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Bounded curvature manifolds without Euclidean isometric immersions of bounded mean curvature
Abstract
For every integer n≥2, we construct a smooth complete Riemannian metric Gn on ℝn with full curvature norm at most one and injectivity radius at least one for which no C2 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.
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BibTeX
@misc{cheng2026bounded,
title = {{Bounded curvature manifolds without Euclidean isometric immersions of bounded mean curvature}},
author = {Haoxuan Cheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19168},
eprint = {2609.19168},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19168},
}
RIS
TY - UNPB
TI - Bounded curvature manifolds without Euclidean isometric immersions of bounded mean curvature
AU - Haoxuan Cheng
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.19168
UR - https://arxiv.org/abs/2609.19168
AB - For every integer n≥2, we construct a smooth complete Riemannian metric G_n on 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.
ER -