Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read Equality of the marked p-energy spectrum of k-surfaces, or of the modified area and energy entropies, forces the ambient metric of a negatively curved 3-manifold to be hyperbolic or constant curvature.

desk verdict Energy-spectrum rigidity for k-surfaces is new and mostly sound, but Theorem 1.6 has a real gap: convergence of conformal currents is not enough, as written, to guarantee the laminar-measure convergence the proof needs. read the letter →

arxiv 2412.14389 v2 pith:WA6JQ4WB submitted 2024-12-18 math.DG math.DS

Sébastien Alvarez , Ben Lowe , Graham Smith This is my paper · ORCID
classification math.DGmath.DS MSC 53C4253C2453C4057K32
keywords k-surfacesmarkedenergyspectrumentropyrigidityhyperbolic3-manifoldfoliatedPlateauproblemquasi-Fuchsiansurfacesconformalcurrentssectionalcurvature
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

This paper studies k-surfaces—surfaces of constant positive extrinsic curvature k—in closed negatively curved 3-manifolds, and asks how their growth rates, when counted by energy H^p dA, detect the ambient curvature. It establishes three rigidity statements: equality of the marked p-energy spectrum with the hyperbolic spectrum forces the metric to be isometric to hyperbolic; equality of the normalized area and energy spectra forces constant sectional curvature; and a modified energy entropy of almost-round quasi-Fuchsian k-surfaces is maximized by the hyperbolic metric, with equality only for the hyperbolic metric. Together these give the first counting results for closed k-surfaces by energy, and they make the energy spectrum a complete rigidity detector in the pinched-curvature regime. The proofs use the foliated Plateau problem to replace surfaces by invariant measures on the unit sphere bundle, then pass from pointwise curvature estimates to full-support integrals via equidistribution.

What carries the argument

The machinery is the foliated Plateau problem and its measure-theoretic shadow. For every round circle at infinity in the hyperbolic compactification, there is a unique k-disk spanning it; as the circle varies, the Gauss lifts of these disks foliate the unit sphere bundle, and their frame bundles carry a free right PSL2(R)-action. In the hyperbolic metric this foliation is the homogeneous foliation of PSL2(C)/SO(2) by PSL2(R)-orbits, whose leaves are totally umbilical. PSL2(R)-invariant measures on frame and sphere bundles correspond to conformal currents on round circles; a classification theorem says the only ergodic ones are the Lebesgue current with full support and currents supported on closed Fuchsian surfaces. The proofs use equidistribution of almost-round quasi-Fuchsian subgroups: a sequence whose laminar measures converge to the unique fully supported ergodic laminar measure. On the geometric side, a key lemma shows that in hyperbolic space k-disks spanned by (1+epsilon)-quasicircles have mean curvature uniformly close to $k^{{1/2}}$; this supplies the strict energy comparison that makes the hyperbolic metric maximal, and equality cases are fed into full-support integrals to force rigidity.

What would settle it

A concrete refutation of Theorem 1.6 would be a closed hyperbolic 3-manifold (X,h0) and a non-isometric metric h with -1 <= sect_h <= -a for which Ent^p_{k,hat m}(X,h) = Ent^p_{k,hat m}(X,h0) for some 0 < k < a and p >= 0. A more targeted test is to check the companion paper's equidistribution theorem directly: if there is a sequence of (1+epsilon_n)-quasi-Fuchsian subgroups whose limiting laminar measure is supported on a closed Fuchsian surface rather than the full-support measure, then the step from pointwise energy equality to full-support integrals would break.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.6, is a rigid inequality for the modified p-energy entropy Ent^p_{k,hat m}. For a closed hyperbolic 3-manifold (X,h0) and any metric h with -1 <= sect_h <= -a for 0 < a < 1, and for 0 < k < a and p >= 0, Ent^p_{k,hat m}(X,h) <= Ent^p_{k,hat m}(X,h0), with equality if and only if h and h0 are isometric. Theorem 1.3 is the marked-spectrum rigidity: if the marked p-energy spectra of h and h0 coincide, then h is hyperbolic, hence isometric to h0. Theorem 1.5 characterizes constant curvature for metrics with sectional curvature at most -a: the normalized marked p- and q-energy spectra are asymptotic for some p != q if and only if h has constant sectional curvature. Theorem 1.8 is the same rigidity at the level of entropies: the modified area entropy equals the normalized modified p-energy entropy for some p != 0 if and only if the sectional curvature is constant. In the equality cases, the arguments force the mean curvature of the k-surface foliation to be constant along the relevant full-support measure, and then use the axiom-of-spheres rigidity to conclude the ambient curvature is constant.

