Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Complex harmonic maps and rank 2 higher Teichm\"uller theory

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that every rank-2 Hitchin component carries a mapping-class-group-invariant pseudo-Kähler structure combining the Goldman symplectic form with Labourie's complex structure, via a new theory of complex harmonic maps.

desk verdict Serious, likely-correct paper on rank-2 Hitchin components; the Assumption 4.16 worry dissolves once you read Theorem C as the construction. read the letter →

arxiv 2506.11746 v1 pith:GUS7Y3JP submitted 2025-06-13 math.DG math.CVmath.GTmath.RT

Christian El Emam , Nathaniel Sagman This is my paper
classification math.DGmath.CVmath.GTmath.RT MSC 53C4332G1553D3014H60
keywords complexharmonicmapsHitchincomponentshigherTeichmüllertheorypseudo-KählerstructureGoldmansymplecticformLabourieaffineTodaequationsopers
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper initiates a complex analogue of harmonic maps: maps from a surface with two oppositely oriented complex structures into the holomorphic Riemannian symmetric space $G^{\mathbb C}/K^{\mathbb C}$. With this tool it proves that every rank-2 Hitchin component $\mathrm{Hit}(S,G)$ admits a mapping-class-group-invariant pseudo-Kähler structure whose compatible almost complex structure is Labourie's and whose symplectic form is Goldman's, of signature $(6g-6, 2(2d_G-1)(g-1))$. It also establishes a Bers-type simultaneous uniformization theorem: an invariant open set of pairs of rank-2 Hitchin representations maps locally biholomorphically into the analytic character variety of the complexified group, with points realized as holonomies of conformal complex harmonic maps, and with the two marginal loci landing in $G^{\mathbb C}$-opers. The existence of these maps is reduced to solving a new system of complex elliptic equations, the complex affine Toda equations. If correct, this settles the compatibility question for Goldman's form and Labourie's complex structure that had been open at least since the two parametrizations of the $\mathrm{PSL}(3,\mathbb R)$ Hitchin component.

What carries the argument

The central machine is a complex harmonic map $f:(\widetilde S,c_1,c_2)\to G^{\mathbb C}/K^{\mathbb C}$: a map from a surface carrying two oppositely oriented complex structures into a holomorphic Riemannian symmetric space, equivalently a complex harmonic $G$-bundle $(P_{K^{\mathbb C}},A_{K^{\mathbb C}},\phi_1,\phi_2)$ with a $K^{\mathbb C}$-connection and two Higgs fields satisfying $\partial_1\phi_2=\partial_2\phi_1=0$ and flatness $F(A)+[\phi_1,\phi_2]=0$. The construction seeks an isomorphism $I$ between the adjoint bundles built from $(c_1,q_1)$ and $(c_2,q_2)$ that solves the flatness equation (13); under the paper's Assumption 4.16, that the restriction of $I$ to the Cartan subbundle is multiplication by $-1$, the flatness equation reduces to the complex affine Toda equations $\Delta_h\log U_\alpha = 2\sum_{\beta\in\Pi} a_{\alpha\beta}r_\beta U_\beta - 2a_{\alpha\delta}\,(q_1q_2/h^d)\,U_{-\delta} - 2$, with $U_\alpha=U_{\xi(\alpha)}$. These equations are the engine: their solutions produce the complex harmonic $G$-bundles whose holonomies define the maps $B_G$ and $L^C_G$, and an explicit description of the flat connections then yields the compatibility of the symplectic form and complex structure.

What would settle it

A concrete test: take $G=\mathrm{PSL}(3,\mathbb R)$, choose distinct complex structures $c_1,c_2$ and a generic pair of nonzero cubic differentials $(q_1,q_2)$, and check whether every solution to the flatness equation (13) for the corresponding bi-Hitchin data restricts to $-1$ on the Cartan subbundle; a solution with different Cartan restriction would disprove Assumption 4.16, while a simultaneous failure of the complex affine Toda equations (2) on data where Theorem C asserts existence would falsify the theorem's sufficiency statement.

Watch

Extended reading notes

Core claim

The central claim is that, for every split real adjoint Lie group $G$ of rank 2, the Hitchin component $\mathrm{Hit}(S,G)$, endowed with the Goldman symplectic form $\omega_G$ and Labourie's complex structure $J_G$, is pseudo-Kähler of signature $(6g-6, 2(2d_G-1)(g-1))$, and the mapping class group preserves this structure. The proof is built on a second main theorem, a Bers-type theorem: there is a connected invariant subset $\Omega_G\subset \mathrm{Hit}(S,G)\times \mathrm{Hit}(S,G)$ containing the diagonal, $\mathrm{Hit}(S,G)\times T(S)$, and $T(S)\times \mathrm{Hit}(S,G)$, on which the diagonal identification extends uniquely to a continuous, equivariant local biholomorphism $B_G:\Omega_G\to \chi^{\mathrm{an}}(\pi_1(S),G^{\mathbb C})$. The image points are holonomies of conformal complex harmonic maps to $G^{\mathbb C}/K^{\mathbb C}$; on the two products with Teichmüller space the images are exactly holonomies of $G^{\mathbb C}$-opers, and on $T(S)\times T(S)$ the map is Bers' simultaneous uniformization. The paper also states partial generalizations to higher rank (Theorems A' and B'), proves existence of complex harmonic maps from solutions of complex affine Toda equations (Theorem C), and shows a relation between complex harmonic maps and opers (Theorem D).

Load-bearing premise

The load-bearing premise is Assumption 4.16, that a certain bundle isomorphism between the two adjoint bundles constructed from the two complex structures and top differentials acts as multiplication by $-1$ on the Cartan subbundle; the paper proves this on the real locus and argues heuristically for holomorphic families, but does not prove it in general, and if it fails the reduction to the solvable affine Toda equations breaks.

