Harmonic maps from 𝕊2n+1 to ℂPn with low Morse index
Abstract
We study the Jacobi spectrum and low-index rigidity of complex Hopf fibrations. We first give a representation-theoretic description of the Jacobi operator of a homogeneous Riemannian fibration and apply it to π:(𝕊2n+1,gε)⟶(ℂPn,h), where gε is a Berger metric and h is the Fubini--Study metric. For n≥3, we determine all negative and zero eigenvalues and obtain explicit formulas for the Morse index and nullity for every ε>0. In particular, for the unit round metric, Ind(π)=2n+2 and Null(π)=n(3n+5). We then extend Rivière's rigidity argument from 𝕊3→𝕊2 to higher dimensions. If ϕ:𝕊2n+1→ℂPn is harmonic and Ind(ϕ)=2n+2, then dϕx∘ dϕx∗ commutes with the complex structure of ℂPn at every point. Consequently, up to an isometry of the domain, ϕ factors through the Hopf fibration as ϕ=P∘π∘ f, where P:ℂPn→ℂPn is holomorphic.
Related
- Instability and Morse Index of Exponentially Subelliptic Harmonic Maps
- Harmonic maps from 𝕊2n+1 into ℂℙn with least Morse index
- Spectral and convex geometry of α-harmonic maps and sharp Morse index rigidity
- Harmonic maps from spheres with lowest possible index and their properties
- Dirac-harmonic maps from manifolds with boundary via index theory
BibTeX
@misc{lara2026harmonic,
title = {{Harmonic maps from $\mathbbS^2n+1$ to $\mathbbCP^n$ with low Morse index}},
author = {Luiz Lara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07815},
eprint = {2610.07815},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07815},
}
RIS
TY - UNPB
TI - Harmonic maps from 𝕊^2n+1 to ℂP^n with low Morse index
AU - Luiz Lara
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07815
UR - https://arxiv.org/abs/2610.07815
AB - We study the Jacobi spectrum and low-index rigidity of complex Hopf fibrations. We first give a representation-theoretic description of the Jacobi operator of a homogeneous Riemannian fibration and apply it to π:(𝕊^2n+1,g_ε)⟶(ℂP^n,h), where g_ε is a Berger metric and h is the Fubini--Study metric. For n≥3, we determine all negative and zero eigenvalues and obtain explicit formulas for the Morse index and nullity for every ε>0. In particular, for the unit round metric, Ind(π)=2n+2 and Null(π)=n(3n+5). We then extend Rivière's rigidity argument from 𝕊^3→𝕊^2 to higher dimensions. If ϕ:𝕊^2n+1→ℂP^n is harmonic and Ind(ϕ)=2n+2, then dϕ_x∘ dϕ_x^∗ commutes with the complex structure of ℂP^n at every point. Consequently, up to an isometry of the domain, ϕ factors through the Hopf fibration as ϕ=P∘π∘ f, where P:ℂP^n→ℂP^n is holomorphic.
ER -