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Proposition 4.10 (Harmonic functions with slow exponential growth) . [03HK]

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Proposition 4.10 (Harmonic functions with slow exponential growth).

Suppose uu is harmonic on the Calabi model space (𝒞,g𝒞)(\mathcal{C},g_{\mathcal{C}}), i.e.,

(4.93) Δg𝒞​u=0,\Delta_{g_{\mathcal{C}}}u=0,

If u=O⁡(eδ​z)u=O(e^{\delta z}) for some δ∈(0,δ¯)\delta\in(0,\underline{\delta}), where δ¯>0\underline{\delta}>0 depends only on the Calabi model space (𝒞,g𝒞)(\mathcal{C},g_{\mathcal{C}}). then there are constants a0,b0∈ℝa_{0},b_{0}\in\mathbb{R} such that

(4.94) u⁡(z)=a0​z+b0+O⁡(e−δ¯​z).u(z)=a_{0}z+b_{0}+O(e^{-\underline{\delta}z}).

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