Load-bearing premise

The rigidity step that upgrades pointwise comparisons to statements about the whole manifold relies on a theorem from the companion paper: there exists a sequence of almost-round quasi-Fuchsian surface subgroups whose laminar measures converge to a unique fully supported ergodic measure on the sphere bundle; if this equidistribution theorem fails, the equality cases in Theorems 1.3, 1.6, and 1.8 cannot be concluded.

Editorial extensions

If this is right

  • Equality of marked p-energy spectra for any single p >= 0 is a complete rigidity test among metrics with curvature pinched between -1 and -a.
  • Equality of the modified area entropy with the normalized modified p-energy entropy is a necessary and sufficient condition for constant curvature, giving a purely exponential-growth criterion.
  • The hyperbolic metric maximizes the modified p-energy entropy for k-surfaces among metrics with sectional curvature at least -1 and at most -a, extending entropy-domination ideas from geodesic flows and minimal surfaces to k-surfaces.
  • These are the first growth-rate results for closed k-surfaces counted by energy, partially answering the open question of whether a thermodynamical formalism exists for this class of surfaces.
  • Under an auxiliary condition that the manifold contains no closed totally geodesic surfaces, the same rigidity holds for unmodified entropy functionals that count only almost-round surfaces without imposing equidistribution, as the paper notes.

Reading between the lines

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

  • A natural next step, not taken in this paper, would be to test stability: if the modified entropies of h are within delta of those of h0, is h forced to be geometrically close to h0? The proof scheme suggests a quantitative version should hold via the same full-support integrals.
  • The near-totally-umbilical lemma suggests a local converse: in a metric close to hyperbolic, the ratio of normalized p- and q-energies over almost-round quasi-Fuchsian surfaces should deviate from 1 by an amount comparable to the sectional-curvature variation; measuring this ratio would probe Theorem 1.5 without computing entropies.
  • If the equidistribution theorem could be replaced by a softer argument, the rigidity would extend to broader families of compact k-surfaces beyond quasi-Fuchsian ones, which is exactly the regime of the original open question about thermodynamical formalism.
  • The equality of area and energy entropies is an extremely strong global detector; one could try to use it to define a k-surface pressure functional whose derivative at a hyperbolic metric annihilates only conformal directions, connecting to marked length spectrum rigidity.
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 / 5 minor

Summary. The paper studies, for closed negatively curved 3-manifolds, the asymptotic counting and rigidity properties of compact k-surfaces counted by p-energy. The main results are: Theorem 1.3, marked p-energy spectrum rigidity for metrics pinched between -1 and -a; Theorem 1.5, asymptotic coincidence of normalized marked area and energy spectra characterizes constant sectional curvature; Theorem 1.6, a rigid upper bound for a modified p-energy entropy, with equality characterizing the hyperbolic metric; and Theorem 1.8, equality of modified area and energy entropies again characterizing constant curvature. The arguments combine the foliated Plateau problem, equidistribution of Kahn-Markovic sequences, Ratner's measure classification, and energy and area estimates for nearly Fuchsian k-surfaces. The proofs are for the most part detailed and the geometric lemmas are cleanly stated. The paper is careful in stating hypotheses and makes good use of the companion paper [2].

