Coincidence of the two holonomy maps on the full regular minimal-immersion space

Establish that the holonomy map defined using equivariant minimal immersions into $PSL(2,\mathbb C)$ coincides on the full space $\mathrm M^*(S)$ with the holonomy map arising from complex harmonic $PSL(2,\mathbb C)\times PSL(2,\mathbb C)$-bundles, by means of an analogue of the Gauss-map construction, while clarifying its relationship with the complex harmonic Higgs-bundle framework.

Background

The paper constructs a deformation space of regular minimal immersions into PSL(2,C)PSL(2,\mathbb C) and proves that its holonomy map is holomorphic and locally biholomorphic after passing to the finite-dimensional quotient M(S)\mathbb M(S). The authors compare this construction with an earlier framework based on complex harmonic maps into a product of spaces of unoriented geodesics, or equivalently complex harmonic Higgs bundles, noting that the two approaches produce the same immersion-data space and quotient.

The holonomy maps are shown to agree on the totally real Anti-de Sitter locus MGH(S)\mathcal{MGH}(S), but the paper does not establish their equality on all of M∗(S)\mathrm M^*(S). The authors suggest that an analogue of the Gauss map should provide the comparison, subject to resolving compatibility issues with the complex harmonic Higgs-bundle formulation.

References

In fact, we believe that the two holonomy maps coincide on $\mathrm M*(S)$ via an analogue approach using the Gauss map, even if attention should be paid to the relation with the complex harmonic Higgs bundles setting treated in . Since these aspects are beyond the scope of this paper, we defer further clarification to some future work of the first author and Sagman.

— Extending Donaldson's HyperKähler Structure via Minimal Surfaces into $\mathrm{PSL}(2,\mathbb{C})$  (2610.22288 - Emam et al., 15 Oct 2026) in Remark following Theorem 3.12, Section 3 (The holonomy map)