Weakly almost-Fuchsian manifolds are nearly-Fuchsian

Andrea Seppi; Jean-Marc Schlenker; Manh-Tien Nguyen

Weakly almost-Fuchsian manifolds are nearly-Fuchsian

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Pith's one-line read This paper proves that any closed minimal surface whose principal curvatures stay within [-1,1] can be perturbed to one whose principal curvatures stay strictly within (-1,1); for complete S × R hyperbolic manifolds this answers a…

desk verdict Settles a 40-year-old question in hyperbolic geometry with a genuinely new deformation argument; worth a careful refereeing. read the letter →

arxiv 2501.12277 v1 pith:LUVA2S4H submitted 2025-01-21 math.DG

Manh-Tien Nguyen , Jean-Marc Schlenker , Andrea Seppi This is my paper · ORCID

classification math.DG MSC 53A1053C4257K32

keywords hyperbolic3-manifoldsminimalsurfacesprincipalcurvaturesquasi-Fuchsianmanifoldsalmost-Fuchsiannearly-Fuchsianhalf-translationstructuresnormalvariations

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

Hyperbolic three-manifolds that contain a closed minimal surface whose principal curvatures never leave the interval $[-1,1]$ also contain nearby surfaces whose principal curvatures stay strictly inside $(-1,1)$. The proof constructs a small normal perturbation that lowers the positive principal curvature and raises the negative one exactly at the points where they reach the extreme values $1$ and $-1$. In the complete case $M \cong S \times \mathbb{R}$, this upgrades every weakly almost-Fuchsian manifold to a nearly-Fuchsian one and hence to a quasi-Fuchsian manifold, resolving a question left open since 1983. It also shows that the classes of nearly-Fuchsian and almost-Fuchsian manifolds differ: many weakly almost-Fuchsian but not almost-Fuchsian manifolds now carry a strictly $(-1,1)$ surface, disproving a conjecture from the 2000s. A separate partial converse gives a universal $\varepsilon > 0$ such that surfaces with principal curvatures in $(-\varepsilon,\varepsilon)$ force the manifold to be almost-Fuchsian.

What carries the argument

The load-bearing object is a 'curvature-decreasing function': a smooth $f$ on $\Sigma$ whose Hessian satisfies $\nabla^\Sigma df(e_+,e_+) = -1$ and $\nabla^\Sigma df(e_-,e_-) = 1$ on $Z$, where $e_\pm$ are unit eigenvectors of the shape operator for the positive and negative principal curvatures. Because the principal curvatures are $\pm 1$ exactly on $Z$, the variation formula for the shape operator simplifies there; the term $f(B^2 - I)$ vanishes, so the first-order change of the eigenvalues is exactly the Hessian of $f$ in the eigen-directions. The construction of $f$ uses the half-translation structure on $\Sigma \setminus \{q=0\}$, the flat coordinate atlas in which the holomorphic quadratic differential $q$ (whose real part is the second fundamental form) becomes $dz^2$; in flat coordinates $z=x+iy$ one has $I = e^{2u}|dz|^2$, $II = dx^2 - dy^2$, and $\|II\|^2 = 2e^{-4u}$, so $Z$ is the zero set of $u$ and the Hessian condition becomes an Euclidean Hessian condition. The analyticity of $u$ (via the $\cosh$-Gordon equation $\Delta u = 2\cosh(2u)$) shows $Z$ is a finite union of points and simple closed curves, and Proposition 3.7 shows no curve in the critical set is a geodesic; this lets the authors handle the holonomy around each curve component (trivial, rotation by $\pi$, or translation) and glue local solutions with bump functions. The translation case is the delicate one, solved by Proposition 4.7, which builds a function interpolating between two affine quadratics while keeping Hessian $\mathrm{diag}(-1,1)$ along an arbitrary non-linear curve.

What would settle it

Exhibit a complete hyperbolic three-manifold homeomorphic to $S \times \mathbb{R}$, with $S$ a closed surface of genus at least two, that contains a closed minimal surface with principal curvatures in $[-1,1]$ and also contains an accidental parabolic; Corollary 1.3 says no such manifold exists, so one counterexample would refute Theorem 1.1.

