Let (S,g₀) be a hyperbolic surface, \({\rho}\) be a Hitchin representation for \({PSL(n,{\mathbb{R}})}\), and f be the unique \({\rho}\)-equivariant harmonic map from \({({\widetilde{S}}, \widetilde g_0)}\) to the corresponding symmetric space. We show its energy density satisfies \({e(f)\geq 1}\) and equality holds at one point only if \({e(f)\equiv 1}\) and \({\rho}\) is the base \({n}\)-Fuchsian representation of (S,g₀). In particular, we show given a Hitchin representation \({\rho}\) for \({PSL(n,{\mathbb{R}})}\), every \({\rho}\)-equivariant minimal immersion f from the hyperbolic plane \({{\mathbb{H}}^2}\) into the corresponding symmetric space X is distance-increasing, i.e. \({f^*g_{X}\geq g_{{\mathbb{H}}^2}}\). Equality holds at one point only if it holds everywhere and \({\rho}\) is an n-Fuchsian representation.

中文翻译：

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希钦表示形式的谐波图

令（S，g ₀）为双曲曲面，\（{\ rho} \）为\（{PSL（n，{\ mathbb {R}}）} \）的Hitchin表示，f为唯一\ （{\ rho} \）-等价谐波图，从\（{（{\ widetilde {S}}，\ widetilde g_0）} \）到对应的对称空间。我们证明其能量密度满足\（{e（f）\ geq 1} \），并且仅当\（{e（f）\ equiv 1} \）和\（{\ rho} \）为基本\（{n} \） -（S，g ₀）的Fuchsian表示。特别是，我们展示了给定的Hitchin表示形式\（{\ RHO} \）为\（{PSL（N，{\ mathbb {R}}）} \），每\（{\ RHO} \） -equivariant极小浸入˚F从双曲平面\（{{ \ mathbb {H}} ^ 2} \）到对应的对称空间X中的距离增加，即\（{f ^ * g_ {X} \ geq g _ {{\\ mathbb {H}} ^ 2}} \）。平等只有在任何地方都具有且\（{\ rho} \）是n-品红的表示形式时，才在某一点成立。