We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, Z ( s ), on Teichmüller space. We then use this formula to determine the asymptotic behavior as $$\mathfrak {R}s \rightarrow \infty $$ R s → ∞ of the second variation. As a consequence, for $$m \in {\mathbb {N}}$$ m ∈ N , we obtain the complete expansion in m of the curvature of the vector bundle $$H^0(X_t, {\mathcal {K}}_t)\rightarrow t\in {\mathcal {T}}$$ H 0 ( X t , K t ) → t ∈ T of holomorphic m-differentials over the Teichmüller space $${\mathcal {T}}$$ T , for m large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $$O(m^2 \mathrm{e}^{-l_0 m}),$$ O ( m 2 e - l 0 m ) , where $$l_0$$ l 0 is the length of the shortest closed hyperbolic geodesic.

中文翻译：

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Selberg zeta 函数和曲率渐近的第二个变体

我们给出了在 Teichmüller 空间上 Selberg zeta 函数 Z ( s ) 对数的第二个变体的明确公式。然后我们使用这个公式来确定渐近行为为 $$\mathfrak {R}s \rightarrow \infty $$ R s → ∞ 的第二个变化。因此，对于 $$m \in {\mathbb {N}}$$ m ∈ N ，我们获得了向量丛 $$H^0(X_t, {\mathcal {K }}_t)\rightarrow t\in {\mathcal {T}}$$ H 0 ( X t , K t ) → t ∈ T 在 Teichmüller 空间上的全纯 m 微分 $${\mathcal {T}}$ $ T ，对于 m 大。此外，我们表明该曲率与 Quillen 曲率一致，直至指数衰减项 $$O(m^2 \mathrm{e}^{-l_0 m}),$$ O ( m 2 e - l 0 m ) ，其中 $$l_0$$l 0 是最短闭合双曲测地线的长度。