The scalar curvature of a projectively invariant metric on a convex domain
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 \[ K(x)=-1+O(\mathrm{dist}(x,\partial \Omega)^3). \]
scientific article; zbMATH DE number 6876746
scientific article