Significance. If the main theorems are correct, they provide the first counting results for closed k-surfaces according to energy, partially answering Labourie's question, and they establish a new form of entropy rigidity in negative curvature by characterizing hyperbolic metrics through equality of modified k-surface energy entropies. The paper is careful in stating hypotheses and makes good use of the companion paper [2]; it also contains a self-contained proof of the Mueller-Puchta lower bound (Lemma 5.6) and a clean almost-umbilical estimate (Lemma 3.2). The main concern is a load-bearing step in the entropy-rigidity proofs, where convergence of conformal currents is asserted to imply convergence of the associated laminar measures.

major comments (2)
  1. [Section 5.2.1, proof of Theorem 1.6] The proof asserts: "By definition of the sequence ([Gamma_n])_n, the sequence (mu_{k,h}(partial_infinity Gamma_n))_n of laminar measures converges to mu, the unique ergodic laminar measure that has full support in ShX." This is not justified by the definition of QF_hat_m(eta), which is a condition on the conformal currents hat_m(partial_infinity Gamma_n) being eta-close to Leb, not on the laminar measures mu_{k,h}(partial_infinity Gamma_n). The equidistribution results Theorem 2.16 and Theorem 2.17 are stated for specially constructed sequences, and Lemma 2.15 requires the quasiconformal constants epsilon_n to tend to 0; the sequence produced by Lemma 5.8 has no such control. To make the step rigorous, the authors must prove, or cite from [2], a continuity and tightness statement for the map from conformal currents to laminar measures, including that accumulation points are laminar measures with conformal current Leb and hence, by Theorem 2.12, equal to mu. The same gap occurs in the proof of Theorem 1.8 in Section 5.2.2, where the phrase "Here again, by definition..." repeats the assertion. This point is load-bearing for the equality case of the entropy rigidity theorems.
  2. [Section 5.1.3, definition of modified entropy] The definition of QF_hat_m(eta) and hence of Ent^p_{k,hat_m}(X,h) is not well-posed as written. The text says that for all C >= 1 the space of conformal currents over QC^+(C) is metrizable and then "Let rho be such a metric." But the distance between hat_m(partial_infinity Gamma) and hat_m depends on the choice of C and on the choice of metric rho on QC^+(C); different choices may give different sets QF_hat_m(eta) and hence different entropies. Since QF is the union over all C, one cannot compare a measure supported on QC^+(C) with Leb using a single fixed metric unless a convention is specified. The authors should either define the entropy by fixing a suitable category of conformal currents on the increasing union QC^+(C), or prove that the double limit in eta approaching 0 is independent of the auxiliary choices. This is central, as Theorems 1.6 and 1.8 are statements about this quantity.
minor comments (5)
  1. [Section 5.1.3] The sentence "This entropy counts only quasi-Fuchsian k-surfaces that accumulate to the fully-supported, ergodic laminar measure" is presented as an interpretation, but it is not immediate from the definition, which is in terms of conformal currents; it should either be proved or flagged as a separate lemma.
  2. [Section 2.3.4, Lemma 2.15] The phrase "If lim_{n to infinity} epsilon_n = 0" is redundant since epsilon_n was already introduced as a sequence converging to zero; this is a minor wording issue.
  3. [Section 5.2.1] In the proof of Theorem 1.6, the sentence "we get that sect_h = -1 almost everywhere for mu" should read "mu-almost everywhere".
  4. [Section 1.4.3 and Section 5.2.2] Question 1.11 and Question 5.10 are identical; the duplication should be removed or cross-referenced explicitly.
  5. [Section 5.2.1, proof of Lemma 5.8] The contradiction argument in Lemma 5.8 implicitly uses the pointwise bound W^p_h >= W^p_{h0}, which follows from Lemma 3.1 for the same topological type; this should be stated explicitly so that the contradiction step is transparent.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main rigidity arguments reduce to independent geometric inequalities and external equidistribution theorems; heavy self-citation and one unproved continuity step do not make the derivation circular.

