Equivariant Minimal Surfaces for Semisimple Representations

Definitions and mathematical statement

Let $\Sigma$ be a closed surface of genus at least $2$, let $\widetilde\Sigma$ be its universal cover, and let $\pi_1(\Sigma)$ act on $\widetilde\Sigma$ by deck transformations. Let $G$ be a semisimple Lie group and let $\rho:\pi_1(\Sigma)\to G$ be a group homomorphism. The source writes $X=G/K$, with the left action $g\cdot(hK)=(gh)K$, but does not specify $K$ or a Riemannian metric on $X$.

A map $f:\widetilde\Sigma\to X$ is $\rho$-equivariant if \[ f(\gamma\cdot p)=\rho(\gamma)\cdot f(p) \qquad\text{for every }\gamma\in\pi_1(\Sigma)\text{ and }p\in\widetilde\Sigma. \] In a Riemannian manifold, a smooth immersed two-dimensional surface is minimal when its mean-curvature vector vanishes everywhere. Applying this definition here requires geometric data on $G/K$ that the record does not supply.

Source existence question. For every closed surface $\Sigma$ of genus at least $2$, every semisimple Lie group $G$, and every representation $\rho:\pi_1(\Sigma)\to G$, does there exist a smooth $\rho$-equivariant map \[ f:\widetilde\Sigma\longrightarrow X=G/K \] whose image is a minimal surface?

Unresolved qualifications. This retains the source's target but is not a fully specified Riemannian existence problem until $K$ and the metric are identified. In particular, taking $K$ to be maximal compact and taking a symmetric-space metric would be additional interpretations. The source requires a smooth map with minimal image; it does not explicitly require that $f$ be an immersion or specify whether branched minimal immersions, self-intersections, or singular images are allowed. A constant image is not a two-dimensional surface under the definition above, and no convention permitting such degeneracy is supplied. The record also does not explicitly state orientability, restrictions on $G$ beyond semisimplicity, or additional hypotheses on $\rho$. None are imposed here. The question asks for existence; uniqueness appears only in the supplied literature assessment.

Short English statement

The source asks whether every representation of the fundamental group of a closed surface of genus at least $2$ into a semisimple Lie group admits an equivariant smooth map into $G/K$ with minimal-surface image. The subgroup $K$, ambient metric, and permitted regularity or degeneracy of that image need clarification. The supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry, permanent ID $30003523$, source problem code catalogue entry, titled Equivariant Minimal Surfaces for Semisimple Representations, in the Oberwolfach Reports Open Problems collection. The record lists proposed year 2017 and supplies no named proposer.
• Supplied report citation: Differentialgeometrie im Großen (2018), Oberwolfach Reports, pp. 1917--1971, DOI 10.4171/owr/2017/31. https://doi.org/10.4171/owr/2017/31. The collection describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied literature list labels this same reference Oberwolfach Report 31/2017; the differing year labels are retained without reconciliation.
• Supplied literature reference: Collier--Tholozan--Toulisse, Maximal representations. https://arxiv.org/abs/1702.08799.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that existence and uniqueness of equivariant minimal immersions are proved for quasi-isometric, Hitchin, and maximal representations in several rank-two settings, while the question for arbitrary semisimple representations remains open. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.