caa a22 4500 605495955 CHVBK 20210128100534.0 cr unu---uuuuu 210128e20150201xx s 000 0 eng 10.1007/s10231-013-0362-6 doi (NATIONALLICENCE)springer-10.1007/s10231-013-0362-6 Fox Daniel Departamento de Matemática Aplicada, EUIT Industrial, Universidad Politécnica de Madrid, Ronda de Valencia 3, 28012, Madrid, Spain aut A Schwarz lemma for Kähler affine metrics and the canonical potential of a proper convex cone [Elektronische Daten] [Daniel Fox] 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 $$n$$ n -dimensional cone, a rescaling of the canonical potential is an $$n$$ 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. Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2013 Convex cones nationallicence Kähler affine metrics nationallicence Self-concordant barrier nationallicence Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/1(2015-02-01), 1-42 0373-3114 194:1<1 2015 194 10231 https://doi.org/10.1007/s10231-013-0362-6 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-013-0362-6 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Fox Daniel Departamento de Matemática Aplicada, EUIT Industrial, Universidad Politécnica de Madrid, Ronda de Valencia 3, 28012, Madrid, Spain aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/1(2015-02-01), 1-42 0373-3114 194:1<1 2015 194 10231