Bulletin of the
Korean Mathematical Society BKMS

We find a $C^{\infty}$-continuous path of Riemannian metrics $g_t$ on $\mathbb{R}^{k}$, $k \geq 3$, for $0 \leq t \leq \varepsilon$ for some number $\varepsilon>0$ with the following property: $g_0$ is the Euclidean metric on $\mathbb{R}^{k}$, the scalar curvatures of $g_t$ are strictly decreasing in $t$ in the open unit ball and $g_t$ is isometric to the Euclidean metric in the complement of the ball. Furthermore we extend the discussion to the Fubini-Study metric in a similar way.

Keywords: scalar curvature

MSC numbers: 53B20, 53C20, 53C21