Extending Donaldson's HyperKähler Structure via Minimal Surfaces into PSL(2,ℂ)

@misc{emam2026extending,
  title = {{Extending Donaldson's HyperKähler Structure via Minimal Surfaces into $\mathrmPSL(2,\mathbbC)$}},
  author = {Christian El Emam and Filippo Mazzoli and Nicholas Rungi and Andrea Seppi and Andrea Tamburelli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22288},
  eprint = {2610.22288},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22288},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Extending Donaldson's HyperKähler Structure via Minimal Surfaces into PSL(2,ℂ)
AU  - Christian El Emam
AU  - Filippo Mazzoli
AU  - Nicholas Rungi
AU  - Andrea Seppi
AU  - Andrea Tamburelli
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22288
UR  - https://arxiv.org/abs/2610.22288
AB  - We construct a deformation space for minimal immersions into PSL(2,ℂ) with its holomorphic Killing form, and define an explicit, mapping class group invariant, complex hyperKähler structure on such deformation space. Such complex hyperKähler structure restricts to: Donaldson's hyperKähler structure on a totally real copy of almost-Fuchsian space, Mazzoli-Seppi-Tamburelli's para-hyperKähler structure on a totally real copy of maximally globally hyperbolic 𝔸d𝕊^3 manifolds, and Loustau-Sanders' complex bi-Kähler structure on a holomorphic copy of quasi-Fuchsian space. Moreover, all these loci are totally geodesic. We discuss other geometric aspects of this space, such as the Hamiltonian property of a natural ℂ^*-action, involving the Goldman symplectic form and a complex-valued area functional.
ER  -