Teichmüller空间中与Hitchin表示相关的能量的严格多重亚谐性
Strict plurisubharmonicity of the energy on Teichmüller space associated to Hitchin representations
摘要：
设$Σ$是亏格至少为2的闭曲面，$ρ\colon π_1(Σ) \to G$是$G=\text{PSL}(n,\mathbb{R})$，$\text{PSp}(2n,\mathbb{R})$，$\text{PSO}(n,n+1)$或$\text{G}_2$的Hitchin表示.本文考虑$Σ$的Teichmüller空间上的能量泛函$E$，它赋予$\mathcal{T}(Σ)$中每一点相应的$ρ$ -等变调和映射的能量.本文的主要结果是：$E$是严格多重次调和的.作为推论，我们得到了能量泛函的任意临界点的指数上的$3 \cdot \text{genus}(Σ) -3$的上界。
英文摘要：
Let $Σ$ be a closed surface of genus least two and $ρ\colon π_1(Σ) \to G$ a Hitchin representation into $G=\text{PSL}(n,\mathbb{R})$, $\text{PSp}(2n,\mathbb{R})$, $\text{PSO}(n,n+1)$ or $\text{G}_2$. We consider the energy functional $E$ on the Teichmüller space of $Σ$ which assigns to each point in $\mathcal{T}(Σ)$ the energy of the associated $ρ$-equivariant harmonic map. The main result of this paper is that $E$ is strictly plurisubharmonic. As a corollary we obtain an upper bound of $3 \cdot \text{genus}(Σ) -3$ on the index of any critical point of the energy functional.
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