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

Fox, Daniel Jeremy Forrest (2015). A Schwarz lemma for Kahler affine metrics and the canonical potential of a proper convex cone. "Annali di Matematica Pura ed Applicata", v. 194 (n. 1); pp. 1-42. ISSN 0373-3114. https://doi.org/10.1007/s10231-013-0362-6.

Description

Title: A Schwarz lemma for Kahler affine metrics and the canonical potential of a proper convex cone
Author/s:
  • Fox, Daniel Jeremy Forrest
Item Type: Article
Título de Revista/Publicación: Annali di Matematica Pura ed Applicata
Date: 1 February 2015
ISSN: 0373-3114
Volume: 194
Subjects:
Faculty: E.U.I.T. Industrial (UPM)
Department: Matemática Aplicada
Creative Commons Licenses: Recognition - No derivative works - Non commercial

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Abstract

This is an account of some aspects of the geometry of Kahler 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 Kahler affine metrics of Yau s Schwarz lemma for volume forms. By a theorem of Cheng and Yau, there is a canonical Kahler 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-Kahler space.

More information

Item ID: 33432
DC Identifier: http://oa.upm.es/33432/
OAI Identifier: oai:oa.upm.es:33432
DOI: 10.1007/s10231-013-0362-6
Official URL: http://link.springer.com/article/10.1007%2Fs10231-013-0362-6
Deposited by: Memoria Investigacion
Deposited on: 29 Jan 2015 13:27
Last Modified: 29 Jan 2015 13:27
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