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PreprintCC BY 4.0arXiv

Classification of Isometric Immersions of Spherical and HCMU Footballs into E3

  • Zhiqiang Wei
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Abstract

We study global isometric immersions of rotational metrics with two conical ends into Euclidean three-space. For a metric g=du2+f(u)2dθ2 of positive Gauss curvature, we prove that vanishing endpoint traces of the weighted mixed shape coefficient f2b force every smooth isometric immersion to be rotational. The resulting congruence classes are parametrized by an integer angular covering degree bounded below by the two cone-angle parameters. The proof combines a scalar elliptic equation derived from the Gauss--Codazzi system, the maximum principle, and a global angular closing condition. For spherical footballs, isolated-singularity theory and a direct estimate in coordinates conformal for the second fundamental form establish the trace condition for every smooth immersion of the regular part. We also classify all rotational isometric immersions of HCMU footballs, including those whose curvature changes sign. When the minimum curvature is nonnegative, this classification exhausts the immersions satisfying the trace condition, and hence all immersions admitting a C2 polar extension at both tips.

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Cite

BibTeXRIS
BibTeX
@misc{wei2026classification,
  title = {{Classification of Isometric Immersions of Spherical and HCMU Footballs into $\E^3$}},
  author = {Zhiqiang Wei},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25232},
  eprint = {2610.25232},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25232},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Classification of Isometric Immersions of Spherical and HCMU Footballs into E^3
AU  - Zhiqiang Wei
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25232
UR  - https://arxiv.org/abs/2610.25232
AB  - We study global isometric immersions of rotational metrics with two conical ends into Euclidean three-space. For a metric g=du^2+f(u)^2dθ^2 of positive Gauss curvature, we prove that vanishing endpoint traces of the weighted mixed shape coefficient f^2b force every smooth isometric immersion to be rotational. The resulting congruence classes are parametrized by an integer angular covering degree bounded below by the two cone-angle parameters. The proof combines a scalar elliptic equation derived from the Gauss--Codazzi system, the maximum principle, and a global angular closing condition. For spherical footballs, isolated-singularity theory and a direct estimate in coordinates conformal for the second fundamental form establish the trace condition for every smooth immersion of the regular part. We also classify all rotational isometric immersions of HCMU footballs, including those whose curvature changes sign. When the minimum curvature is nonnegative, this classification exhausts the immersions satisfying the trace condition, and hence all immersions admitting a C^2 polar extension at both tips.
ER  -