Scalar and Mean Curvature Comparison on Compact Cylinder

Let $X$ be a closed, oriented Riemannian manifold. Denote by $(M=X\times I,\mathrm{\partial }M=X\times \{0\}\cup X\times \{1\},g)$ a compact cylinder with smooth boundary, $\dim M⩾3$. In this article, we address the following question: If $g$ is a Riemannian metric having (i) positive scalar curvature (PSC metric) on $M$ and nonnegative mean curvature on $\mathrm{\partial }M$; and (ii) the $g$-angle between normal vector field ${\nu }_g$ along $\mathrm{\partial }M$ and ${\mathrm{\partial }}_{\xi }\in \mathrm{\Gamma }(TI)$ being less than $\frac{\pi }{4}$, then there exists a metric $\tilde{g}$ on $M$ such that $\tilde{g}{\mathrm{\mid }}_{X\times \{0\}}$ is a PSC metric on $X\cong X\times \{0\}$. Equivalently, we show that if $X$ admits no PSC metric, but $M$ admits a PSC metric $g$ satisfying the angle condition, then the mean curvature on $\mathrm{\partial }M$ must be negative somewhere. This generalizes a result of Gromov and Lawson (Ann. of Math. (2), 1980) for $X={\mathbb{T}}^n$.