full rationale

The paper's central claims are not circular. The rigidity of the marked p-energy spectrum (Theorem 1.3) derives a pointwise curvature conclusion from integral inequalities: equality of spectra plus the hyperbolic-surface asymptotics of Corollary 3.3 force ∫(1+σ)dμ_{S_n,h}→0, and then Lemma 4.1 combined with Theorem 2.16 yields σ=-1 almost everywhere. The asymptotic energy/area spectrum theorem (Theorem 1.5) uses the same structure to obtain H=k^{1/2} and then invokes the external Leung-Nomizu axiom-of-spheres result; this is independent of the assumptions. Theorem 1.6 and Theorem 1.8 are built on the same inequalities, with Lemma 5.8 extracting a sequence of surfaces whose energy ratio tends to 1. The main external dependence is on the authors' companion paper [2] for the foliated Plateau problem, the bijective correspondence between conformal currents and laminar measures, and the equidistribution of Kahn-Marković sequences (Theorem 2.16), as well as the Lowe-Neves equidistribution theorem [47]. These are separate mathematical results with their own proofs, not restatements of the conclusions of this paper, so their use is load-bearing self-citation but not circular. One sentence in the proof of Theorem 1.6 deserves note: "By definition of the sequence([Γ_n])n∈N, the sequence(µk,h(∂∞Γn))n∈N of laminar measures converges toµ" is not a purely definitional consequence of membership in QF_hat m(η_n), which is formulated using conformal currents rather than laminar measures; the implication depends on the bijective correspondence of §2.3.1 and an unstated continuity/tightness step. This is a gap or correctness risk, not a circular reduction: the target conclusion (constant sectional curvature) is not built into the definition of the modified entropy. Overall, the derivation is self-contained relative to its stated external inputs, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's central claims rest on several external or prior results that are not reproved: the asymptotic, fibered and foliated Plateau problems for k-disks, the equidistribution of Kahn-Marković sequences, Ratner's classification, Kahn-Marković topological counting, and the Leung-Nomizu characterization of constant curvature. These are cited from [2], [31], [43], and [53]. They are standard tools or results from the authors' own prior work, and no data-fitting parameters are introduced.

assumptions (6)
  • domain assumption Existence, uniqueness, and continuity of the k-disk spanning each oriented Jordan curve in the asymptotic Plateau problem (Theorem 2.1, from [2]).
    Defines the k-surface Sk,h([Gamma]) for every quasi-Fuchsian subgroup; used throughout Sections 2 and 4.
  • domain assumption Fibered and foliated Plateau problems: the spaces of k-disks spanned by round circles form continuous PSL2(R)-equivariant bundles and a foliation of the frame and sphere bundles (Theorems 2.2 and 2.3, from [2]).
    Provides the lamination structure and PSL2(R)-action needed for measure classifications and equidistribution.
  • domain assumption Equidistribution of Kahn-Marković sequences: existence of (1+epsilon_n)-quasi-Fuchsian subgroups whose associated laminar measures converge to the unique fully supported ergodic laminar measure (Theorem 2.16, from [2]).
    Converts pointwise energy comparisons into full-support measure integrals in the proofs of Theorems 1.3, 1.5, 1.6, and 1.8.
  • standard math Ratner's classification of measures invariant under PSL2(R) on PSL2(C) (Theorem 2.11, citing [53]).
    Yields the dichotomy for ergodic conformal currents: either proportional to Lebesgue or supported on a closed Fuchsian surface.
  • standard math Kahn-Marković topological counting of quasi-Fuchsian surface subgroups (Theorem 2.7, citing [31]).
    Used to compute the hyperbolic energy entropy in Lemma 5.1 and to prove the entropy inequality in Lemma 5.2.
  • standard math Leung-Nomizu theorem: a 3-manifold in which every tangent plane is tangent to a totally umbilical constant mean curvature surface has constant sectional curvature (cited in Section 4.2, [43]).
    Used to conclude constant curvature from total umbilicity of the leaves of the k-surface foliation in Theorem 1.5.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces." pith.science (2026). https://pith.science/paper/WA6JQ4WB

@misc{pith2026241214389,
  author       = {Pith},
  title        = {Pith review of: Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WA6JQ4WB}},
  note         = {Machine review of arXiv:2412.14389}
}
abstract

Labourie raised the question of determining the possible asymptotics for the growth rate of compact $k$-surfaces, counted according to energy, in negatively curved $3$-manifolds, indicating the possibility of a theory of thermodynamical formalism for this class of surfaces. Motivated by this question and by analogous results for the geodesic flow, we prove a number of results concerning the asymptotic behavior of high energy $k$-surfaces, especially in relation to the curvature of the ambient space. First, we determine a rigid upper bound for the growth rate of quasi-Fuchsian $k$-surfaces, counted according to energy, and with asymptotically round limit set, subject to a lower bound on the sectional curvature of the ambient space. We also study the marked energy spectrum for $k$-surfaces, proving a number of domination and rigidity theorems in this context. Finally, we show that the marked area and energy spectra for $k$-surfaces in $3$-dimensional manifolds of negative curvature are asymptotic if and only if the sectional curvature is constant.