Editorial extensions

If this is right

  • Every rank-2 Hitchin component carries an $\mathrm{MCG}(S)$-invariant pseudo-Kähler structure whose metric has signature $(6g-6, 2(2d_G-1)(g-1))$; the Goldman form and Labourie's complex structure can therefore be used interchangeably in the study of these components.
  • A Bers-type theorem holds in rank 2: an invariant connected open set of pairs of Hitchin representations maps locally biholomorphically into the complex character variety, with image holonomies coming from conformal complex harmonic maps; in particular the diagonal identification extends much further than formal analytic continuation alone would guarantee.
  • On the marginal loci $\mathrm{Hit}(S,G)\times T(S)$ and $T(S)\times \mathrm{Hit}(S,G)$, the map $B_G$ produces $G^{\mathbb C}$-opers, giving a direct bridge between higher Teichmüller theory and the oper side of geometric Langlands.
  • The complex affine Toda equations supply an explicit existence mechanism: solving them produces complex harmonic $G$-bundles, and the resulting flat connections are explicit enough to compute the pullback of Goldman's form and its signature; the same mechanism yields partial higher-rank theorems A' and B' for the cyclic locus.
  • On the anti-conjugate locus $AC(S,G)$ a genuine Kähler structure exists (Theorem E), and for $G=\mathrm{PU}(2,1)$ the Loftin-McIntosh neighbourhood of Fuchsian representations becomes Kähler (Corollary E).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If Assumption 4.16 were proved in full, the bi-Hitchin section strategy would yield a holomorphic extension of Hitchin's section over the entire bi-Hitchin base, producing a complex manifold of complex harmonic $G$-bundles; the paper explicitly leaves this as future work.
  • The convergence picture surrounding Gaiotto's conformal limit suggests a complex analogue: suitably scaled families of complex harmonic $G$-bundles with one Higgs field degenerating should converge to $G^{\mathbb C}$-opers, and Theorem D gives the endpoint of exactly such a degeneration; this is an inference, not a theorem in the paper.
  • The $\mathbb C^*$-action $(P,A,\phi_1,\phi_2)\mapsto (P,A,\xi\phi_1,\xi^{-1}\phi_2)$ on complex harmonic bundles should produce twistor families for the pseudo-Kähler structure, extending the ordinary hyperKähler twistor lines of Higgs bundle moduli; this is implicit in Section 5's discussion rather than proved.
  • A testable extension is numerical: for $G=\mathrm{PSp}(4,\mathbb R)$ or $G_2'$, compute the singular locus $\mathrm{Sing}_G$ of the Bers Laplacian linearization; if it is empty, the same argument as for $\mathrm{PSL}(3,\mathbb R)$ would extend $B_G$ to all of $\mathrm{Hit}(S,G)\times T(S)\cup T(S)\times \mathrm{Hit}(S,G)$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces complex harmonic maps to holomorphic Riemannian symmetric spaces and develops the associated theory of complex harmonic G-bundles. Its main results are: Theorem C, which reduces the flatness equation for such bundles to a system of complex affine Toda equations on a cyclic locus; Theorem D, which identifies marginal-locus solutions with GC-opers; Theorems B and B', which construct Bers-type maps BG and LG^C extending Labourie's immersion/parametrization; and Theorems A and A', which prove that Goldman's symplectic form is compatible with Labourie's complex structure, yielding a mapping-class-group-invariant pseudo-Kähler structure of the stated signature on rank-2 Hitchin components. A further result, Theorem E, constructs Kähler structures on spaces related to the q2=-q1 locus. The proof architecture is global analysis: Banach/Fréchet implicit function theorems, elliptic estimates for the Bers Laplacian, the Analytic Fredholm theorem, and a maximum principle. The paper is carefully organized and gives explicit connection forms and equation (2) in detail.

Significance. If the main chain of arguments is completed, Theorem A settles the long-standing compatibility question between Goldman's symplectic form and Labourie's complex structure for all rank-2 Hitchin components, with the predicted signature; Theorems B and D give a geometric Bers-type theory with connections to GC-opers; and Theorems A', B', and E provide substantial partial generalizations to higher rank and to related Kähler structures. The paper is also valuable for its detailed analytic setup, explicit flat-connection computations, the use of the maximum principle to prove injectivity of the linearization on the real locus, and the signature computation reduced to a single point. The main caveat is that the central reduction from the bundle flatness equation (13) to the scalar system (2) rests on Assumption 4.16, whose verification is incomplete away from the real and marginal loci.

major comments (2)
  1. [Section 4.5.1, Assumption 4.16; Section 6.5, Theorem 6.20; Section 7.2, Proposition 7.1] Assumption 4.16 is load-bearing. The proof of Theorem C reduces the flatness equation (13) to the complex affine Toda equations (2) only after imposing I restricted to S times h^C equals multiplication by -1. Proposition 4.18 then forces the connection A_I to be valued in S times h^C, and this is what makes the curvature computation in Section 4.5.2 close. The paper verifies Assumption 4.16 on the real locus c1=c2, q1=q2 via cyclic Higgs bundle theory and on the marginal loci q1=0 or q2=0 via Proposition 4.23. However, Section 6 defines SOL_G as the zero set of the reduced scalar operator (26), not as the set of solutions to (13). The implicit-function-theorem branches produced near the real and marginal loci in Sections 6.5-6.6 are not shown to come from a Lie-bundle isomorphism I satisfying Assumption 4.16 and the full flatness equation (13). Consequently, for points of Omega'_G away from the explicitly checked loci, the holomorphic map L_G^C built in Theorem 6.20 may not be the holomorphic extension of Labourie's immersion, and the Lagrangian computation of Proposition 7.1 would concern a different object. This is a verification gap in the logical chain, not a demonstrated contradiction, but it must be closed or the statements of Theorems B' and A must be restricted accordingly.
  2. [Theorem C, stated in Section 1.3 and proved in Section 4.5.2] Theorem C is stated unconditionally as an implication from a solution U of (2) to the existence of a complex harmonic map with the given bi-Hitchin basepoint. The proof, however, requires Assumption 4.16 in addition to the existence of U, and Assumption 4.16 is not listed among the hypotheses of Theorem C. If the assumption is intended to be part of the hypothesis, the theorem statement must be amended; if it is intended to follow automatically from the existence of U, that implication is not proved. This is not merely a cosmetic issue because Theorem C is the engine for Theorems D, B, B', and A.
minor comments (4)
  1. [Section 1.1] The phrase "we incorrectly refer to the parametrization as Labourie's" is confusing; it would be clearer to say that, for simplicity, the authors use "Labourie's parametrization" also for the product case PSL(2,R)^2.
  2. [Section 6.2, Proposition 6.7] The phrase "the diagonal action of Diff_0(S) times Diff_0(S)" is ambiguous; the intended meaning is the diagonal action of Diff_0(S) on the product, so the wording should be adjusted.
  3. [Section 4.7.4] The notation b\sigma_0 for the Gauss map is very close to the notation b\sigma used for the Cartan involution in Section 2.4, which may cause confusion; a different symbol for the Gauss map would help.
  4. [Section 6.6, proof of Theorem B'] The sentence "Since L_G is MCG(S)-invariant, the uniqueness of analytic continuation implies that L_G^C is MCG(S)-invariant" should more precisely say "equivariant" or "intertwines the two actions," since L_G is equivariant rather than invariant.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central theorems derived from explicit flat-connection descriptions and independent analytic lemmas; Assumption 4.16 is a stated ansatz with a verification gap, not a circular reduction.

