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

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 \(\mathbb C^n\).