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

2007/06/28 by Morgan Sherman, Sherman, Morgan
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Numerical Analysis (math.NA) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.0706.4338

openalex publication_date 2007/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a recent paper Donaldson defines three operators on a space of Hermitian metrics on a complex projective manifold: T, T_ν, 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 \mathbbCPⁿ.

Citations

• Scalar Curvature and Stability of Toric Varieties
• Scalar Curvature and Projective Embeddings, I
• Some numerical results in complex differential geometry
• Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics
• Numerical approximations to extremal metrics on toric surfaces

Related

• A variational approach to the Hermitian-Einstein metrics and the Quot-scheme limit of Fubini-Study metrics
• Remarks on the Donaldson metric
• Gradient estimates for Donaldson's equation on a compact Kähler manifold
• Fekete points and convergence towards equilibrium measures on complex manifolds
• A remark on 'Some numerical results in complex differential geometry'