full rationale

Walking the derivation chain, Theorem C reduces the bundle flatness equation (13) to the complex affine Toda system (2) by a direct computation under the explicit ansatz Assumption 4.16. The ansatz is not obtained from the conclusion being proved; the paper verifies it on the real locus via cyclic Higgs-bundle theory and on the marginal loci via the constant solution of Proposition 4.23, and the heuristic about holomorphic families is only a heuristic. Theorem D reduces to Beilinson-Drinfeld opers and a direct verification of the relative position condition, so it does not presuppose its own conclusion. Theorem B' is built with the implicit-function/analytic-Fredholm apparatus applied to the reduced scalar operator (26), with holomorphicity supplied by the authors' earlier holomorphic dependence results for Bers metrics ([31]) and complex Lie derivatives ([30]); these are parameter-free analytic lemmas that do not assume Theorems A, B, C, or D. Theorem A and A' follow from the explicit flat-connection form and the Lagrangian property (Proposition 7.1), with the signature computed at one point and extended, rather than from the statement being proved. The main self-citations are technical and non-load-bearing for the central compatibility claim. A genuine caveat, which is a correctness risk rather than circularity: Assumption 4.16 is established only on real and marginal loci, and the implicit-function-theorem branches used in Section 6 are not shown to integrate back to a bundle isomorphism satisfying Assumption 4.16 and the full flatness equation (13); this is a verification gap in the logical chain, not a circular reduction.

Assumptions & free parameters 2 free parameters · 8 assumptions · 4 invented entities

The central claims rest on a large body of external theory: the non-abelian Hodge correspondence (Theorem 2.6), Labourie's rank-2 parametrization (Theorem 2.7), Bers' theorem, and Beilinson-Drinfeld opers. The authors' own prior work supplies the analytic backbone: holomorphic dependence of Bers metrics ([31]), elliptic estimates and complex Lie derivatives ([30]), which are load-bearing for the holomorphicity of the holonomy map and the Fredholm analysis. The one genuinely ad hoc input is Assumption 4.16 restricting the isomorphism I on the Cartan subalgebra; without it the derivation of the complex affine Toda equations does not go through. The fixed constants in the paper (Coxeter numbers, affine Cartan numbers, the solution U0 = A_g^{-1} e) are determined by Lie theory or by unique solvability, not fitted to any target result. No data fitting, error bars, or code are involved, being a pure mathematics paper.

