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On the degenerate Weyl problem on isometric immersions

  • Siran Li
  • Xiangxiang Su
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Abstract

This paper is concerned with the Weyl problem, \emph{i.e.}, the existence of isometric immersions or embeddings of two-spheres into the three-dimensional Euclidean space or general ambient three-manifolds. We establish two results on the degenerate Weyl problem, namely when the Gaussian curvature Kg is only nonnegative rather than strictly positive. First, for a smooth metric g on the two-sphere, if Kg is strictly positive except at finitely many points where the Hessian of Kg is positive definite, then g admits a global C2,1-isometric embedding into ℝ3. It appears to be the first result on the degenerate Weyl problem with purely intrinsic conditions on g. Second, for a general simply-connected ambient three-manifold (ℳ,g̅), if Kg ≥ K0 ≥ secg̅ for some constant K0, (Kg−K0)−1/2 ∈ Lp with p ≥ 2, and a certain uniform pinching condition holds for approximate nondegenerate isometric immersions, then there exists a W3,p-isometric immersion. Alongside we also resolve the nondegenerate Weyl problem (\emph{i.e.}, when Kg>0) into general simply-connected ambient three-manifolds for g, g̅ ∈ C2,1.

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Cite

BibTeXRIS
BibTeX
@misc{li2026degenerate,
  title = {{On the degenerate Weyl problem on isometric immersions}},
  author = {Siran Li and Xiangxiang Su},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28905},
  eprint = {2609.28905},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28905},
}
RIS
TY  - UNPB
TI  - On the degenerate Weyl problem on isometric immersions
AU  - Siran Li
AU  - Xiangxiang Su
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.28905
UR  - https://arxiv.org/abs/2609.28905
AB  - This paper is concerned with the Weyl problem, \emph{i.e.}, the existence of isometric immersions or embeddings of two-spheres into the three-dimensional Euclidean space or general ambient three-manifolds. We establish two results on the degenerate Weyl problem, namely when the Gaussian curvature K_g is only nonnegative rather than strictly positive. First, for a smooth metric g on the two-sphere, if K_g is strictly positive except at finitely many points where the Hessian of K_g is positive definite, then g admits a global C^2,1-isometric embedding into ℝ^3. It appears to be the first result on the degenerate Weyl problem with purely intrinsic conditions on g. Second, for a general simply-connected ambient three-manifold (ℳ,g̅), if K_g ≥ K_0 ≥ sec_g̅ for some constant K_0, (K_g−K_0)^−1/2 ∈ L^p with p ≥ 2, and a certain uniform pinching condition holds for approximate nondegenerate isometric immersions, then there exists a W^3,p-isometric immersion. Alongside we also resolve the nondegenerate Weyl problem (\emph{i.e.}, when K_g>0) into general simply-connected ambient three-manifolds for g, g̅ ∈ C^2,1.
ER  -