A note on characteristic functions and projectively invariant metrics on a bounded convex domain
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.
scientific article; zbMATH DE number 4021590