free parameters (2)
  • Choice of invariant metric parameters (a, b) for G = PSL(2,R)^2 = a = b = 1
    Section 2.5 restricts the product metric on the two hyperbolic factors to equal weights; the authors state the theorems extend to all a, b > 0 'without any added difficulty', so this is a normalization, not a fitted parameter.
  • Normalization of the Ad(G)-invariant bilinear form ν = arbitrary (up to scale on each simple factor)
    Section 2.1: harmonicity and the theorems are independent of the normalization of ν; the specific scale does not enter the conclusions.
assumptions (8)
  • standard math Hitchin-Simpson: a G-Higgs bundle is polystable iff a reduction solving the self-duality equations exists, and stable implies uniqueness (Theorem 2.6).
    Used in Proposition 4.21 to assert existence and uniqueness of real solutions to (2) on the real locus c1 = c2, q1 = q2, via non-abelian Hodge theory.
  • standard math Labourie's rank-2 theorem: L_G : M(S,G) → Hit(S,G) is a diffeomorphism (Theorem 2.7, from [47]).
    Converts Theorems A' and B' (proved for M(S,G)) into Theorems A and B on the Hitchin component; also the source of Labourie's complex structure definition (Definition 2.8).
  • standard math Bers' Simultaneous Uniformization Theorem (Theorem 3.5): quasi-Fuchsian representations are parametrized by T(S) × T(S̄).
    The external benchmark defining the map B_G on T(S) × T(S̄) and the Bers metrics used throughout Sections 3 to 6.
  • standard math Beilinson-Drinfeld: GC-opers on (S,c) are parametrized by the Hitchin base, and oper holonomy is irreducible and simple (Proposition 5.4, Theorem 5.5).
    Pins part (2) of Theorem D to an existing object and guarantees the image of L^C_G lies in the smooth character variety.
  • standard math Analytic Fredholm Theorem and Rellich-Kondrachov compactness, used with elliptic estimates for the Bers Laplacian to control Sing_L (Theorem 6.17, Proposition 6.18).
    Load-bearing for the marginal locus analysis and the codimension-1 conclusion for Sing_G.
  • standard math Maximum principle for cooperative elliptic systems (Lemma 6.12, from [18, Lemma 3.1]).
    Proves injectivity of the linearized operator on the real locus (Proposition 6.11), hence that the real locus sits inside the regular solution space.
  • domain assumption Holomorphic dependence of Bers metrics on (c1,c2) in Fréchet spaces, and the elliptic and transport machinery of complex Lie derivatives ([31, Theorem B], [30, Sections 4-6]).
    Imported from the authors' prior work; used in Propositions 6.8-6.10 and Theorem 6.13 to get holomorphicity of T_G, of f_hol, and the density and connectedness of the co-real analytic locus. A gap here would break the holomorphicity of B_G and L^C_G.
  • ad hoc to paper Assumption 4.16: the isomorphism I solving the flatness equation restricts to −1 on S × h^C.
    Reduces the bi-Hitchin strategy to the complex affine Toda equations (2); verified on the real locus by cyclic Higgs bundle theory and argued heuristically to persist in holomorphic families (Remark 4.17), but not proved for the full generality of Theorem C.
invented entities (4)
  • Complex harmonic maps and complex harmonic G-bundles (Definitions 4.1, 4.7) independent evidence
    purpose: Complexification of harmonic maps: maps from (S, c1, c2) to G^C/K^C governed by two opposite complex structures; the objects that geometrically realize B_G and L^C_G.
    Anchored externally: equivalence with ordinary harmonic maps on the real locus, flat connections that reduce to Beilinson-Drinfeld opers (Theorem D) and to Bers' map on T(S) × T(S̄) (Section 4.7.1), and the classical equations obtained by specialization.
  • Complex affine Toda equations (Equation (2)) independent evidence
    purpose: Sufficient PDE condition for existence of complex harmonic G-bundles with cyclic bi-Hitchin basepoint.
    Specializes to the classical affine Toda system, the Bochner formula (3), the Tzitzeica equation for PSL(3,R) (16), and Gauss equations for minimal surfaces in H^3 (Section 4.7.4), giving independent anchors.
  • Bi-Hitchin fibration and bi-Hitchin base (Definition 4.11) independent evidence
    purpose: Two-complex-structure analogue of the Hitchin fibration for complex harmonic G-bundles.
    Restricts to the classical Hitchin base when c1 = c2, and the holomorphicity of the differentials is proved in Proposition 4.12.
  • The bi-Hitchin section (Remark 1.2)
    purpose: Proposed holomorphic extension of Hitchin's section over pairs (c1, c2); not constructed.
    Explicitly left for future work ('It would be all quite technical... we leave these considerations for the future'); purely conjectural at this stage.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Complex harmonic maps and rank 2 higher Teichm\"uller theory." pith.science (2026). https://pith.science/paper/GUS7Y3JP

@misc{pith2026250611746,
  author       = {Pith},
  title        = {Pith review of: Complex harmonic maps and rank 2 higher Teichm\"uller theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GUS7Y3JP}},
  note         = {Machine review of arXiv:2506.11746}
}
abstract

We initiate and develop the theory of complex harmonic maps to holomorphic Riemannian symmetric spaces, which we make use of to study complex analytic aspects of higher Teichm\"uller theory, with a focus on rank $2$ Hitchin components. Complex harmonic maps lead to various generalizations of objects from the theory of Higgs bundles; for instance, the Hitchin fibration, cyclic Higgs bundles, and the affine Toda equations. Beyond such generalizations, we also find a relation between complex harmonic maps and opers. Within the realm of higher Teichm\"uller theory, for any rank $2$ Hitchin component, we prove a Bers-type theorem, which extends and improves our previous work on $\mathrm{SL}(3,\mathbb R)$, and we prove that Goldman's symplectic form is compatible with Labourie's complex structure, so that the two determine a mapping class group invariant pseudo-K\"ahler structure. We obtain partial generalizations in higher rank, and we construct K\"ahler structures on other spaces that are related to the Hitchin components.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Global comparison of pseudo-K\"ahler structures on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component

    math.DG 2026-07 conditional novelty 7.0 of 10

    The Rungi–Tamburelli 2-form on the SL(3,R)-Hitchin component equals Goldman's symplectic form, so it is non-degenerate everywhere.

Reference graph

Works this paper leans on

83 extracted references · 59 canonical work pages · cited by 1 Pith paper

  1. [30]

    Complex affine spheres and a Bers theorem for SL(3,C)

    Christian El Emam and Nathaniel Sagman. Complex affine spheres and a Bers theorem for SL(3,C) . 2025. arXiv: 2406.15287 [math.DG]. url: https://arxiv.org/abs/2406. 15287

    work page Pith review arXiv 2025

  2. [31]

    Holomorphic dependence for the Beltrami equation in Sobolev spaces

    Christian El Emam and Nathaniel Sagman. Holomorphic dependence for the Beltrami equation in Sobolev spaces . 2024. arXiv: 2410.06175 [math.CV]. url: https://arxiv. org/abs/2410.06175

    work page Pith review arXiv 2024

  3. [1]

    The Yang-Mills equations over Riemann sur- faces

    Michael Francis Atiyah and Raoul Bott. “The Yang-Mills equations over Riemann sur- faces”. In: Philosophical Transactions of the Royal Society of London. Series A, Mathemat- ical and Physical Sciences 308 (1983), pp. 523–615. url: https://api.semanticscholar. org/CorpusID:13601126

    1983

  4. [2]

    G2 geometry and integrable systems

    D. Baraglia. “ G2 geometry and integrable systems”. PhD thesis. Oxford University, 2009

    2009

  5. [3]

    Cyclic Higgs bundles and the affine Toda equations

    D. Baraglia. “Cyclic Higgs bundles and the affine Toda equations”. In: Geometriae Ded- icata 174 (2010), pp. 25–42. url: https : / / api . semanticscholar . org / CorpusID : 119320524

    2010

  6. [4]

    Quantization of Hitchin’s fibration and Langlands’ program

    A. A. Beilinson and V. G. Drinfeld. “Quantization of Hitchin’s fibration and Langlands’ program”. In: Algebraic and geometric methods in mathematical physics (Kaciveli, 1993) . Vol. 19. Math. Phys. Stud. Kluwer Acad. Publ., Dordrecht, 1996, pp. 3–7. isbn: 0-7923- 3909-6

    work page 1993

  7. [5]

