Convergence of the calabi flow on toric varieties and related Kaehler manifolds

2012/07/25 by Hongnian Huang, Huang, Hongnian
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG

paper · pdf · doi:10.48550/arxiv.1207.5969

arxiv created 2012/07/25 · openalex publication_date 2012/07/25 · arxiv updated 2012/07/26 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

Let X be a toric variety and u be a normalized symplectic potential of the corresponding polytope P. Suppose that the Riemannian curvature is bounded by 1 and ∫_{∂ P} u ~ d σ< C₁, then there exists a constant C₂ depending only on C₁ and P such that max_P u < C₂. As an application, we show that if (X,P) is analytic uniform K-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Riemannian curvature is uniformly bounded along the Calabi flow. Also we provide a proof of a conjecture of Donaldson. Finally, assuming that the curvature is bounded along the Calabi flow, our method would provide a proof of a conjecture due to Apostolov, Calderbank, Gauduchon and Tonnesen-Friedman.

Citations

• Interior estimates for solutions of Abreu's equation
• The Calabi flow on toric Fano surface
• Uniform K-stability for extremal metrics on toric varieties
• On the Extension of the Calabi Flow on Toric Varieties
• Filtrations and test-configurations

Related

• Toric Surfaces, K-Stability and Calabi Flow
• Calabi flow on toric varieties with bounded Sobolev constant, I
• Calabi flow with bounded L^p scalar curvature
• The Calabi flow on toric Fano surface
• On the Extension of the Calabi Flow on Toric Varieties