无临界点的分数Toledo表示能量
Critical-point-free energy for fractional-Toledo representations
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中文总结 AI 辅助
针对亏格$g\ge2$的闭定向曲面，构造满足非整数Toledo不变量且能量函数无临界点的不可约约化表示，证明对应分支极小曲面遗忘映射非满射。
中文摘要 AI 辅助
设$S_g$为亏格$g\ge2$的闭定向曲面，对约化表示$\rho:\pi_1(S_g)\to\PU(2,1)$，$E_\rho$是Teichmüller空间上与到$\mathbb{CH}^2$的等变调和映射关联的能量函数。对所有满足$3\nmid d$的正整数$d$、所有足够大的$h$及每个$g>h$，我们构造不可约约化表示$\rho_{g,h,d}:\pi_1(S_g)\to\PU(2,1)$，满足$\tau(\rho_{g,h,d})=2h-2-\frac{2d}{3}\notin\mathbb Z$且$\operatorname{Crit}(E_{\rho_{g,h,d}})=\varnothing$。因此，这些非整数Toledo分支中关联的分支极小曲面遗忘映射不是满射。
英文摘要
Let $S_g$ be a closed oriented surface of genus $g\ge2$. For a reductive representation $ρ:π_1(S_g)\to\PU(2,1)$, let $E_ρ$ be the energy function on Teichmüller space associated to equivariant harmonic maps into $\CH^2$. For every positive integer $d$ with $3\nmid d$, all sufficiently large $h$, and every $g>h$, we construct an irreducible reductive representation \[ ρ_{g,h,d}:π_1(S_g)\to\PU(2,1) \] with \[ τ(ρ_{g,h,d})=2h-2-\frac{2d}{3}\notin\mathbb Z, \qquad \operatorname{Crit}(E_{ρ_{g,h,d}})=\varnothing. \] Consequently, the associated branched-minimal-surface forgetful map is not surjective in these nonintegral Toledo components.
