A Schwarz lemma for Kähler affine metrics and the canonical potential of a proper convex cone

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Abstract

This is an account of some aspects of the geometry of Kähler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry–Emery Ricci tensors. Such techniques yield a version for Kähler affine metrics of Yau’s Schwarz lemma for volume forms. By a theorem of Cheng and Yau, there is a canonical Kähler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an \(n\)-dimensional cone, a rescaling of the canonical potential is an \(n\)-normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Ampère metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-Kähler space.

Keywords

Convex cones Kähler affine metrics Self-concordant barrier 

Mathematics Subject Classification

53A15 53C42 90C25 
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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada, EUIT IndustrialUniversidad Politécnica de MadridMadridSpain

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