A note on characteristic functions and projectively invariant metrics on a bounded convex domain
type: Item
instance of: scholarly article
title: A note on characteristic functions and projectively invariant metrics on a bounded convex domain
zbMATH Open document ID: 0628.32027
published in: Tokyo Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Two important biholomorphically invariant metrics on bounded pseudoconvex domains in \({\mathbb{C}}^ n\) are the Bergman metric and the complete Einstein-Kähler metric of Cheng and Yau. The present article considers real analogues of these, defined on bounded convex domains D in \({\mathbb{R}}^ n\). The results are interesting for their own sake, but also because there is an important interplay between the real and complex cases via the following two constructions: (1) given D, one can consider the tube domain over it in \({\mathbb{C}}^ n\); (2) one can construct a cone in \({\mathbb{R}}^{n+1}\) with base D and consider the tube domain over the cone. It is these constructions that motivate the author's basic definitions. Denoting by \(D^*\) the polar set of D, he defines the characteristic function of D by \(\chi_ D(x)=(n!)\int_{D^*}(1+<x,\xi >)^{-n-1}d\xi\) and the kernel function by \(k_ D(x)=((2n+1)!)\int_{D^*}(1+<x,\xi >)^{-2n- 2}\chi_{D^*}(\xi)^{-1}d\xi.\) He proves that the formula \(-u^{- 1}d^ 2u\), where u is \(\chi_ D\) or \(k_ d\), defines two projectively invariant metrics on D. Furthermore, if D is strictly convex and has \(C^ 2\) boundary, then both metrics are complete. In what follows the author gives asymptotic expansions of \(\chi_ D\) and \(k_ D\) near the boundary, analogous to Fefferman's expansion of the Bergman kernel. There are a number of other results, too complicated to state here in detail. Connections with the Monge-Ampère equation and with known results about homogeneous cones are pointed out and several examples are computed. There is much in the paper to inspire further work on the subject.
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 32F45
zbMATH DE Number: 4021590
zbMATH Keywords: bounded convex domains
zbMATH Keywords: projectively invariant metrics
author: Takeshi Sasaki
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270151570
OpenAlex ID: W1997878584
DOI: 10.3836/TJM/1270151570
description: scientific article; zbMATH DE number 4021590