A counterexample to Yau's bounded-mean-curvature embedding question
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 ℝ4 with bounded Ricci curvature, positive injectivity radius, and unbounded full curvature. The Gauss equation then forces every C2 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.
Related
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BibTeX
@misc{cheng2026counterexample,
title = {{A counterexample to Yau's bounded-mean-curvature embedding question}},
author = {Haoxuan Cheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00153},
eprint = {2610.00153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00153},
}
RIS
TY - UNPB
TI - A counterexample to Yau's bounded-mean-curvature embedding question
AU - Haoxuan Cheng
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00153
UR - https://arxiv.org/abs/2610.00153
AB - 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 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.
ER -