AI 中文总结
本文研究复Hopf纤维化的Jacobi谱，给出低指标刚性定理：若调和映射的Morse指标为2n+2，则其必通过Hopf纤维化分解，且分解中的映射为全纯。
AI 中文摘要
我们研究复Hopf纤维化的Jacobi谱和低指标刚性。首先，我们给出齐次黎曼纤维化的Jacobi算子的表示论描述，并将其应用于\\[ \pi:(\mathbb{S}^{2n+1},g_\epsilon)\longrightarrow(\mathbb{C}P^n,h), \\] 其中$g_\epsilon$是Berger度量，$h$是Fubini–Study度量。对于$n\geq3$，我们确定所有负特征值和零特征值，并对每个$\epsilon>0$得到Morse指标和零度的显式公式。特别地，对于单位圆度量，$\mathrm{Ind}(\pi)=2n+2$且$\mathrm{Null}(\pi)=n(3n+5)$。然后我们将Rivière的刚性论证从$\mathbb{S}^3\to\mathbb{S}^2$推广到更高维。如果$\phi:\mathbb{S}^{2n+1}\to\mathbb{C}P^n$是调和的且$\mathrm{Ind}(\phi)=2n+2$，则$d\phi_x\circ d\phi_x^\ast$在每一点与$\mathbb{C}P^n$的复结构交换。因此，在定义域的一个等距变换下，$\phi$通过Hopf纤维化分解为$\phi=P\circ\pi\circ f$，其中$P:\mathbb{C}P^n\to\mathbb{C}P^n$是全纯的。
英文摘要
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 \[ π:(\mathbb{S}^{2n+1},g_ε)\longrightarrow(\mathbb{C}P^n,h), \] where $g_ε$ is a Berger metric and $h$ is the Fubini--Study metric. For $n\geq3$, 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, $\mathrm{Ind}(π)=2n+2$ and $\mathrm{Null}(π)=n(3n+5)$. We then extend Rivière's rigidity argument from $\mathbb{S}^3\to\mathbb{S}^2$ to higher dimensions. If $ϕ:\mathbb{S}^{2n+1}\to\mathbb{C}P^n$ is harmonic and $\mathrm{Ind}(ϕ)=2n+2$, then $dϕ_x\circ dϕ_x^\ast$ commutes with the complex structure of $\mathbb{C}P^n$ at every point. Consequently, up to an isometry of the domain, $ϕ$ factors through the Hopf fibration as $ϕ=P\circπ\circ f$, where $P:\mathbb{C}P^n\to\mathbb{C}P^n$ is holomorphic.
