Existence problem of extremal Kaehler metrics

2013/07/19 by Toshiki Mabuchi, Mabuchi, Toshiki
Mathematics · #14L24 #32Q26 #53C25 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:14L24 #msc:32Q26 #msc:53C25

paper · pdf · doi:10.48550/arxiv.1307.5203

arxiv created 2013/07/19 · openalex publication_date 2013/07/19 · arxiv updated 2013/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we shall give some affirmative answer to an extremal Kaehler version of the Yau-Tian-Donaldson Conjecture. For a polarized algebraic manifold (X,L), we choose a maximal algebraic torus T in the group of holomorphic automorphisms of X. Then the polarization class c₁(L) will be shown to admit an extremal Kaehler metric if (X,L) is strongly K-stable relative to T.

Related

• The Yau-Tian-Donaldson Conjecture for general polarizations
• Extremal metrics and stabilities on polarized manifolds
• Asymptotic polybalanced kernels on extremal Kaehler manifolds
• A splitting theorem for extremal Kaehler metrics
• A uniform version of the Yau-Tian-Donaldson correspondence for extremal Kähler metrics on polarized toric manifolds