Bounded curvature manifolds without Euclidean isometric immersions of bounded mean curvature

Abstract

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, bounded second fundamental form would give uniformly controlled finite Jacobian representations of the Laplacian of the conformal factor. We construct disjoint conformal blocks for which the curvature remains bounded while the finite Jacobian representation cost tends to infinity. Taking Euclidean products gives all higher dimensions. The result answers Yau's Problem~52 negatively under the stronger assumption of bounded full curvature.