Melting of the Euclidean metric to negative scalar curvature (Q2843784)

\textit{Y. Kang} et al. considered in [Bull. Korean Math. Soc. 49, No. 3, 581--588 (2012; Zbl 1242.53038)] the scalar-curvature melting of Euclidean metric in dimension \(3\). In the paper under review the author completes these considerations in any dimension \(n \geq 3\). Namely, he finds a \(C^\infty\)-continuous path of Riemannian metrics \(g_t\) on \(\mathbb R^n\), where \(t\in[0,\varepsilon]\), for some \(\varepsilon>0\) satisfying the following property: \(g_0\) is the Euclidean metric, the scalar curvature of the path metrics is decreasing with respect to \(t\) in the open unit ball and \(g_t\) is isometric to the Euclidean metric outside of the ball. The author extends these investigations to the Fubini-Study metric.