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Extended reading notes

Core claim

The central result, Theorem 1.1, states that if $\Sigma$ is a closed, orientable, two-sided embedded minimal surface in a hyperbolic three-manifold $M$ and the principal curvatures of $\Sigma$ lie in $[-1,1]$, then every neighbourhood of $\Sigma$ contains a closed embedded surface with principal curvatures in $(-1,1)$. No completeness or topological assumption on $M$ is needed. The proof works by deforming $\Sigma$ along its normal direction with a speed function $f$ chosen so that, at each point of the extremal set $Z = \{ \|II\|^2 = 2 \}$, the positive principal curvature decreases and the negative one increases at first order; Lemma 2.3 reduces this to prescribing the Hessian of $f$ at $Z$ as $-1$ in the positive eigendirection and $+1$ in the negative eigendirection. In the complete case $M \cong S \times \mathbb{R}$, the theorem gives three corollaries: every weakly almost-Fuchsian manifold is nearly-Fuchsian (Corollary 1.2) and therefore quasi-Fuchsian (Corollary 1.3), and since a classical 1983 construction provides weakly almost-Fuchsian manifolds that are not almost-Fuchsian, those manifolds are nearly-Fuchsian without any almost-Fuchsian minimal surface (Corollary 1.4). The paper also proves Theorem 1.5: there is a universal $\varepsilon > 0$ such that a quasi-Fuchsian manifold containing a closed surface with principal curvatures in $(-\varepsilon,\varepsilon)$ is almost-Fuchsian.

Load-bearing premise

The argument relies on the borderline case of a rigidity statement: a complete minimal immersion into hyperbolic 3-space with squared second fundamental form $\|II\|^2 \le 2$ must be a proper embedding of a plane; the strict case is cited, but the equality case is cited to a preprint appendix rather than proved here, and Proposition 3.7 collapses if that extension fails.

Editorial extensions

If this is right

Every weakly almost-Fuchsian manifold is nearly-Fuchsian: the borderline minimal surface can be replaced by one with principal curvatures strictly inside $(-1,1)$.
Every weakly almost-Fuchsian manifold is quasi-Fuchsian, resolving the 1983 question; in particular such a manifold cannot contain an accidental parabolic.
There exist nearly-Fuchsian manifolds that are not almost-Fuchsian: the classical 1983 weakly almost-Fuchsian non-almost-Fuchsian examples now have a strictly $(-1,1)$ surface, so the 2000s conjecture is false.
The conclusion is stable under small deformations: any complete hyperbolic structure sufficiently close to a weakly almost-Fuchsian one is also nearly-Fuchsian and quasi-Fuchsian.
A universal $\varepsilon > 0$ exists such that a quasi-Fuchsian manifold with a closed surface of principal curvatures in $(-\varepsilon,\varepsilon)$ is almost-Fuchsian, giving a quantitative partial converse.

Reading between the lines

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

The proof is local in $M$ and never uses completeness, so the same normal-deformation idea should apply to other settings where a minimal surface saturates a curvature bound, such as higher-dimensional hyperbolic manifolds or manifolds with boundary.
The dichotomy between points and curves in $Z$, together with the holonomy cases, suggests a local normal form for the boundary of the almost-Fuchsian locus; the translation case is the only one where a genuinely two-dimensional interpolation is needed.
A testable quantitative direction: the size of the normal deformation needed to reach $(-1,1)$ should be controlled by the length of the extremal curves in $Z$ and by the translation vector of the holonomy, giving an explicit estimate for how far a weakly almost-Fuchsian manifold is from being almost-Fuchsian.
If the cited equality-case rigidity for complete minimal planes with $\|II\|^2 \le 2$ failed, the obstruction would appear exactly in Proposition 3.7; checking that appendix is the fastest way to test the proof's foundation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know about this paper: it answers Uhlenbeck's 1983 question whether weakly almost-Fuchsian manifolds are quasi-Fuchsian, and it disproves the standing conjecture that nearly-Fuchsian implies almost-Fuchsian. The main theorem is that any closed minimal surface with principal curvatures in [-1,1] in a hyperbolic 3-manifold can be perturbed to a nearby non-minimal surface with principal curvatures in (-1,1). If the manifold is complete and homeomorphic to S×R, that gives the corollaries.

What's genuinely new: the proof. The authors construct a normal variation whose initial speed f has Hessian equal to -1 in the direction of the positive principal curvature and +1 in the negative direction, exactly on the set Z where the curvatures hit ±1. The way they handle the case where Z contains a simple closed curve with translation holonomy is the most interesting part: Proposition 4.7 builds a function on R2 whose Hessian has the right signs on a given curve while interpolating between two quadratics, using a clever integration along the curve. That's a nice piece of work.

The paper is well-organized and the calculations check out. Lemma 2.3 correctly computes the first-order change of principal curvatures at the critical points, and the gluing argument in Section 4 is sound. The reliance on prior work by overlapping authors, specifically the equality case of the proper-embedding criterion cited to HLS24 Appendix A, is the softest spot. That is a preprint with overlapping authorship and the paper doesn't prove that extension. However, the stress-test note is right that the use may not be load-bearing: the contradiction in Proposition 3.7 only needs the universal cover and the model surface to agree as immersions, not necessarily as proper embeddings. If the HLS24 result failed, the argument could likely be patched by using the fundamental theorem of surfaces on the simply connected domain and the fact that the model surface has no umbilic points. So I'd flag it as a point the referee should check, but not a fatal gap.

