Convergence properties of Donaldson's $T$-iterations on the Riemann sphere

TL;DR

This paper studies the convergence behavior of Donaldson's iterative operators on Hermitian metrics, focusing on the Riemann sphere and higher-dimensional projective spaces, to understand their role in approximating constant scalar curvature metrics.

Contribution

It provides a detailed analysis of the convergence properties of Donaldson's $T$-iterations on the Riemann sphere and extends the study to higher-dimensional complex projective spaces.

Findings

Convergence of $T$-iterations to balanced metrics established.

Insights into the rate of convergence and stability properties.

Extension of results from the Riemann sphere to $ ext{CP}^n$.

Abstract

In a recent paper Donaldson defines three operators on a space of Hermitian metrics on a complex projective manifold: $T, T_{\nu}, T_K.$ Iterations of these operators converge to balanced metrics, and these themselves approximate constant scalar curvature metrics. In this paper we investigate the convergence properties of these iterations by examining the case of the Riemann sphere as well as higher dimensional $\mathbb{CP}^n$.