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PreprintCC BY 4.0HAL

Remarks on the critical catenoid and an overdetermined eigenvalue problem in Bbb S2

  • Rabah Souam
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Abstract

It was recently proved that an embedded minimal annulus with free boundary in the unit ball Bbb B3 of the Euclidean space is, up to a rotation, the critical catenoid. We prove that an immersed, possibly branched, free boundary minimal annulus in Bbb B3 with embedded boundary components is necessarily free of branch points and is embedded. This shows the uniqueness of the critical catenoid holds under these weaker hypotheses. We then apply these results to prove that if Ω is a sufficiently regular annulus in Bbb S2 for which there exists a smooth function satisfying the overdetermined eigenvalue problem Δ u+ 2u = 0 on Ω u=0 in ∂Ω |∇ u|=1 in ∂Ω. then Ω is, up to rotation, a rotational annulus with equatorial symmetry.

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Cite

BibTeXRIS
BibTeX
@misc{souam2026remarks,
  title = {{Remarks on the critical catenoid and an overdetermined eigenvalue problem in $\Bbb S^2$}},
  author = {Rabah Souam},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06386},
  eprint = {2610.06386},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05775098v2},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Remarks on the critical catenoid and an overdetermined eigenvalue problem in Bbb S^2
AU  - Rabah Souam
PY  - 2026
DA  - 2026/10/02
PB  - HAL
DO  - 10.48550/arxiv.2610.06386
UR  - https://hal.science/hal-05775098v2
AB  - It was recently proved that an embedded minimal annulus with free boundary in the unit ball Bbb B^3 of the Euclidean space is, up to a rotation, the critical catenoid. We prove that an immersed, possibly branched, free boundary minimal annulus in Bbb B^3 with embedded boundary components is necessarily free of branch points and is embedded. This shows the uniqueness of the critical catenoid holds under these weaker hypotheses. We then apply these results to prove that if Ω is a sufficiently regular annulus in Bbb S^2 for which there exists a smooth function satisfying the overdetermined eigenvalue problem Δ u+ 2u = 0 on Ω u=0 in ∂Ω |∇ u|=1 in ∂Ω. then Ω is, up to rotation, a rotational annulus with equatorial symmetry.
ER  -