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

Antonio Alarcon; Franc Forstneric

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arXiv · 2610.31847

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 $\mathbb R^n$, $n\ge 3$, 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 $F:X\to\mathbb R^n$, 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 $\mathbb R^3$ of finite topology. Under the same conditions, we also construct a $G$-equivariant proper conformal minimal immersion of $X$ to the unit ball of $\mathbb R^n$ 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 $\mathbb C^n$.

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BibTeX RIS

Antonio Alarcon, Franc Forstneric. 2026-10-22. The Calabi-Yau and Sullivan conjectures for minimal surfaces with symmetries. https://arxiv.org/abs/2610.31847

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