AI 中文总结
针对亏格≥2的闭定向曲面的辛同胚群单位连通分支，作者发展调和坐标方法，引入调和通量范数并证明其非退化，推导相关性质与不变量，推进辛拓扑领域研究。
AI 中文摘要
针对亏格 $g\ge2$ 的闭定向曲面的辛同胚群的单位连通分支 $G_\omega(\Sigma_g)$，我们发展了一种调和坐标方法。通过将阿贝尔-雅可比位移内禀分解为整体通量部分与零平均调和涨落，我们引入调和通量范数，并在不使用弗洛尔理论的情况下证明其在整个单位连通分支上非退化。我们还证明该范数在 $C^0$ 拓扑中连续，由此推导出 $\mathrm{Ham}(\Sigma_g,\omega)$ 在 $G_\omega(\Sigma_g)$ 内是 $C^0$ 闭的，并构造了单位附近的局部单射调和坐标图。在此过程中，我们推导了该范数的一阶展开式、定量不动点障碍，并提出了与调和位移滤过相关的有限维持续不变量——调和条形码。
英文摘要
We develop a harmonic-coordinate approach to the identity component $G_ω(Σ_g)$ of the symplectomorphism group of a closed oriented surface of genus $g\ge2$. Using an intrinsic decomposition of the Abel-Jacobi displacement into a global flux part and a zero-average harmonic fluctuation, we introduce the harmonic flux norm and prove its non-degeneracy on the full identity component without Floer theory. We also show the norm is continuous in the $C^0$-topology, deduce that $\mathrm{Ham}(Σ_g,ω)$ is $C^0$-closed inside $G_ω(Σ_g)$, and produce a locally injective harmonic-coordinate chart near the identity. Along the way we derive first-order expansions for the norm, quantitative fixed-point obstructions, and propose a finite-dimensional persistence invariant (the harmonic barcode) associated to the harmonic displacement filtration.
