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PreprintarXiv licencearXiv

Harmonic maps from π•Š2n+1 into β„‚β„™n with least Morse index

  • Qun Chen
  • Guofang Wang
  • Mingwei Zhang
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Abstract

Let u:π•Š2n+1β†’β„‚β„™n be a smooth nonconstant harmonic map. We prove that its Morse index is equal to $ 2n+2$ if and only if u = h ∘ Ο€βˆ˜ Ξ, where Ξ:π•Š2n+1β†’π•Š2n+1 is an isometric transformation, Ο€:π•Š2n+1β†’β„‚β„™n is the Hopf map and h:β„‚β„™nβ†’β„‚β„™n is a holomorphic map with degree one. When n=1, it refines a classical result of Urakawa and a recent result of RiviΓ¨re. Moreover, we prove that hβˆ˜Ο€βˆ˜ Ξ has nullity 3n2+5n.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026harmonic,
  title = {{Harmonic maps from $\mathbbS^2n+1$ into $\mathbbCP^n$ with least Morse index}},
  author = {Qun Chen and Guofang Wang and Mingwei Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37945},
  eprint = {2609.37945},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37945},
}
RIS
TY  - UNPB
TI  - Harmonic maps from π•Š^2n+1 into β„‚β„™^n with least Morse index
AU  - Qun Chen
AU  - Guofang Wang
AU  - Mingwei Zhang
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37945
UR  - https://arxiv.org/abs/2609.37945
AB  - Let u:π•Š^2n+1β†’β„‚β„™^n be a smooth nonconstant harmonic map. We prove that its Morse index is equal to $ 2n+2$ if and only if u = h ∘ Ο€βˆ˜ Ξ, where Ξ:π•Š^2n+1β†’π•Š^2n+1 is an isometric transformation, Ο€:π•Š^2n+1β†’β„‚β„™^n is the Hopf map and h:β„‚β„™^nβ†’β„‚β„™^n is a holomorphic map with degree one. When n=1, it refines a classical result of Urakawa and a recent result of RiviΓ¨re. Moreover, we prove that hβˆ˜Ο€βˆ˜ Ξ has nullity 3n^2+5n.
ER  -