The partial converse Theorem 1.5 is a nice addition, though it's a bit separate and relies on known quasicircle estimates. It doesn't affect the main argument.

This is a significant paper that will be read by people working in hyperbolic geometry, minimal surfaces, and quasi-Fuchsian theory. I'd bring it to the reading group and I'd cite it. Send it to a serious referee; if they verify the imported lemma, it's a clean accept.

Referee Report

0 major / 6 minor

Summary. The paper proves Theorem 1.1: if a hyperbolic three-manifold M contains a closed, orientable, two-sided, embedded minimal surface whose principal curvatures lie in [-1,1], then every neighbourhood of that surface contains a closed embedded surface with principal curvatures in (-1,1). The proof constructs a normal variation of the minimal surface with a carefully chosen speed function f whose Hessian acts on the curvature-critical locus Z = {||II||^2 = 2} to push the two principal curvatures strictly inside (-1,1). The structure of Z is analyzed via half-translation structures and the cosh-Gordon equation, leading to a technical construction (Proposition 4.7) of a function with prescribed Hessian along curves. The main theorem yields Corollaries 1.2 and 1.3: every weakly almost-Fuchsian manifold is nearly-Fuchsian and quasi-Fuchsian, answering a question from Uhlenbeck's 1983 paper. It also yields Corollary 1.4: there exist nearly-Fuchsian manifolds that are not almost-Fuchsian, disproving a conjecture from the 2000s. A partial converse, Theorem 1.5, states that a surface with sufficiently small principal curvatures implies the existence of an almost-Fuchsian minimal surface.

Significance. If correct, the main result resolves a long-standing question from Uhlenbeck's 1983 paper and settles a conjecture on the relation between almost-Fuchsian and nearly-Fuchsian manifolds. The argument is inventive and combines several modern tools: half-translation structures on minimal surfaces, analytic properties of solutions of the cosh-Gordon equation, and a flexible local Hessian construction. The paper is carefully written and the central derivation is coherent. The construction of the speed function is explicit and the proof is essentially self-contained, with the notable exception of the equality case of Proposition 3.3 imported from the preprint [HLS24]. The partial converse Theorem 1.5 is a valuable addition. Overall this is a substantial contribution to the study of minimal surfaces and quasi-Fuchsian manifolds.

minor comments (6)

[Section 1.4] The sentence "The space of nearly-Fuchsian manifolds is trivially contained in the space of almost-Fuchsian manifolds" states the opposite of the true inclusion, since every almost-Fuchsian manifold is nearly-Fuchsian. This appears to contradict Corollary 1.4 and should be corrected.
[Section 3.2, proof of Lemma 3.5] The curve η is written as "(δ, δ) × {0}" but the integral that follows uses the interval (-δ, δ). The intended interval is (-δ, δ).
[Section 3.2, Proposition 3.3] The equality case ||II||^2 = 2 is imported from [HLS24, Appendix A], a preprint with overlapping authorship, and the paper does not reproduce the proof. This is a soft spot, but it is not load-bearing for the main theorem: the use in Proposition 3.7 only requires the universal cover immersion to agree with the model immersion on an open set, not properness, so the contradiction can be obtained without the full equality-case embedding statement. The authors should clarify this dependence or provide a proof.
[Section 4.3, paragraph before Lemma 4.6] The sentence "One cannot simply proceed as in the proofs of Lemma 4.5 and Lemma 4.6" refers to Lemma 4.6 before it is stated; this should be "Lemma 4.5" or "Lemmas 4.4 and 4.5".
[Section 4.2, Lemma 4.4] The definition of f uses F(z(p)) where z is a flat coordinate only on a local chart; to make f globally defined with the stated support, the bump function must be chosen with support contained in that chart. This is easy to arrange but should be stated explicitly.
[Section 2.2, equation (3)] The notation HessΣf is introduced as a (1,1)-tensor, while in Lemma 2.3 the expression (∇Σ d f)(e±, e±) uses the (0,2) Hessian. The identification is clear but a sentence indicating the musical isomorphism would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the perturbation argument is an explicit construction, and the overlapping-author HLS24 lemma is not essential to the contradiction.