Discussion (0). Continue with ORCID 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. Foliated Plateau problems, geometric rigidity and equidistribution of closed $k$-surfaces

    math.DG 2025-02 unverdicted

    A survey of foliated Plateau problems showing that area-entropy and marked-area-spectrum rigidity for k-surfaces mirror classical geodesic-flow rigidity.

Reference graph

Works this paper leans on

57 extracted references · 51 canonical work pages · cited by 1 Pith paper

  1. [2]

    Alvarez, B

    S. Alvarez, B. Lowe, and G. Smith. Foliated Plateau problems and asymptotic counting of surface subgroups.to appear in Ann. Sci. École Norm. Sup., 2024. 26

    work page 2024

  2. [1]

    Ahlfors.Lectures on quasiconformal mappings

    L. Ahlfors.Lectures on quasiconformal mappings. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987. With the assistance of Clifford J. Earle, Jr., Reprint of the 1966 original

    work page 1987

  3. [3]

    Alvarez and G

    S. Alvarez and G. Smith. Prescription de courbure des feuilles des laminations: retour sur un théorème de Candel.Ann. Inst. Fourier (Grenoble), 71(6):2549–2593, 2021

    work page 2021

  4. [4]

    Alvarez and J

    S. Alvarez and J. Yang. Physical measures for the geodesic flow tangent to a transversally conformal foliation.Ann. Inst. H. Poincaré C Anal. Non Linéaire, 36(1):27–51, 2019

    work page 2019

  5. [5]

    Dominating surface-group representations via Fock-Goncharov coordinates

    P. Barman and S. Gupta. Dominating surface-group representations via Fock-Goncharov coor- dinates. Preprint, [arXiv:2405.15378], 2024

    work page Pith review arXiv 2024

  6. [6]

    Basilio, C

    B. Basilio, C. Lee, and J. Malionek. Totally geodesic surfaces in hyperbolic 3-manifolds: Algo- rithms and examples.Preprint, [arXiv:2403.12397], 2024

    arXiv 2024

  7. [7]

    Besson, G

    G. Besson, G. Courtois, and S. Gallot. Entropies et rigidités des espaces localement symétriques de courbure strictement négative.Geom. Funct. Anal., 5(5):731–799, 1995

    work page 1995

  8. [8]

    Besson, G

    G. Besson, G. Courtois, and S. Gallot. Minimal entropy and Mostow’s rigidity theorems.Ergodic Theory and Dynamical Systems, 16(4):623–649, 1996

    work page 1996

Show all 57 references
  1. [9]

    F. Bonahon. Geodesic laminations with transverse Hölder distributions.Ann. Sci. École Norm. Sup. (4), 30(2):205–240, 1997

    1997

  2. [10]

    Brody and E

    N. Brody and E. Reyes. Approximating hyperbolic lattices by cubulations. Preprint, [arXiv:2404.01511], 2024

    2024 arXiv

  3. [11]

    Ergodic Theory Dynam

    K.BurnsandA.Katok.Manifoldswithnonpositivecurvature. Ergodic Theory Dynam. Systems, 5(2):307–317, 1985

    1985

  4. [12]

    Calegari, F

    D. Calegari, F. Marques, and A. Neves. Counting minimal surfaces in negatively curved 3- manifolds. Duke Math. J., 171(8):1615–1648, 2022

    2022

  5. [13]

    C. Croke. Rigidity for surfaces of nonpositive curvature.Comment. Math. Helv., 65(1):150–169, 1990

    1990

  6. [14]

    Croke and N

    C. Croke and N. Dairbekov. Lengths and volumes in Riemannian manifolds.Duke Math. J., 125(1):1–14, 2004

    2004

  7. [15]

    Croke, N

    C. Croke, N. Dairbekov, and V. Sharafutdinov. Local boundary rigidity of a compact Riemann- ian manifold with curvature bounded above.Trans. Amer. Math. Soc., 352(9):3937–3956, 2000

