Bounded curvature manifolds without Euclidean isometric immersions of bounded mean curvature

For every integer $n\ge2$, we construct a smooth complete Riemannian metric $G_n$ on $\mathbb{R}^n$ with full curvature norm at most one and injectivity radius at least one for which no $C^2$ isometric immersion into a finite-dimensional Euclidean space has bounded mean curvature. In dimension two,