A Schwarz lemma for K\"ahler affine metrics and the canonical potential of a proper convex cone

TL;DR

This paper explores the geometry of K"ahler affine metrics on convex cones, establishing a Schwarz lemma, uniqueness of the canonical potential, and connections to conic programming and Monge-Ampère metrics.

Contribution

It introduces a Schwarz lemma for K"ahler affine metrics, characterizes the canonical potential of convex cones, and links it to barrier functions and Monge-Ampère metrics.

Findings

A Schwarz lemma for K"ahler affine metrics is established.

The canonical potential is uniquely characterized and related to hyperbolic affine spheres.

Rescaled canonical potential serves as an n-normal barrier in conic programming.

Abstract

This is an account of some aspects of the geometry of K\"ahler 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 K\"ahler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical K\"ahler 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-Amp\`ere metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-K\"ahler space.