    Alexander Beilinson and Vladimir Drinfeld. Opers. 2005. arXiv: math/0501398 [math.AG]. url: https://arxiv.org/abs/math/0501398

    arXiv 2005

  8. [6]

    Holomorphic differentials as functions of moduli

    Lipman Bers. “Holomorphic differentials as functions of moduli”. In: Bull. Amer. Math. Soc. 67 (1961), pp. 206–210. issn: 0002-9904. doi: 10.1090/S0002-9904-1961-10569-7 . url: https://doi.org/10.1090/S0002-9904-1961-10569-7

    work page doi:10.1090/s0002-9904-1961-10569-7 1961

Show all 83 references
  1. [7]

    Simultaneous uniformization

    Lipman Bers. “Simultaneous uniformization”. In: Bull. Amer. Math. Soc. 66 (1960), pp. 94–

    1960

  2. [8]

    Spaces of Riemann surfaces as bounded domains

    Lipman Bers. “Spaces of Riemann surfaces as bounded domains”. In: Bulletin of the Amer- ican Mathematical Society 66.2 (1960), pp. 98–103

    1960

  3. [9]

    On immersions of surfaces into SL(2, C) and geometric consequences

    Francesco Bonsante and Christian El Emam. “On immersions of surfaces into SL(2, C) and geometric consequences”. In: Int. Math. Res. Not. IMRN 12 (2022), pp. 8803–8864. issn: 1073-7928,1687-0247. doi: 10.1093/imrn/rnab189 . url: https://doi.org/10. 1093/imrn/rnab189

    2022 doi

  4. [10]

    The pressure metric for Anosov representations

    Martin J. Bridgeman et al. “The pressure metric for Anosov representations”. In: Geomet- ric and Functional Analysis 25 (2013), pp. 1089–1179.url: https://api.semanticscholar. org/CorpusID:6691244

    2013

  5. [11]

    Burstall and J.H

    F.E. Burstall and J.H. Rawnsley. Twistor Theory for Riemannian Symmetric Spaces: With Applications to Harmonic Maps of Riemann Surfaces . Lecture Notes in Mathematics. Springer, 1990. isbn: 9780387526027. url: https : / / books . google . de / books ? id = H8EZAQAAIAAJ

    1990

  6. [12]

    Finite order automorphisms of Higgs bundles: theory and application

    Brian Collier. “Finite order automorphisms of Higgs bundles: theory and application”. In: PhD thesis (2016)

    2016

  7. [13]

    Conformal limits for parabolic SL(n,C)-Higgs bundles

    Brian Collier, Laura Fredrickson, and Richard Wentworth. Conformal limits for parabolic SL(n,C)-Higgs bundles . 2024. arXiv: 2407.16798 [math.DG]. url: https://arxiv.org/ abs/2407.16798

    2024 arXiv

  8. [14]

    (G,P)-Opers and global Slodowy slices

    Brian Collier and Andrew Sanders. “(G,P)-Opers and global Slodowy slices”. In: Advances in Mathematics 377 (2021), p. 107490. issn: 0001-8708. doi: https://doi.org/10.1016/ j.aim.2020.107490 . url: https://www.sciencedirect.com/science/article/pii/ S0001870820305181

    2021

  9. [15]

    Conformal limits and the Bia lynicki-Birula strat- ification of the space of λ-connections

    Brian Collier and Richard Wentworth. “Conformal limits and the Bia lynicki-Birula strat- ification of the space of λ-connections”. In: Adv. Math. 350 (2019), pp. 1193–1225. issn: 0001-8708,1090-2082. doi: 10.1016/j.aim.2019.04.034 . url: https://doi.org/10. 1016/j.aim.2019.04.034

    2019 doi

  10. [16]

    Copyright Page

    “Copyright Page”. In: Differential Geometry: Bundles, Connections, Metrics and Curva- ture. Oxford University Press, Oct. 2011. isbn: 9780199605880. doi: 10.1093/acprof: oso/9780199605880.002.0004 . eprint: https://academic.oup.com/book/0/chapter/ 66 REFERENCES 161651467/chapt...

    2011

  11. [17]

    Flat G-bundles with canonical metrics

    Kevin Corlette. “Flat G-bundles with canonical metrics”. In: J. Differential Geom. 28.3 (1988), pp. 361–382. issn: 0022-040X. url: http://projecteuclid.org/euclid.jdg/ 1214442469

    1988

  12. [18]

    On cyclic Higgs bundles

    Song Dai and Qiongling Li. “On cyclic Higgs bundles”. In: Mathematische Annalen 376.3–4 (Nov. 2020), pp. 1225–1260. issn: 1432-1807. doi: 10.1007/s00208-018-1779-4

    2020 doi

  13. [19]

    The Covariance Metric in the Blaschke Locus

    Xian Dai and Nikolaos Eptaminitakis. “The Covariance Metric in the Blaschke Locus”. In: The Journal of Geometric Analysis 34 (Mar. 2024). doi: 10.1007/s12220-024-01586-w

    2024 doi

  14. [20]

    Deformation spaces of convex real- projective structures and hyperbolic affine structures

    Mehdi-Reza Darvishzadeh and William M. Goldman. “Deformation spaces of convex real- projective structures and hyperbolic affine structures”. In: J. Korean Math. Soc. 33.3 (1996), pp. 625–639. issn: 0304-9914,2234-3008

    1996

  15. [21]

    Twisted harmonic maps and the self-duality equations

    S. K. Donaldson. “Twisted harmonic maps and the self-duality equations”. In: Proc. Lon- don Math. Soc. (3) 55.1 (1987), pp. 127–131. issn: 0024-6115. doi: 10.1112/plms/s3- 55.1.127. url: https://doi.org/10.1112/plms/s3-55.1.127

    1987 doi

  16. [22]

    Moment maps in differential geometry

    Simon K. Donaldson. “Moment maps in differential geometry”. In: Surveys in differential geometry 8 (2003), pp. 171–189. url: https://api.semanticscholar.org/CorpusID: 124403816

