Daskalopoulos and Mese proved the holomorphic rigidity conjecture of Teichmüller space: if the mapping class group Γ\Gamma of a surface acts discretely on a contractible Kähler manifold M~\tilde M whose finite-index quotient is a smooth quasi-projective variety admitting an algebraic compactification with boundary of codimension at least 3, then M~\tilde M is equivariantly biholomorphic or conjugate biholomorphic to Teichmüller space. The codimension hypothesis is an artifact of 1990s finite-energy harmonic-map technology (Jost–Yau), and the authors explicitly posed its removal as an open problem, announcing an infinite-energy approach. The authors' infinite-energy program has now produced existence and uniqueness of ρ\rho-equivariant pluriharmonic maps of possibly infinite (logarithmic-growth) energy from smooth quasi-projective varieties into the Weil–Petersson completion of Teichmüller space, and the companion regularity conjecture (harmonic maps from arbitrary-dimensional domains land in a single stratum) has been settled in a Johns Hopkins thesis. However, the codimension-free rigidity theorem itself — the equivariant biholomorphism conclusion under the quasi-projectivity hypothesis alone — has not been proved or claimed in the literature.

Background

Teichmüller space T\mathcal{T} is the space of conformal (equivalently, complete hyperbolic) structures on an oriented surface SS of genus gg with pp marked points, up to isotopy, with 3g−3+p>03g-3+p>0. It is a contractible complex manifold on which the mapping class group Γ\Gamma of SS (the group of isotopy classes of orientation-preserving diffeomorphisms) acts by holomorphic automorphisms. The Weil–Petersson metric GWPG_{WP} on T\mathcal{T} is the Kähler metric obtained by L2L^2-pairing of infinitesimal deformations with quadratic differentials; it has negative sectional curvature and strongly negative curvature tensor in the sense of Siu, but is incomplete. Its metric completion (T‾,dT‾)(\overline{\mathcal{T}}, d_{\overline{\mathcal{T}}}) is the augmented Teichmüller space: set-theoretically T‾\overline{\mathcal{T}} adds boundary points representing stable nodal surfaces obtained by pinching disjoint simple closed curves, and it is a stratified metric space of non-positive curvature in the sense of Alexandrov (an NPC or CAT(0) space, i.e. a geodesic space in which triangles are no fatter than in Euclidean space). Each boundary stratum Tc\mathcal{T}^{c}, indexed by a system cc of pinched curves, is itself a lower-dimensional Teichmüller space; T\mathcal{T} is the top-dimensional open stratum. Unlike Euclidean buildings, T‾\overline{\mathcal{T}} is not locally compact near its boundary, which is the source of the main analytic difficulties. A harmonic map from a Riemannian domain into an NPC space is defined, following Gromov–Schoen and Korevaar–Schoen, as an energy-minimizing (or stationary) map in the Sobolev class; its energy is finite when the map is locally L2L^2-controlled. A map from a Kähler domain is pluriharmonic if its restriction to every holomorphic disc is harmonic. Siu's celebrated Bochner technique shows that a harmonic map from a compact Kähler manifold into a target with strongly negative curvature tensor, having real rank at least 3 somewhere, is holomorphic or conjugate holomorphic; this underlies Siu's strong rigidity theorem, which settled Yau's conjecture for compact Kähler manifolds. Jost and Yau attempted to extend such rigidity to Teichmüller space — the statement that the mapping-class-group action determines T\mathcal{T} as a complex manifold — but their proof was incorrect, and their corrected framework required, as input, an equivariant map of finite energy whose existence was guaranteed only under an auxiliary compactification hypothesis. Daskalopoulos and Mese (Invent. Math. 224, 2021) proved the holomorphic rigidity conjecture by developing the regularity theory of harmonic maps u:Ω→(T‾,dT‾)u : \Omega \to (\overline{\mathcal{T}}, d_{\overline{\mathcal{T}}}): they showed the singular set S(u)S(u) (points with no neighborhood mapping into a single stratum) has Hausdorff dimension at most n−2n-2 for nn-dimensional domains, with decay estimates enabling the Bochner technique. Their Theorem 1.1 assumes that Γ\Gamma acts as a discrete automorphism group on a contractible Kähler manifold M~\tilde M such that for a finite-index subgroup Γ′\Gamma', the quotient M=M~/Γ′M = \tilde M/\Gamma' is a smooth quasi-projective variety (a Zariski-open subset of a projective variety) admitting an algebraic compactification M‾\overline{M} with codim⁡(M‾∖M)≥3\operatorname{codim}(\overline{M}\setminus M) \ge 3; the conclusion is that M~\tilde M is Γ\Gamma-equivariantly biholomorphic or conjugate biholomorphic to T\mathcal{T}. The codimension hypothesis, inherited from Jost–Yau, exists only to produce a finite-energy ρ\rho-equivariant map M~→T\tilde M \to \mathcal{T} through a homotopy equivalence; the authors explicitly flag it as undesirable and state their intent to circumvent it using maps of infinite energy; that program has since yielded the existence of such maps of possibly infinite (logarithmic-growth) energy into the Weil–Petersson completion, for quasi-projective quotients and sufficiently large ρ\rho (Theorem 26 of their ICM 2026 survey). A homomorphism ρ\rho into Γ\Gamma is called sufficiently large if its image contains two independent pseudo-Anosov mapping classes (elements with no power represented by a periodic or reducible map; independence means their fixed point sets on the sphere of projective measured foliations — the boundary compactification relevant to pseudo-Anosov dynamics — do not coincide); finite-index subgroups of Γ\Gamma are sufficiently large.

