  In this work, we present a comprehensive study of holographic stress tensor
correlators on general Riemann surfaces, extending beyond the previously
well-studied torus cases to explore higher genus conformal field theories
(CFTs) within the framework of the Anti-de Sitter/conformal field theory
(AdS/CFT) correspondence. We develop a methodological approach to compute
holographic stress tensor correlators, employing the Schottky uniformization
technique to address the handlebody solutions for higher genus Riemann
surfaces. Through rigorous calculations, we derive four-point stress tensor
correlators, alongside recurrence relations for higher-point correlators,
within the $\mathrm{AdS}_3/\mathrm{CFT}_2$ context. Additionally, our research
delves into the holography of cutoff $\mathrm{AdS}_3$ spaces, offering novel
insights into the lower-point correlators of the $T\bar{T}$-deformed theories
on higher genus Riemann surfaces up to the first deformation order.
