PreprintCC BY 4.0arXiv
High-Dimensional Families of Minimal Surfaces of Arbitrary Genus in Round Spheres
Abstract
We prove the existence of arbitrarily high-dimensional families of minimal surfaces of any prescribed genus and conformal structure in even-dimensional round spheres. More precisely, let n≥2 and let Σ be any closed Riemann surface of genus g. We construct a sequence of degrees dℓ→+∞ such that, for every ℓ, there exists a complex manifold of complex dimension 2dℓ+n2(1−g) consisting of linearly full branched superminimal immersions of Σ into 𝕊2n of degree dℓ. In particular, this yields parametrized families of linearly full branched minimal immersions whose dimensions tend to infinity.
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BibTeX
@misc{caniato2026highdimensional,
title = {{High-Dimensional Families of Minimal Surfaces of Arbitrary Genus in Round Spheres}},
author = {Riccardo Caniato},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09016},
eprint = {2609.09016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09016},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - High-Dimensional Families of Minimal Surfaces of Arbitrary Genus in Round Spheres
AU - Riccardo Caniato
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.09016
UR - https://arxiv.org/abs/2609.09016
AB - We prove the existence of arbitrarily high-dimensional families of minimal surfaces of any prescribed genus and conformal structure in even-dimensional round spheres. More precisely, let n≥2 and let Σ be any closed Riemann surface of genus g. We construct a sequence of degrees d_ℓ→+∞ such that, for every ℓ, there exists a complex manifold of complex dimension 2d_ℓ+n^2(1−g) consisting of linearly full branched superminimal immersions of Σ into 𝕊^2n of degree d_ℓ. In particular, this yields parametrized families of linearly full branched minimal immersions whose dimensions tend to infinity.
ER -