On the connected components of the moduli space of equivariant minimal surfaces in CH^2

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An equivariant minimal surface in $\CH^n$ is a minimal map of the Poincar\'{e} disc into $\CH^n$ which intertwines
two actions of the fundamental group of a closed surface $\Sigma$: a Fuchsian representation on the disc and an
irreducible action by isometries on $\CH^n$. The moduli space of these can been studied by relating
it to the nilpotent cone in each moduli space of $PU(n,1)$-Higgs bundles over the conformal surface corresponding to
the map. By providing a necessary condition for points
on this nilpotent cone to be smooth this article shows that away from the points corresponding to branched minimal immersions
or $\pm$-holomorphic immersions the moduli space is smooth. The argument is easily adapted to show that for $\RH^n$ the
full space of (unbranched) immersions is smooth.
For $\CH^2$ we show that the connected components of the moduli space of minimal
immersions are indexed by the Toledo invariant and the Euler number of the normal bundle of the immersion.
This is achieved by studying the limit points of the $\Ct$-action on the nilpotent cone.
It is shown that the limit points as $t\to 0$ lead only to branched minimal immersions or $\pm$-holomorphic immersions.
In particular, the Euler number of the normal bundle can only jump by passing through branched minimal maps.
Original languageEnglish
Article number56
Number of pages19
JournalGeometriae Dedicata
Publication statusPublished - 17 Apr 2023

Bibliographical note

© The Author(s) 2023


  • Minimal surface, Higgs bundle, complex hyperbolic plane, nilpotent cone

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