Noncritical multi-bubbles and hyperbolic double bubbles

Abstract

We prove that standard k-bubbles uniquely minimize perimeter in ℝⁿ and 𝕊ⁿ for n≥2 and 1≤ k≤ n, and that standard double bubbles uniquely minimize perimeter in ℍⁿ for n≥3. For nonnegative curvature, the proof combines the structure and spectral theory of Milman--Neeman with a uniform semialgebraic exceptional set, an almost-everywhere profile comparison, and localized variational arguments. A curvature-variable lift gives a uniformly elliptic comparison at the Euclidean limit. The independent hyperbolic argument uses the rotational structure of Cotton--Freeman, a scalar meridian force potential, and a transverse conformal variation. The critical range k=n+1 in ℝⁿ and 𝕊ⁿ, and hyperbolic clusters with more than two finite cells, are not covered.

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BibTeX RIS

BibTeX

@misc{wang2026noncritical,
  title = {{Noncritical multi-bubbles and hyperbolic double bubbles}},
  author = {Zihao Wang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00252},
  eprint = {2611.00252},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00252},
}

RIS

TY  - UNPB
TI  - Noncritical multi-bubbles and hyperbolic double bubbles
AU  - Zihao Wang
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00252
UR  - https://arxiv.org/abs/2611.00252
AB  - We prove that standard k-bubbles uniquely minimize perimeter in ℝ^n and 𝕊^n for n≥2 and 1≤ k≤ n, and that standard double bubbles uniquely minimize perimeter in ℍ^n for n≥3. For nonnegative curvature, the proof combines the structure and spectral theory of Milman--Neeman with a uniform semialgebraic exceptional set, an almost-everywhere profile comparison, and localized variational arguments. A curvature-variable lift gives a uniformly elliptic comparison at the Euclidean limit. The independent hyperbolic argument uses the rotational structure of Cotton--Freeman, a scalar meridian force potential, and a transverse conformal variation. The critical range k=n+1 in ℝ^n and 𝕊^n, and hyperbolic clusters with more than two finite cells, are not covered.
ER  -

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