AI 中文总结
研究闭黎曼曲面上四次微分射线对应的PSp(4,R)-希钦分量中希钦表示的和乐渐近行为，得到其奇异值与特征值绝对值的显式渐近公式，公式增长率与四次微分局部四次根的积分相关。
AI 中文摘要
设\textrm{X}为闭黎曼曲面，\textrm{q}为\textrm{X}上非零全纯四次微分，属于\textrm{H}^0(\textrm{X},K^4)。对\textrm{t}>0，射线\textrm{tq}确定\textrm{PSp}(4,\textrm{R})-希钦分量中的一族希钦表示。研究当\textrm{t}\to+\text{∞}时，这些表示沿闭曲线的和乐渐近行为，得到所有奇异值及和乐所有特征值绝对值的显式渐近公式，其对数增长率由沿构成曲线测地代表的鞍形连接积分\textrm{q}的局部四次根（相对于奇异平坦度量\textrm{|q|}^{1/2}）给出，且未对\textrm{q}的零点阶数施加任何限制。
英文摘要
Let \(X\) be a closed Riemann surface and let \(q\in H^0(X,K^4)\) be a nonzero holomorphic quartic differential on \(X\). For \(t >0\), the ray \(tq\) determines a family of Hitchin representations in the \(\operatorname{PSp}(4,\mathbb R)\)-Hitchin component. We study, as \(t\to+\infty\), the asymptotic behavior of their holonomy along closed curves. We obtain explicit asymptotic formulas for all singular values and for the absolute values of all eigenvalues of the holonomy. Their logarithmic growth rates are given by integrating the local fourth roots of \(q\) along the saddle connections forming the geodesic representative of the curve with respect to the singular flat metric \(\lvert q\rvert^{1/2}\). No restriction is imposed on the orders of the zeros of \(q\).
Comments75 pages, 5 figures
