Hyperbolic Superspaces and Super-Riemann Surfaces

In this paper, we will generalize some results in Manin’s paper “Three-dimensional hyperbolic geometry as $\mathrm{\infty }$-adic Arakelov geometry” to the supergeometric setting. More precisely, viewing ${\mathbb{C}}^{1\mathrm{\mid }1}$ as the boundary of the hyperbolic superspace ${\mathcal{H}}^{3\mathrm{\mid }2}$, we reexpress the super-Green functions on the supersphere ${\widehat{\mathbb{C}}}^{1\mathrm{\mid }1}$ and the supertorus $T^{1\mathrm{\mid }1}$ by some data derived from the supergeodesics in ${\mathcal{H}}^{3\mathrm{\mid }2}$.