PreprintCC BY 4.0arXiv
Ramification and Finiteness of Equivariant Minimal Surfaces
Abstract
Let S be a closed oriented surface of genus g, with g ≥ 2, and let 𝒳0good be the good locus of PU(2, 1) representations with Toledo invariant zero. For 1 ≤ b ≤ 2g−3, we show that representations admitting an equivariant minimal map of total ramification at least b, for some marked complex structure, form a countable union of compact null sets of Hausdorff dimension at most $16g-16-2b$. For larger b, this locus is empty. Consequently, on a dense Gδ subset of full measure in 𝒳0good, every nonconstant equivariant minimal map is an immersion, for every marked complex structure. On the convex cocompact locus, a dense Gδ subset of full measure consists of representations admitting a finite nonempty set of marked equivariant minimal immersions.
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BibTeX
@misc{biswas2026ramification,
title = {{Ramification and Finiteness of Equivariant Minimal Surfaces}},
author = {Indranil Biswas and Satyajit Guin and Pradip Kumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29263},
eprint = {2610.29263},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29263},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ramification and Finiteness of Equivariant Minimal Surfaces
AU - Indranil Biswas
AU - Satyajit Guin
AU - Pradip Kumar
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29263
UR - https://arxiv.org/abs/2610.29263
AB - Let S be a closed oriented surface of genus g, with g ≥ 2, and let 𝒳_0^good be the good locus of PU(2, 1) representations with Toledo invariant zero. For 1 ≤ b ≤ 2g−3, we show that representations admitting an equivariant minimal map of total ramification at least b, for some marked complex structure, form a countable union of compact null sets of Hausdorff dimension at most $16g-16-2b$. For larger b, this locus is empty. Consequently, on a dense G_δ subset of full measure in 𝒳_0^good, every nonconstant equivariant minimal map is an immersion, for every marked complex structure. On the convex cocompact locus, a dense G_δ subset of full measure consists of representations admitting a finite nonempty set of marked equivariant minimal immersions.
ER -