Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $g\ge2$. Denote $\omega_1, \dots , \omega_g$ a basis of holomorphic differentials, and consider the Riemaniann metric $g = \omega_1 \circ \overline{\omega_1}+\dots + \omega_1 \circ \overline{\omega_1}$ on $X$. I am interested in comparing $g$ with the hyperbolic metric $h$ (i.e. with constant curvature -1) on $X$ which is conformal to $g$.

I am interested in results comparing $h$ and $g$. In particular, there exist constants $c_1, c_2$ sucht that $c_1 \ell_h(\gamma) \le \ell_g(\gamma) \le c_2 \ell_h(\gamma)$, where $\gamma$ is a curve in $X$ and $\ell_{h}, \ell_g$ denotes the lengths with respect to the two different Riemaniann structures?

I am not an expert on this topic, and I believe such comparisons should be well-studied in the literature, but I couldn't find an appropriate reference or keyword to help my search.

I am in particular interested in metrics of the form $g = \omega_1 \circ \overline{\omega_1}+\dots + \omega_1 \circ \overline{\omega_1}$ because this is the metric on $X$ induced by the embedding of the curve in the Jacobian $X \to J(X) \cong \mathbb{C}^g\big/\Lambda$ given by the Abel-Jacobi map, expressed using the basis $\omega_1, \dots , \omega_g$.

Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $a\ge2$. Denote $\omega_1, \dots , \omega_a$ a basis of holomorphic differentials, and consider the Riemaniann metric $g = \omega_1 \circ \overline{\omega_1}+\dots + \omega_a \circ \overline{\omega_a}$ on $X$. I am interested in comparing $g$ with the hyperbolic metric $h$ (i.e. with constant curvature -1) on $X$ which is conformal to $g$.

I am interested in results comparing $h$ and $g$. In particular, there exist constants $c_1, c_2$ (depending on $a$ and $\omega_1, \dots , \omega_a$) such that $c_1 \ell_h(\gamma) \le \ell_g(\gamma) \le c_2 \ell_h(\gamma)$, where $\gamma$ is a curve in $X$ and $\ell_{h}, \ell_g$ denotes the lengths with respect to the two different Riemaniann structures?

I am not an expert on this topic, and I believe such comparisons should be well-studied in the literature, but I couldn't find an appropriate reference or keyword to help my search.

I am in particular interested in metrics of the form $g = \omega_1 \circ \overline{\omega_1}+\dots + \omega_a \circ \overline{\omega_a}$ because this is the metric on $X$ induced by the embedding of the curve in the Jacobian $X \to J(X) \cong \mathbb{C}^a\big/\Lambda$ given by the Abel-Jacobi map, expressed using the basis $\omega_1, \dots , \omega_a$.