    2003

  17. [23]

    Lie algebras and equations of Korteweg-de Vries type

    V. G. Drinfeld and V. V. Sokolov. “Lie algebras and equations of Korteweg-de Vries type”. In: Current problems in mathematics, Vol. 24 . Itogi Nauki i Tekhniki. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 81–180

    1984

  18. [24]

    From the Hitchin section to opers through nonabelian Hodge

    Olivia Dumitrescu et al. “From the Hitchin section to opers through nonabelian Hodge”. In: J. Differential Geom. 117.2 (2021), pp. 223–253. issn: 0022-040X,1945-743X. doi: 10. 4310/jdg/1612975016. url: https://doi.org/10.4310/jdg/1612975016

    2021

  19. [25]

    Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

    Sorin Dumitrescu and Abdelghani Zeghib. “Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds”. In: Math. Ann. 345.1 (2009), pp. 53–81. issn: 0025-5831,1432-1807. doi: 10.1007/s00208- 009- 0342- 8. url: https://doi.org/10. 1007/s00208-009-0342-8

    2009 doi

  20. [26]

    A fibre bundle description of Teichm¨ uller theory

    Clifford J. Earle and James Eells. “A fibre bundle description of Teichm¨ uller theory”. In: Journal of Differential Geometry 3.1-2 (1969), pp. 19–43. doi: 10.4310/jdg/1214428816. url: https://doi.org/10.4310/jdg/1214428816

    1969

  21. [27]

    Deformations of metrics and associated harmonic maps

    J. Eells and L. Lemaire. “Deformations of metrics and associated harmonic maps”. In: Proc. Indian Acad. Sci. Math. Sci. 90.1 (1981), pp. 33–45. issn: 0253-4142. doi: 10.1007/ BF02867016. url: https : / / doi - org . clsproxy . library . caltech . edu / 10 . 1007 / BF02867016

    1981

  22. [28]

    On the Gauss map of equivariant immersions in hyperbolic space

    Christian El Emam and Andrea Seppi. “On the Gauss map of equivariant immersions in hyperbolic space”. In: J. Topol. 15.1 (2022), pp. 238–301. issn: 1753-8416,1753-8424

    2022

  23. [29]

    A metric uniformization model for the Quasi-Fuchsian space

    Christian El Emam. A metric uniformization model for the Quasi-Fuchsian space . 2023. arXiv: 2307.07388 [math.DG]

    2023 arXiv

  24. [32]

    Opers and TBA

    Davide Gaiotto. “Opers and TBA”. In: (Mar. 2014). arXiv: 1403.6137 [hep-th]

    2014 arXiv

  25. [33]

    Higgs bundles for real groups and the Hitchin-Kostant-Rallis section

    Oscar Garcia-Prada, Ana Pe´ on-Nieto, and S. Ramanan. “Higgs bundles for real groups and the Hitchin-Kostant-Rallis section”. In: Transactions of the American Mathematical Society 370 (Nov. 2015). doi: 10.1090/tran/7363. REFERENCES 67

    2015 doi

  26. [34]

    Higgs bundles and surface group representations

    Oscar Garc ´ ıa-Prada. “Higgs bundles and surface group representations”. In:Moduli spaces and vector bundles. Vol. 359. London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 2009, pp. 265–310

    2009

  27. [35]

    Involutions and higher order automorphisms of Higgs bundle moduli spaces

    Oscar Garc ´ ıa-Prada and S. Ramanan. “Involutions and higher order automorphisms of Higgs bundle moduli spaces”. In: Proc. Lond. Math. Soc. (3) 119.3 (2019), pp. 681–732. issn: 0024-6115,1460-244X. doi: 10 . 1112 / plms . 12242. url: https : / / doi . org / 10 . 1112/plms.12242

    2019

  28. [36]

    The symplectic nature of fundamental groups of surfaces

    William M. Goldman. “The symplectic nature of fundamental groups of surfaces”. In: Adv. in Math. 54.2 (1984), pp. 200–225. issn: 0001-8708. doi: 10.1016/0001-8708(84)90040-

    1984 doi

  29. [37]

    Differential topology

    Victor Guillemin and Alan Pollack. Differential topology. Reprint of the 1974 original. AMS Chelsea Publishing, Providence, RI, 2010, pp. xviii+224. isbn: 978-0-8218-5193-7. doi: 10.1090/chel/370. url: https://doi.org/10.1090/chel/370

    1974 doi

  30. [38]

    url: https://doi.org/10.1016/0001-8708(84)90040-9

    doi

  31. [39]

    The Selfduality equations on a Riemann surface

    Nigel J. Hitchin. “The Selfduality equations on a Riemann surface”. In: Proc. Lond. Math. Soc. 55 (1987), pp. 59–131. doi: 10.1112/plms/s3-55.1.59

    1987 doi

  32. [40]

    Lie groups and Teichm¨ uller space

    N. J. Hitchin. “Lie groups and Teichm¨ uller space”. In: Topology 31.3 (1992), pp. 449–473. issn: 0040-9383. doi: 10.1016/0040- 9383(92)90044- I. url: https://doi.org/10. 1016/0040-9383(92)90044-I

    1992 doi

  33. [41]

    K¨ ahler metric on the space of convex real projective structures on surface

    Inkang Kim and Genkai Zhang. “K¨ ahler metric on the space of convex real projective structures on surface”. In: Journal of Differential Geometry 106.1 (2017), pp. 127–137

    2017

  34. [42]

    Humphreys

    James E. Humphreys. Introduction to Lie algebras and representation theory. Vol. 9. Grad- uate Texts in Mathematics. Second printing, revised. Springer-Verlag, New York-Berlin, 1978, pp. xii+171. isbn: 0-387-90053-5

    1978

  35. [43]

    Anthony W. Knapp. Lie groups beyond an introduction . Second. Vol. 140. Progress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA, 2002, pp. xviii+812. isbn: 0-8176- 4259-5

    2002

  36. [44]

