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

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.

Las capacidades primitivas del hormigón. Rudolf Steiner, Gottfried Böhm, Rudolf Olgiati: tres aproximaciones.

RESUMEN. El objeto de la tesis es encontrar y profundizar en el estudio de arquitecturas al margen de la historia de la arquitectura moderna. En concreto aquellos edifi cios en los que la experimentación con el hormigón se convierte en el tema central del proyecto. Son obras singulares, en ocasiones herméticas y enigmáticas, cuyo carácter primitivo es debido necesariamente a las propiedades del material escogido Estas arquitecturas primitivas en hormigón tienen un tratamiento elemental, tosco. El estudio parte de la premisa de que estos rasgos primitivos pueden aparecer de manera voluntaria. Este momento germinal, también presente en los inicios de cada época, no se analizará por tanto como una fase torpe y arcaica por desconocimiento de una técnica más refi nada sino, al contrario, por la clara voluntad de sus arquitectos de utilizar este material antiguo y nuevo. ABSTRACT. The purpose of the thesis is to fi nd and study examples of architecture outside the history of modern architecture, specifi cally those buildings in which experimentation with concrete becomes the focus of the project. Works are sometimes enigmatic, the primitive character is necessarily due to material properties chosen primitive concrete. These architectures have a basic treatment, primary. The study starts from the premise that these primitive features can appear voluntarily. This germinal moment, also present at the beginning of each period, thus will not be discussed as a an archaic and ignorant phase of more refi ned technique, but by the clear will of its architects to use this material old and new.

Letter from Schwarz to William Tudor, undated

One letter from Paris regarding Tudor’s departure. In French.