Lower bounds on the Calabi functional

2005/06/24 by Simon Donaldson, S. K. Donaldson, Donaldson, S. K. · 4 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG

paper · pdf · doi:10.48550/arxiv.math/0506501

arxiv created 2005/06/24 · openalex publication_date 2005/06/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The main result of this paper shows that "test configurations" give new lower bounds on the L² norm of the scalar curvature on a Kahler manifold. This is closely analogous to the analysis of the Yang-Mills functional over Riemann surfaces by Atiyah and Bott. The proof uses asymptotic approximation by finite-dimensional problems: the essential ingredient being the Tian-Zelditch-Lu expansion of the "density of states" function.

Cited by

• Energy functionals on almost Kähler manifolds: I
• The maximal destabilizers for Chow and K-stability
• The CR Yamabe invariant and constant scalar curvature Sasaki metrics
• Relative uniform K-stability over models implies existence of extremal metrics

Related

• On the limit of spectral measures associated to a test configuration
• Calabi flow with bounded L^p scalar curvature (II)
• Calabi flow with bounded L^p scalar curvature
• Remarks on a result of Chen-Cheng
• On the lower bounds of the L²-norm of the Hermitian scalar curvature