    Holomorphic extensions of Laplacians and their determinants

    Young-Heon Kim. “Holomorphic extensions of Laplacians and their determinants”. In: Adv. Math. 211.2 (2007), pp. 517–545. issn: 0001-8708,1090-2082. doi: 10.1016/j.aim. 2006.09.009. url: https://doi.org/10.1016/j.aim.2006.09.009

    2007 doi

  37. [45]

    Andreas Kriegl and Peter W. Michor. The convenient setting of global analysis . Vol. 53. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997, pp. x+618. isbn: 0-8218-0780-3. doi: 10.1090/surv/053. url: https://doi.org/ 10.1090/surv/053

    1997 doi

  38. [46]

    Sur certaines structures fibr´ ees complexes

    J.L. Koszul and B Malgrange. “Sur certaines structures fibr´ ees complexes”. In:Arch. Math 9 (1958), pp. 102–109

    1958

  39. [47]

    Cyclic surfaces and Hitchin components in rank 2

    Fran¸ cois Labourie. “Cyclic surfaces and Hitchin components in rank 2”. In: Ann. of Math. (2) 185.1 (2017), pp. 1–58. issn: 0003-486X,1939-8980. doi: 10.4007/annals.2017.185. 1.1. url: https://doi.org/10.4007/annals.2017.185.1.1

    2017 doi

  40. [48]

    Cross ratios, Anosov representations and the energy functional on Teichm¨ uller space

    Fran¸ cois Labourie. “Cross ratios, Anosov representations and the energy functional on Teichm¨ uller space”. In:Ann. Sci. ´Ec. Norm. Sup´ er. (4) 41.3 (2008), pp. 437–469. issn: 0012-9593. doi: 10.24033/asens.2072. url: https://doi.org/10.24033/asens.2072

    2008 doi

  41. [49]

    An introduction to Higgs bundles via harmonic maps

    Qiongling Li. “An introduction to Higgs bundles via harmonic maps”. In: SIGMA Sym- metry Integrability Geom. Methods Appl. 15 (2019), Paper No. 035, 30. issn: 1815-0659. doi: 10.3842/SIGMA.2019.035. url: https://doi.org/10.3842/SIGMA.2019.035

    2019 doi

  42. [50]

    Flat projective structures on surfaces and cubic holomorphic differ- entials

    Fran¸ cois Labourie. “Flat projective structures on surfaces and cubic holomorphic differ- entials”. In: Pure Appl. Math. Q. 3.4 (2007), pp. 1057–1099. issn: 1558-8599,1558-8602. doi: 10.4310/PAMQ.2007.v3.n4.a10 . url: https://doi.org/10.4310/PAMQ.2007.v3. n4.a10

    2007 doi

  43. [51]

    Equivariant minimal surfaces in CH2 and their Higgs bundles

    John Loftin and Ian McIntosh. “Equivariant minimal surfaces in CH2 and their Higgs bundles”. In: Asian J. Math. 23.1 (2019), pp. 71–106. issn: 1093-6106,1945-0036. doi: 10.4310/AJM.2019.v23.n1.a5. url: https://doi.org/10.4310/AJM.2019.v23.n1.a5

    2019 doi

  44. [52]

    Teichm¨ uller space is totally geodesic in Goldman space

    Qiongling Li. “Teichm¨ uller space is totally geodesic in Goldman space”. In:Asian J. Math. 20.1 (2016), pp. 21–46. issn: 1093-6106,1945-0036. doi: 10.4310/AJM.2016.v20.n1.a2 . url: https://doi.org/10.4310/AJM.2016.v20.n1.a2. 68 REFERENCES

    2016 doi

  45. [53]

    Affine spheres and convex RPn-manifolds

    John C. Loftin. “Affine spheres and convex RPn-manifolds”. In: Amer. J. Math. 123.2 (2001), pp. 255–274. issn: 0002-9327,1080-6377. url: http://muse.jhu.edu/journals/ american_journal_of_mathematics/v123/123.2loftin.pdf

    2001

  46. [54]

    Minimal Lagrangian surfaces in CH2 and representations of surface groups into SU (2, 1)

    John Loftin and Ian McIntosh. “Minimal Lagrangian surfaces in CH2 and representations of surface groups into SU (2, 1)”. In: Geom. Dedicata 162 (2013), pp. 67–93. issn: 0046- 5755,1572-9168. doi: 10.1007/s10711-012-9717-1 . url: https://doi.org/10.1007/ s10711-012-9717-1

    2013 doi

  47. [55]

    Para-hyperK¨ ahler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds

    Filippo Mazzoli, Andrea Seppi, and Andrea Tamburelli. “Para-hyperK¨ ahler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds”. In: to appear in Mem. Amer. Math. Soc. (July 2021). doi: 10.48550/arXiv.2107.10363 . arXiv: 2107.10363 [math.DG]

    2021 doi arXiv

  48. [56]

    Bi-Lagrangian structures and Teichm¨ uller theory

    Brice Loustau and Andrew Sanders. Bi-Lagrangian structures and Teichm¨ uller theory

  49. [57]

    Cyclic Higgs bundles and minimal surfaces in pseudo-hyperbolic spaces

    Xin Nie. “Cyclic Higgs bundles and minimal surfaces in pseudo-hyperbolic spaces”. In: Advances in Mathematics 436 (2024), p. 109402. issn: 0001-8708. doi: https : / / doi . org/10.1016/j.aim.2023.109402 . url: https://www.sciencedirect.com/science/ article/pii/S0001870823005455

    2024

  50. [58]

    Lie Groups and Lie Algebras III

    Arkadij L. Onishchik and Ernest Borisovich Vinberg. “Lie Groups and Lie Algebras III”. In: 1993. url: https://api.semanticscholar.org/CorpusID:124687276

    1993

  51. [59]

    Nicolaescu

    Liviu I. Nicolaescu. Lectures on the geometry of manifolds . Second. World Scientific Pub- lishing Co. Pte. Ltd., Hackensack, NJ, 2007, pp. xviii+589. isbn: 978-981-277-862-8; 981- 277-862-4. doi: 10.1142/9789812770295 . url: https://doi-org.clsproxy.library. caltech.edu/10.11...

