A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem

2011/03/04 by Bo Berndtsson, Berndtsson, Bo · 11 citations
Mathematics · #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1103.0923

openalex publication_date 2011/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For ϕ a metric on the anticanonical bundle, -K_X, of a Fano manifold X we consider the volume of X ∫_X e^-ϕ. We prove that the logarithm of the volume is concave along continuous geodesics in the space of positively curved metrics on -K_X and that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on X. As consequences we get a simplified proof of the Bando-Mabuchi uniqueness theorem for Kähler - Einstein metrics and a generalization of this theorem to 'twisted' Kähler-Einstein metrics.

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• A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger\n inequalities and Kahler-Einstein metrics
• The conical Kähler-Ricci flow with weak initial data on Fano manifold
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• Stability of the conical Kähler-Ricci flows on Fano manifolds
• On the existence of conic Kahler-Einstein metrics

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• A Bando-Mabuchi Uniqueness Theorem
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• On the existence of conic Kahler-Einstein metrics
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