The scalar curvature of a projectively invariant metric on a convex domain
type: Item
instance of: scholarly article
title: The scalar curvature of a projectively invariant metric on a convex domain
zbMATH Open document ID: 1398.53020
author: Xianqiang Yang
published in: Journal of Geometry
publication date: 2018-05-30T00:00:00Z
review text: The following results are proved. Theorem 1. Let \(\Omega\subset \mathbb R^n\) be a strongly convex bounded domain with smooth boundary, \(\omega\) a projectively invariant metric on \(\Omega\) and \(R\) be the scalar curvature of \((\Omega,\omega)\). For \(x\) near \(\partial \Omega\), choose \(y(x)\) so that \(\mathrm{dist}(x,y)=\mathrm{dist}(x,\partial \Omega)\), then \[ R(x)=-n(n-1)+2^{{-2}\over{n+1}}J(y)\cdot \mathrm{dist}(x,\partial \Omega)+O(\mathrm{dist}(x,\partial \Omega)^2), \] where \(J\) is the Fubini-Pick invariant of the boundary \(\partial \Omega\). Theorem 2. Let \(\Omega\subset \mathbb R^2\) be a strongly convex bounded domain with smooth boundary, and \(K\) be the sectional curvature of \((\Omega,\omega)\), then \[ K(x)=-1+O(\mathrm{dist}(x,\partial \Omega)^3). \]
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 6876746
zbMATH Keywords: scalar curvature
zbMATH Keywords: Fubini-Pick invariant
zbMATH Keywords: projectively invariant metric
reviewed by: Huili Liu
MaRDI profile type: Publication
full work available at URL: s00022-018-0416-4
OpenAlex ID: W2797351695
Wikidata QID: Q124879719
cites work: Global affine differential geometry of hypersurfaces
cites work: Q3333677
cites work: A note on characteristic functions and projectively invariant metrics on a bounded convex domain
cites work: Sectional curvature of projective invariant metrics on a strictly convex domain
cites work: Q3195803
cites work: Boundary estimates of solutions for a class of Monge-Ampère equations
DOI: 10.1007/S00022-018-0416-4
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description: scientific article; zbMATH DE number 6876746