An Introduction to Higgs Bundles via Harmonic Maps

Li, Shuyu

doi:10.3842/sigma.2019.035

Cited by 16 publications

(23 citation statements)

References 52 publications

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“…It follows from Proposition 2.12 that this map is real analytic. It is conjectured (see for example [12,Conjecture 9.3]) that equivariant harmonic maps associated to Hitchin representations are immersions which would correspond to the set I being empty.…”

Section: Hitchin Representationsmentioning

confidence: 99%

Equivariant harmonic maps depend real analytically on the representation

Slegers

2021

manuscripta math.

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We consider harmonic maps into symmetric spaces of non-compact type that are equivariant for representations that induce a free and proper action on the symmetric space. We show that under suitable non-degeneracy conditions such equivariant harmonic maps depend in a real analytic fashion on the representation they are associated to. The main tool in the proof is the construction of a family of deformation maps which are used to transform equivariant harmonic maps into maps mapping into a fixed target space so that a real analytic version of the results in [4] can be applied.

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Section: Hitchin Representationsmentioning

confidence: 99%

Equivariant harmonic maps depend real analytically on the representation

Slegers

2021

manuscripta math.

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“…so the Hitchin section is parametrized by the quadratic differentials q 2 ∈ H 0 (X, K 2 X ). Hit 2 describes the Teichmüller space Teich(X ) in terms of Higgs bundles, and the quadratic differential q 2 measures the non-conformality of the harmonic differeomorphism (X , g 0 ) → (X , g) [27].…”

Section: Conformal Limits and Gaiotto's Conjecturementioning

confidence: 99%

Non-Abelian Hodge Theory and Related Topics

Huang¹

2020

SIGMA

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“…Holomorphic differentials on Riemann surfaces are central objects in Teichmüller theory, especially quadratic and cubic differentials, due to their close relationship with the study of harmonic maps from closed surfaces. There are many recent important works also relating harmonic maps with representations of surface groups into various characteristic varieties, see for instance a recent survey [Li19] and references within. In this section we will explore some geometric applications of our main results.…”

Section: Geometric Applicationsmentioning

confidence: 99%

Donaldson Functional in Teichmüller Theory

Huang¹,

Lucia²,

Tarantello³

2020

Preprint

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In this paper we define the Donaldson functional on closed Riemann surfaces whose Euler-Lagrange equations are a system of differential equations which generalizes Hitchin's self-dual equations on closed surfaces. We prove that this functional admits a unique critical point which is a global minimum. An immediate consequence is that this system of generalized selfdual equations admits also a unique solution. Among the applications in geometry of this fact, we find closed minimal immersions with the given Higgs data in hyperbolic manifolds, and consequently we present a unified variational approach to construct representations of the fundamental groups of closed surfaces into several character varieties.

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An Introduction to Higgs Bundles via Harmonic Maps

Cited by 16 publications

References 52 publications

Equivariant harmonic maps depend real analytically on the representation

Equivariant harmonic maps depend real analytically on the representation

Non-Abelian Hodge Theory and Related Topics

Donaldson Functional in Teichmüller Theory

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