"ahler-Einstein metrics

## Abstract

We introduce the coupled Ricci-Calabi functional and the coupled H-functional which measure how far from a coupled K\"ahler-Einstein metric in the sense of Hultgren-Witt Nystr\"om. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give corresponding Hessian formulas for these functionals at each critical point, which have an application to a Matsushima type obstruction theorem for the existence of a coupled K\"ahler-Einstein metric.