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Constantly curved minimal immersions of the two-sphere in unitary groups

  • Rui Pacheco
  • Mehmood Ur Rehman
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Abstract

In this article, we investigate rigidity results for constantly curved minimal immersions of the two-sphere S2 into the unitary group U(n). Using loop group methods for harmonic maps, we establish a correspondence between such immersions and a distinguished class of constantly curved holomorphic immersions of S2 into finite-dimensional Grassmannians. In the case U(3), we classify the constantly curved minimal immersions of S2 with uniton number one and prove that, under a natural unramifiedness condition, those of uniton number two are S1-invariant; as a consequence, every constantly curved totally unramified minimal immersion S2→ U(3) of uniton number two is unitarily congruent to the composition of the first Gauss map of the Veronese curve in ℂP2 with the Cartan embedding ℂP2↪ U(3).

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Cite

BibTeXRIS
BibTeX
@misc{pacheco2026constantly,
  title = {{Constantly curved minimal immersions of the two-sphere in unitary groups}},
  author = {Rui Pacheco and Mehmood Ur Rehman},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12962},
  eprint = {2609.12962},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12962},
}
RIS
TY  - UNPB
TI  - Constantly curved minimal immersions of the two-sphere in unitary groups
AU  - Rui Pacheco
AU  - Mehmood Ur Rehman
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12962
UR  - https://arxiv.org/abs/2609.12962
AB  - In this article, we investigate rigidity results for constantly curved minimal immersions of the two-sphere S^2 into the unitary group U(n). Using loop group methods for harmonic maps, we establish a correspondence between such immersions and a distinguished class of constantly curved holomorphic immersions of S^2 into finite-dimensional Grassmannians. In the case U(3), we classify the constantly curved minimal immersions of S^2 with uniton number one and prove that, under a natural unramifiedness condition, those of uniton number two are S^1-invariant; as a consequence, every constantly curved totally unramified minimal immersion S^2→ U(3) of uniton number two is unitarily congruent to the composition of the first Gauss map of the Veronese curve in ℂP^2 with the Cartan embedding ℂP^2↪ U(3).
ER  -