    2000

  8. [16]

    Dai and Q

    S. Dai and Q. Li. Domination results inn-Fuchsian fibers in the moduli space of Higgs bundles. Proc. Lond. Math. Soc. (3), 124(4):427–477, 2022

    2022

  9. [17]

    Deroin and N

    B. Deroin and N. Tholozan. Dominating surface group representations by Fuchsian ones.Int. Math. Res. Not., (13):4145–4166, 2016

    2016

  10. [18]

    Einsiedler

    M. Einsiedler. Ratner’s theorem for SL(2, R)-invariant measures. Preprint, [arXiv:0603483], 2006

    2006

  11. [19]

    Flaminio

    L. Flaminio. Local entropy rigidity for hyperbolic manifolds.Comm. Anal. Geom., 3(3-4):555– 596, 1995

    1995

  12. [20]

    Gogolev and J.M

    A. Gogolev and J.M. Reber. A counterexample to marked length spectrum semi-rigidity. Preprint, [arXiv:2309.10882], 2023

    2023 arXiv

  13. [21]

    M. Gromov. Foliated Plateau problem. I. Minimal varieties.Geom. Funct. Anal., 1(1):14–79, 1991

    1991

  14. [22]

    M. Gromov. Foliated Plateau problem. II. Harmonic maps of foliations.Geom. Funct. Anal., 1(3):253–320, 1991

    1991

  15. [23]

    M. Gromov. Three remarks on geodesic dynamics and fundamental group.Enseign. Math. (2), 46(3-4):391–402, 2000

    2000

  16. [24]

    Math., 275(2):325–359, 2015

    F.Guéritaud, F.Kassel, andM.Wolff.Compactanti–deSitter3-manifoldsandfoldedhyperbolic structures on surfaces.Pacific J. Math., 275(2):325–359, 2015

    2015

  17. [25]

    Guillarmou and T

    C. Guillarmou and T. Lefeuvre. The marked length spectrum of Anosov manifolds.Ann. of Math. (2), 190(1):321–344, 2019

    2019

  18. [26]

    Guillemin and D

    V. Guillemin and D. Kazhdan. Some inverse spectral results for negatively curved2-manifolds. Topology, 19(3):301–312, 1980

    1980

  19. [27]

    Hamenstädt

    U. Hamenstädt. Cocycles, symplectic structures and intersection.Geom. Funct. Anal., 9(1):90– 140, 1999

    1999

  20. [28]

    Hamenstädt

    U. Hamenstädt. Incompressible surfaces in rank one locally symmetric spaces.Geom. Funct. Anal., 25(3):815–859, 2015

    2015

  21. [29]

    T. Humbert. Katok’s entropy conjecture near real and complex hyperbolic metrics.Preprint, [arXiv:2409.11197], 2024

    2024

  22. [30]

    J. Kahn, F. Labourie, and S. Mozes. Surface groups in uniform lattices of some semi-simple groups. Acta Math., 232:79–220, 2024

    2024

  23. [31]

    Kahn and V

    J. Kahn and V. Marković. Counting essential surfaces in a closed hyperbolic three-manifold. Geom. Topol., 16(1):601–624, 2012. 27

    2012

  24. [32]

    Kahn and V

    J. Kahn and V. Marković. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math. (2), 175(3):1127–1190, 2012

    2012

  25. [33]

    J. Kahn, V. Markovic, and I. Smilga. Geometrically and topologically random surfaces in a closed hyperbolic three manifold.Preprint, [arXiv:2309.02847]

    arXiv

  26. [34]

    A. Katok. Entropy and closed geodesics.Ergodic Theory Dynam. Systems, 2(3-4):339–365, 1982

    1982

  27. [35]

    A. Katok. Four applications of conformal equivalence to geometry and dynamics.Ergodic Theory Dynam. Systems, 8∗:139–152, 1988

    1988

  28. [36]

    Labourie

    F. Labourie. Problèmes de Monge-Ampère, courbes holomorphes et laminations.Geom. Funct. Anal., 7(3):496–534, 1997

    1997

  29. [37]

    Labourie

    F. Labourie. Un lemme de Morse pour les surfaces convexes.Invent. Math., 141(2):239–297, 2000

    2000

  30. [38]

