to get a smooth metric with the boundary isometric to the squashed sphere $S^{4k-1}$ coming from the Hopf fibration. Let’s denote this metric on the boundary by $h_\varepsilon$ for a later purpose.

We want to close it up by a $4k$-ball $B^{4k}$ equipped with a metric of positive scalar curvature. To construct such a metric we resort to the Gromov-Lawson surgery again. Take a sphere $S^{4k}$ with any metric of positive scalar curvature and let $p$ be any point on it. As before, we perform a Gromov-Lawson surgery in a sufficiently small neighborhood of $p$ to get a $4k$-ball with the positive scalar curvature and the cylindrical end isometric to $S^{4k-1}(\varepsilon')\times [0,1]$ for a $\varepsilon'>0$. And then we take a homothety of the whole thing by $\frac{1}{\varepsilon'}$ so that the boundary is isometric to the round sphere $(S^{4k-1}(1),h_{1})$. In order to glue this to the above obtained part, we have to homotope the metric on the boundary. We take a homotopy $H_{3}(t)=\lambda(t)h_{1}+(1-\lambda(t))h_{\varepsilon}$ for $t\in
[0,1]$.

The metric $H_{3}(t)$ on $S^{4k-1}$ has positive scalar curvature for every $t\in [0,1]$.

Note that $h_{1}$ and $h_{\varepsilon}$ differ only by the size of the Hopf fiber. So for each $t$, $H_3(t)$ also has the same Riemannian submersion structure with the fiber isometric to the round $3$-sphere of radius $r(t):=\lambda(t)+(1-\lambda(t))\varepsilon$. By the O’Neill’s formula [@besse], $$\begin{aligned}
s_{H_{3}(t)}&=&\frac{1}{r^{2}(t)}s_{f}+s_{b}\circ
\pi-r^{2}(t)|A|^{2},\end{aligned}$$ where $s_{f}$, $s_{b}$, and $A$ denote the scalar curvature of the fiber and the base, and the integrability tensor for $t=0$ respectively. Thus $s_{H_{3}(t)}$ is constant for each $t$ and increases as $t$ increases. From the fact that $s_{H_{3}(0)}\equiv(4k-1)(4k-2)>0$, the result follows.

Nevertheless the metric $H_{3}(t)+dt^2$ on $S^{4k-1}\times [0,1]$ may not have positive scalar curvature in general. But due to Gromov and Lawson’s lemma in [@GL], for a sufficiently large cons