Paper · Sep 19, 2018

Higgs bundles and geometric structures on manifolds

Daniele Alessandrini

math.AGarXiv:1809.07290v3
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Abstract

Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichmüller theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin's equations. Baraglia has shown in his Ph.D. Thesis that Higgs bundles can also be used to construct geometric structures in some interesting cases. In this paper, we will explain the main ideas behind this theory and we will survey some recent results in this direction, which are joint work with Qiongling Li.

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Submitted
Sep 19, 2018
Last revised
May 10, 2019 (v3)
Subjects
math.AG, hep-th, math.DG, math.GT
Journal
SIGMA 15 (2019), 039, 32 pages
DOI
10.3842/SIGMA.2019.039
Geometrization

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    Nikolai Erokhovets

    Topics Geometrization · Shared terms “geometric structure”, “geometrization conjecture”, “Thurston”

  2. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos

    Topics Geometrization · Shared terms “Thurston”, “interesting”

  3. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al.

    Topics Geometrization · Shared terms “geometrization conjecture”, “Thurston”

  4. Geometrization of the local Langlands correspondence: an overview

    Laurent Fargues

    Topics Geometrization · Shared terms “geometrization conjecture”

  5. Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local

    Laurent Fargues

    Topics Geometrization · Shared terms “geometrization conjecture”

  6. Etale cohomology of diamonds

    Peter Scholze

    Topics Geometrization · Shared terms “geometrization conjecture”

  7. A Treatise on Information Geometry

    Christopher John Goddard

    Topics Geometrization · Shared terms “Thurston”

  8. Uniform models and short curves for random 3-manifolds

    Peter Feller, Alessandro Sisto and Gabriele Viaggi

    Topics Geometrization · Shared terms “geometrization conjecture”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982)

In the guidebook

  1. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.