    Labourie

    F. Labourie. The phase space ofk-surfaces. InRigidity in dynamics and geometry (Cambridge, 2000), pages 295–307. Springer, Berlin, 2002

    2000

  31. [39]

    Labourie

    F. Labourie. Randomk-surfaces. Ann. of Math. (2), 161(1):105–140, 2005

    2005

  32. [40]

    Labourie

    F. Labourie. Asymptotic counting of minimal surfaces and of surface groups in hyperbolic 3- manifolds. Séminaire Bourbaki, exposé 1179 Astérisque, (430):Exp. No. 1179, 425–457, 2021

    2021

  33. [41]

    Ledrappier

    F. Ledrappier. Harmonic measures and Bowen-Margulis measures.Israel J. Math., 71(3):275– 287, 1990

    1990

  34. [42]

    Ledrappier

    F. Ledrappier. Structure au bord des variétés à courbure négative. InSéminaire de Théorie Spectrale et Géométrie, No. 13, Année 1994–1995, volume 13 ofSémin. Théor. Spectr. Géom., pages 97–122. Univ. Grenoble I, Saint-Martin-d’Hères, 1995

    1994

  35. [43]

    Leung and K

    D. Leung and K. Nomizu. The axiom of spheres in Riemannian geometry.J. Differential Ge- ometry, 5:487–489, 1971

    1971

  36. [44]

    Q. Li. Harmonic maps for Hitchin representations.Geom. Funct. Anal., 29(2):539–560, 2019

    2019

  37. [45]

    A. N. Livšic. Certain properties of the homology ofY-systems. Mat. Zametki, 10:555–564, 1971

    1971

  38. [46]

    B. Lowe. Deformations of totally geodesic foliations and minimal surfaces in negatively curved 3-manifolds. Geom. Funct. Anal., 31(4):895–929, 2021

    2021

  39. [47]

    Lowe and A

    B. Lowe and A. Neves. Minimal surface entropy and average area ratio.J. Differential Geom., 130(2):443–475, 2025

    2025

  40. [48]

    Maclachlan and A

    C. Maclachlan and A. Reid.The arithmetic of hyperbolic 3-manifolds, volume 219 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 2003

    2003

  41. [49]

    Marques and A

    F. Marques and A. Neves. Conformal currents and the entropy of negatively curved three- manifolds. Preprint, [arXiv:2405.16302], 2024

    2024 arXiv

  42. [50]

    G. D. Mostow. Quasi-conformal mappings inn-space and the rigidity of hyperbolic space forms. Inst. Hautes Études Sci. Publ. Math., (34):53–104, 1968

    1968

  43. [51]

    Müller and J.-C

    T. Müller and J.-C. Puchta. Character theory of symmetric groups and subgroup growth of surface groups.J. London Math. Soc. (2), 66(3):623–640, 2002

    2002

  44. [52]

    J.-P. Otal. Le spectre marqué des longueurs des surfaces à courbure négative.Ann. of Math. (2), 131(1):151–162, 1990

    1990

  45. [53]

    M. Ratner. Raghunathan’s topological conjecture and distributions of unipotent flows.Duke Math. J., 63(1):235–280, 1991

    1991

  46. [54]

    A. Reid. Totally geodesic surfaces in hyperbolic3-manifolds. Proc. Edinburgh Math. Soc. (2), 34(1):77–88, 1991

    1991

  47. [55]

    N. Sagman. Infinite energy equivariant harmonic maps, domination, and anti–de Sitter 3- manifolds. J. Differential Geom., 124(3):553–598, 2023

    2023

  48. [56]

    A. Seppi. Minimal discs in hyperbolic space bounded by a quasicircle at infinity.Comment. Math. Helv., 91(4):807–839, 2016

    2016

  49. [57]

    G. Smith. On the asymptotic Plateau problem in Cartan–Hadamard manifolds. Preprint, [arXiv:2107.14670], 2021. Sébastien Alvarez CMAT, Facultad de Ciencias, Universidad de la República, & IRL-IFUMI (CNRS) Igua 4225 esq. Mataojo. Montevideo, Uruguay. email: salvarez@cmat.edu.uy ...

    2021 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.