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

Ian McIntosh

doi:10.1007/s10711-023-00793-z

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

Ian McIntosh

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 language	English
Article number	56
Number of pages	19
Journal	Geometriae Dedicata
Volume	217
DOIs	https://doi.org/10.1007/s10711-023-00793-z
Publication status	Published - 17 Apr 2023

Bibliographical note

Keywords

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

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10.1007/s10711-023-00793-zLicence: CC BY

On the connected components of the moduli space of equivariant minimal surfaces in CH2
© The Author(s) 2023
Final published version, 380 KBLicence: CC BY

Cite this

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title = "On the connected components of the moduli space of equivariant minimal surfaces in CH^2",

abstract = "An equivariant minimal surface in $\CH^n$ is a minimal map of the Poincar\'{e} disc into $\CH^n$ which intertwinestwo actions of the fundamental group of a closed surface $\Sigma$: a Fuchsian representation on the disc and anirreducible action by isometries on $\CH^n$. The moduli space of these can been studied by relatingit to the nilpotent cone in each moduli space of $PU(n,1)$-Higgs bundles over the conformal surface corresponding tothe map. By providing a necessary condition for pointson this nilpotent cone to be smooth this article shows that away from the points corresponding to branched minimal immersionsor $\pm$-holomorphic immersions the moduli space is smooth. The argument is easily adapted to show that for $\RH^n$ thefull space of (unbranched) immersions is smooth.For $\CH^2$ we show that the connected components of the moduli space of minimalimmersions 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.",

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author = "Ian McIntosh",

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AB - An equivariant minimal surface in $\CH^n$ is a minimal map of the Poincar\'{e} disc into $\CH^n$ which intertwinestwo actions of the fundamental group of a closed surface $\Sigma$: a Fuchsian representation on the disc and anirreducible action by isometries on $\CH^n$. The moduli space of these can been studied by relatingit to the nilpotent cone in each moduli space of $PU(n,1)$-Higgs bundles over the conformal surface corresponding tothe map. By providing a necessary condition for pointson this nilpotent cone to be smooth this article shows that away from the points corresponding to branched minimal immersionsor $\pm$-holomorphic immersions the moduli space is smooth. The argument is easily adapted to show that for $\RH^n$ thefull space of (unbranched) immersions is smooth.For $\CH^2$ we show that the connected components of the moduli space of minimalimmersions 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.

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