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

Morgan Sherman[^1]\
California Polytechnic State University

In [@donaldson5] 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 $\mathbb{CP}^n$.

Introduction
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Let $X$ be a compact complex manifold with a positive holomorphic line bundle $L$. A long-standing open problem in Kähler geometry, building on Yau’s solution of the Calabi conjecture [@yau1], is to find sufficient conditions for the existence of a constant scalar curvature Kähler metric in $c_1(L)$. Another is as follows: can such a metric be obtained naturally as a limit of algebraic metrics via embeddings of $X$ into $\mathbb{P} \mathrm{H}^0 (X, L^k)$?

This idea of approximating Kähler metrics by restricting Fubini-Study metrics, advocated by Yau over the years, has led to the development of a rich theory relating analysis and notions of stability in the sense of geometry invariant theory (see [@yau2], [@tian1], [@tian2], [@donaldson3]). In 