The metric extension problem under positive and negative curvature conditions

Abstract

We first prove that every smooth boundary metric on a compact manifold extends to a metric with any prescribed positive lower bound for Ricci curvature, whereas extensions under stronger positive k^th-intermediate Ricci curvature conditions may fail due to local obstructions. For negative sectional curvature, we identify a global obstruction to extension. We then consider the class of compact manifolds defined by the existence of a metric with negative sectional curvature and boundary index at most one. On every manifold in this class, we prove that any smooth boundary metric extends to a metric with negative sectional curvature and strictly convex umbilical boundary. We further show that every such initial metric admits a complete asymptotically hyperbolic isometric extension with the same curvature condition and any prescribed conformal infinity. This class of manifolds is closed under boundary connected sums. As a geometric application of these extension results, every smooth metric on 𝕊ⁿ admits a strictly convex isometric embedding into ℝⁿ⁺¹ equipped with a complete metric of negative sectional curvature. The proofs combine neck constructions with corner smoothing for upper curvature bounds.

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BibTeX RIS

BibTeX

@misc{wang2026metric,
  title = {{The metric extension problem under positive and negative curvature conditions}},
  author = {Wenlong Wang and Jintian Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21518},
  eprint = {2609.21518},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21518},
}

RIS

TY  - UNPB
TI  - The metric extension problem under positive and negative curvature conditions
AU  - Wenlong Wang
AU  - Jintian Zhu
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21518
UR  - https://arxiv.org/abs/2609.21518
AB  - We first prove that every smooth boundary metric on a compact manifold extends to a metric with any prescribed positive lower bound for Ricci curvature, whereas extensions under stronger positive k^th-intermediate Ricci curvature conditions may fail due to local obstructions. For negative sectional curvature, we identify a global obstruction to extension. We then consider the class of compact manifolds defined by the existence of a metric with negative sectional curvature and boundary index at most one. On every manifold in this class, we prove that any smooth boundary metric extends to a metric with negative sectional curvature and strictly convex umbilical boundary. We further show that every such initial metric admits a complete asymptotically hyperbolic isometric extension with the same curvature condition and any prescribed conformal infinity. This class of manifolds is closed under boundary connected sums. As a geometric application of these extension results, every smooth metric on 𝕊^n admits a strictly convex isometric embedding into ℝ^n+1 equipped with a complete metric of negative sectional curvature. The proofs combine neck constructions with corner smoothing for upper curvature bounds.
ER  -

Abstract

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