    2007 doi

  52. [60]

    Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and SL(3, C)-quasi-Fuchsian representations

    Nicholas Rungi and Andrea Tamburelli. Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and SL(3, C)-quasi-Fuchsian representations . 2024. arXiv: 2406 . 14945 [id=’math.DG’]

    2024

  53. [61]

    Para-complex geometry and cyclic Higgs bundles

    Nicholas Rungi and Andrea Tamburelli. Para-complex geometry and cyclic Higgs bundles

  54. [62]

    Nicholas Rungi and Andrea Tamburelli. 2024. arXiv: 2306.02699 [math.DG]. url: https: //arxiv.org/abs/2306.02699

    2024 arXiv

  55. [63]

    Nathaniel Sagman and Ognjen Toˇ si´ c.On Hitchin ’s equations for cyclic G-Higgs bundles

  56. [64]

    The pre-symplectic geometry of opers and the holonomy map

    Andrew Sanders. The pre-symplectic geometry of opers and the holonomy map . 2020. arXiv: 1804.04716 [math.DG]. url: https://arxiv.org/abs/1804.04716

    2020 arXiv

  57. [65]

    On the Chern correspondence for principal fibre bundles with complex reduc- tive structure group

    S. Sandon. “On the Chern correspondence for principal fibre bundles with complex reduc- tive structure group”. PhD thesis. Universiteit Leiden, 2005

    2005

  58. [66]

    Unstable minimal surfaces in symmetric spaces of non-compact type

    Nathaniel Sagman and Peter Smillie. Unstable minimal surfaces in symmetric spaces of non-compact type. 2025. arXiv: 2208.04885 [math.DG]. url: https://arxiv.org/abs/ 2208.04885

    2025 arXiv

  59. [67]

    Character varieties

    Adam S. Sikora. “Character varieties”. In: Trans. Amer. Math. Soc.364.10 (2012), pp. 5173–

    2012

  60. [68]

    Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

    Carlos T. Simpson. “Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization”. In: J. Amer. Math. Soc. 1.4 (1988), pp. 867–918. issn: 0894-0347. doi: 10.2307/1990994. url: https://doi.org/10.2307/1990994. REFERENCES 69

    1988 doi

  61. [69]

    Multidimensional Analytic Fredholm Theory

    Michael Taylor. Multidimensional Analytic Fredholm Theory. https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/fred. pdf

    2018

  62. [70]

    Infinite dimensional GIT and moment maps in differential geometry

    Samuel Trautwein. “Infinite dimensional GIT and moment maps in differential geometry”. PhD thesis. ETH Zurich, 2018

    2018

  63. [71]

    The role of harmonic mappings in rigidity and deformation prob- lems

    Richard M. Schoen. “The role of harmonic mappings in rigidity and deformation prob- lems”. In: Complex geometry (Osaka, 1990) . Vol. 143. Lecture Notes in Pure and Appl. Math. Dekker, New York, 1993, pp. 179–200

    1990

  64. [72]

    Closed minimal surfaces in hyperbolic 3-manifolds

    Karen K. Uhlenbeck. “Closed minimal surfaces in hyperbolic 3-manifolds”. In: Seminar on minimal submanifolds . Vol. 103. Ann. of Math. Stud. Princeton Univ. Press, Princeton, NJ, 1983, pp. 147–168. isbn: 0-691-08324-X; 0-691-08319-3

    1983

  65. [73]

    Stability of minimal graphs in products of surfaces

    Tom Y. H. Wan. “Stability of minimal graphs in products of surfaces”. In: Geometry from the Pacific Rim (Singapore, 1994) . de Gruyter, Berlin, 1997, pp. 395–401. isbn: 3-11-014792-0

    1994

  66. [74]

    Lecture notes

    Zuoqin Wang. Lecture notes. http://staff.ustc.edu.cn/~wangzuoq/Courses/21F-Manifolds/Notes/Lec10.pdf

  67. [75]

    An invitation to higher Teichm¨ uller theory

    Anna Wienhard. “An invitation to higher Teichm¨ uller theory”. In: Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures . World Sci. Publ., Hackensack, NJ, 2018, pp. 1013–1039

    2018

  68. [76]

    The Teichm¨ uller theory of harmonic maps

    Michael Wolf. “The Teichm¨ uller theory of harmonic maps”. In: J. Differential Geom. 29.2 (1989), pp. 449–479. issn: 0022-040X. url: http://projecteuclid.org/euclid.jdg/ 1214442885. Christian El Emam: University of Torino, Dipartimento di Matematica “Giuseppe Peano”, Via Carlo...

    1989

  69. [77]

    On the group of real analytic diffeomorphisms

    Takashi Tsuboi. “On the group of real analytic diffeomorphisms”. In: Ann. Sci. ´Ec. Norm. Sup´ er. (4)42.4 (2009), pp. 601–651. issn: 0012-9593,1873-2151. doi: 10.24033/asens

    2009 doi

  70. [97]

    doi: 10.1090/S0002-9904-1960-10413-2

    issn: 0002-9904. doi: 10.1090/S0002-9904-1960-10413-2 . url: https://doi.org/ 10.1090/S0002-9904-1960-10413-2

    1960 doi

  71. [2020]

    arXiv: 1708.09145 [id=’math.DG’]

    arXiv

  72. [2024]

    url: https://arxiv.org/abs/2410.20853

    arXiv: 2410.20853 [math.DG]. url: https://arxiv.org/abs/2410.20853

  73. [2025]

    url: https://arxiv.org/abs/2503.01615

    arXiv: 2503.01615 [math.DG]. url: https://arxiv.org/abs/2503.01615

    arXiv

  74. [2104]

    url: https://doi.org/10.24033/asens.2104

    doi

  75. [5208]

    doi: 10.1090/S0002-9947-2012-05448-1

    issn: 0002-9947,1088-6850. doi: 10.1090/S0002-9947-2012-05448-1 . url: https: //doi.org/10.1090/S0002-9947-2012-05448-1

    2012 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.