Year: 2026
Type: preprint
Abstract: Yau asked whether every complete Riemannian manifold with bounded Ricci curvature and positive injectivity radius admits an isometric embedding into some Euclidean space with bounded mean curvature. We construct a counterexample: a smooth complete metric on $\mathbb{R}^4$ with bounded Ricci curvature, positive injectivity radius, and unbounded full curvature. The Gauss equation then forces every $C^2$ isometric immersion of this metric into a finite-dimensional Euclidean space to have unbounded mean curvature. The metric is assembled from disjoint curvature peaks whose Ricci contractions stay uniformly bounded. A short-geodesic curvature estimate and the Jacobi equation give the uniform injectivity-radius bound. The construction works in every dimension at least four, and in dimension four the Ricci curvature may be chosen to tend to zero at infinity.
Source: arXiv (Cornell University)
Author: Cheng, Haoxuan
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215725445