Quasi-isometric embeddings for shrinking maps from surfaces into the moduli space

Yibo Zhang

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arXiv · 2511.09317

Quasi-isometric embeddings for shrinking maps from surfaces into the moduli space

Yibo Zhang

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Abstract

We investigate shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces. For such a map and its lift to the Teichm\"uller space, we consider whether they are quasi-isometric embeddings with respect to natural metrics like the Teichm\"uller distance and the intrinsic distance. Under a mild condition, we prove that these properties are characterised solely by the map's monodromy. These characterisations apply, in particular, to holomorphic maps.

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Yibo Zhang. 2025-11-12. Quasi-isometric embeddings for shrinking maps from surfaces into the moduli space. https://arxiv.org/abs/2511.09317

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