Problem Statement

Let SS be an oriented surface of genus gg with pp marked points and 3g−3+p>03g-3+p>0 (in particular, a closed surface of genus g≥2g\ge 2), let T\mathcal{T} be its Teichmüller space, and let Γ\Gamma be the mapping class group of SS. Suppose Γ\Gamma acts as a discrete automorphism group on a contractible Kähler manifold M~\tilde M, and that there is a finite-index subgroup Γ′\Gamma' of Γ\Gamma such that M=M~/Γ′M = \tilde M/\Gamma' is a smooth quasi-projective variety — with no assumption on the codimension of the boundary of an algebraic compactification of MM. Since M~\tilde M is contractible, it is the universal cover of MM, and the action induces a homomorphism ρ:π1(M)→Γ′⊂Γ\rho:\pi_1(M)\to\Gamma'\subset\Gamma. Prove that M~\tilde M is Γ\Gamma-equivariantly biholomorphic or conjugate biholomorphic to T\mathcal{T}, with Γ\Gamma acting on T\mathcal{T} as the mapping class group. Equivalently: establish Theorem 1.1 of Daskalopoulos–Mese (Invent. Math. 224 (2021) 791–916) without their assumption (ii), by establishing the equivariant biholomorphism conclusion under the quasi-projectivity hypothesis alone.

References

  1. Georgios Daskalopoulos and Chikako Mese, Rigidity of Teichmüller space, Inventiones mathematicae 224 (2021), no. 3, 791–916. DOI 10.1007/s00222-020-01020-2, https://doi.org/10.1007/s00222-020-01020-2 (preprint arXiv:1502.03367, https://arxiv.org/abs/1502.03367)
  2. Georgios Daskalopoulos and Chikako Mese, Infinite energy maps and rigidity, preprint (2021), arXiv:2112.13961, https://arxiv.org/abs/2112.13961
  3. Georgios Daskalopoulos and Chikako Mese, Infinite Energy Harmonic Maps from Riemann Surfaces to CAT(0) Spaces, The Journal of Geometric Analysis 33 (2023), no. 10, article 337. DOI 10.1007/s12220-023-01382-y, https://doi.org/10.1007/s12220-023-01382-y (preprint arXiv:2210.11434, https://arxiv.org/abs/2210.11434)
  4. Georgios Daskalopoulos and Chikako Mese, Infinite energy harmonic maps from quasi-compact Kähler surfaces, Advanced Nonlinear Studies 24 (2024), no. 1, 103–140. DOI 10.1515/ans-2023-0122, https://doi.org/10.1515/ans-2023-0122
  5. Georgios Daskalopoulos and Chikako Mese, Uniqueness of equivariant harmonic maps to symmetric spaces and buildings, Mathematical Research Letters 30 (2023), no. 6, 1639–1655. DOI 10.4310/mrl.2023.v30.n6.a1, https://doi.org/10.4310/mrl.2023.v30.n6.a1 (preprint arXiv:2111.11422, https://arxiv.org/abs/2111.11422)
  6. Georgios Daskalopoulos and Chikako Mese, Harmonic Maps in Singular Geometry and Rigidity, Proceedings of the International Congress of Mathematicians 2026, Volume 4: Invited Lectures (Sections 5–8) (2026), 103–120. DOI 10.1137/25m1807630, https://doi.org/10.1137/25m1807630 (preprint arXiv:2510.13708, https://arxiv.org/abs/2510.13708)
  7. Y. Sun, Regularity of harmonic maps into the Teichmüller space, Ph.D. thesis, Johns Hopkins University; stated as Theorem 24 in Daskalopoulos–Mese, Harmonic Maps in Singular Geometry and Rigidity, arXiv:2510.13708, https://arxiv.org/abs/2510.13708.