Convergence Properties of Donaldson's $T$-Iterations on the Riemann Sphere

Abstract

Donaldson gives 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 $\mathbb{CP}^n$.