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PreprintCC BY 4.0arXiv

The Calabi-Yau and Sullivan conjectures for minimal surfaces with symmetries

  • Antonio Alarcon
  • Franc Forstneric
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Abstract

Let G be a finite group acting on a connected open Riemann surface X by holomorphic automorphisms and acting on a Euclidean space ℝn, n≥ 3, by orthogonal transformations. In a previous paper, the second author gave a necessary and sufficient condition for the existence of a G-equivariant conformal minimal immersion F:X→ℝn, and he asked whether such F can be chosen to be a proper map. Assuming that this condition holds, we give an affirmative answer when X is a bordered Riemann surface and G acts freely on the boundary of X. This contributes to the Sullivan conjecture concerning the existence of proper hyperbolic minimal surfaces in ℝ3 of finite topology. Under the same conditions, we also construct a G-equivariant proper conformal minimal immersion of X to the unit ball of ℝn which is complete in the induced metric, thereby opening a new direction in the Calabi--Yau theory regarding the study of complete bounded minimal surfaces. Analogous results hold for null curves in ℂn.

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Cite

BibTeXRIS
BibTeX
@misc{alarcon2026calabiyau,
  title = {{The Calabi-Yau and Sullivan conjectures for minimal surfaces with symmetries}},
  author = {Antonio Alarcon and Franc Forstneric},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31847},
  eprint = {2610.31847},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31847},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Calabi-Yau and Sullivan conjectures for minimal surfaces with symmetries
AU  - Antonio Alarcon
AU  - Franc Forstneric
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31847
UR  - https://arxiv.org/abs/2610.31847
AB  - Let G be a finite group acting on a connected open Riemann surface X by holomorphic automorphisms and acting on a Euclidean space ℝ^n, n≥ 3, by orthogonal transformations. In a previous paper, the second author gave a necessary and sufficient condition for the existence of a G-equivariant conformal minimal immersion F:X→ℝ^n, and he asked whether such F can be chosen to be a proper map. Assuming that this condition holds, we give an affirmative answer when X is a bordered Riemann surface and G acts freely on the boundary of X. This contributes to the Sullivan conjecture concerning the existence of proper hyperbolic minimal surfaces in ℝ^3 of finite topology. Under the same conditions, we also construct a G-equivariant proper conformal minimal immersion of X to the unit ball of ℝ^n which is complete in the induced metric, thereby opening a new direction in the Calabi--Yau theory regarding the study of complete bounded minimal surfaces. Analogous results hold for null curves in ℂ^n.
ER  -