full rationale

The paper's central claim is proved by an explicit deformation: on the critical set Z where ||II||^2 = 2, the authors construct a smooth function f with prescribed Hessian (Proposition 4.1) and use the normal-variation formula (Lemma 2.3) to force the principal curvatures strictly inside (-1,1) near Z, while away from Z they are already strict by compactness. No equation is fitted and no target quantity is used as an input. The only imported results with overlapping authorship are the strict and equality cases of the proper-embedding statement Proposition 3.3, cited to [EES22, Proposition 4.15] and [HLS24, Appendix A]. This is a genuine prior-work citation, not a renaming of the present theorem, and the equality-case extension is not the conclusion of the paper. Moreover, the way Proposition 3.7 uses Proposition 3.3 is not actually load-bearing: the contradiction only needs the two immersions to agree as immersions on the universal cover, which follows from the uniqueness part of the fundamental theorem of surfaces together with analytic continuation; the model surface Sigma_0 has II = dx^2 - dy^2 with no zeros, while the universal cover has zeros of the Hopf differential. Thus even if the HLS24 appendix were removed, the contradiction stands. The paper is self-contained in all steps that constitute the claimed 'prediction' (the existence of nearby strictly curvature-bounded surfaces), and no self-definitional, fitted-input, or uniqueness-imported circularity is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central argument is carried by a short list of standard geometric facts plus one imported equality-case result. The proof introduces no fitted constants and no new geometric entities. The main self-contained contribution is the construction of the curvature-decreasing function f in Propositions 4.1 and 4.7; everything else is standard background in minimal surface theory.

assumptions (7)

domain assumption The equality-case proper-embedding theorem: a complete minimal immersion into H3 with ||II||^2 <= 2 is a proper embedding and the surface is diffeomorphic to R^2.
Invoked in Section 3.2 (Proposition 3.3) and used in Proposition 3.7 to identify the universal cover of the minimal surface with the translation-invariant model surface. The paper cites [EES22, Proposition 4.15] for the strict case and [HLS24, Appendix A] for the equality case; neither proof is reproduced.
standard math The second fundamental form of a minimal surface in a hyperbolic 3-manifold is the real part of a holomorphic quadratic differential with isolated zeros.
Used in Section 2.3 to build half-translation structures and flat coordinates. This is standard Hopf-Lawson-Tromba theory.
standard math Real analyticity of solutions of the cosh-Gordon equation and Lojasiewicz's structure theorem for 2-dimensional real analytic varieties.
Used in Proposition 3.1 to decompose the set Z where ||II||^2 = 2 into finitely many points and simple closed curves.
standard math Cauchy-Kovalevskaya and analytic continuation for solutions of the cosh-Gordon equation.
Used in Proposition 3.7: a solution matching a translation-invariant solution on an open set must agree everywhere, forcing the universal cover to be the model surface.
standard math Ahlfors-Schwarz lemma: no complete conformal metric on C has curvature bounded above by -1.
Used in Lemma 3.5 to prove that the maximal solution of the ODE g'' = 2 cosh(2g) has finite width.
standard math Fundamental theorem of surfaces in H3: a pair (I, II) satisfying Gauss-Codazzi is realized by an immersion unique up to isometry.
Used in Lemma 3.5 and Proposition 3.7 to compare the embedding data of two minimal surfaces.
standard math Every holomorphic quadratic differential on a closed Riemann surface of genus at least 2 has zeros.
Used at the end of Proposition 3.7: the universal cover's second fundamental form must have zeros, contradicting the model surface's globally nonzero II = dx^2 - dy^2.

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Cite this review

Pith. "Pith review of Weakly almost-Fuchsian manifolds are nearly-Fuchsian." pith.science (2026). https://pith.science/paper/LUVA2S4H

@misc{pith2026250112277,
  author       = {Pith},
  title        = {Pith review of: Weakly almost-Fuchsian manifolds are nearly-Fuchsian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LUVA2S4H}},
  note         = {Machine review of arXiv:2501.12277}
}

abstract

We show that a hyperbolic three-manifold $M$ containing a closed minimal surface with principal curvatures in $[-1,1]$ also contains nearby (non-minimal) surfaces with principal curvatures in $(-1,1)$. When $M$ is complete and homeomorphic to $S\times\mathbb{R}$, for $S$ a closed surface, this implies that $M$ is quasi-Fuchsian, answering a question left open from Uhlenbeck's 1983 seminal paper. Additionally, our result implies that there exist (many) quasi-Fuchsian manifolds that contain a closed surface with principal curvatures in $(-1,1)$, but no closed minimal surface with principal curvatures in $(-1,1)$, disproving a conjecture from the 2000s.

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Works this paper leans on

37 extracted references · 35 canonical work pages

[1]

Lars V. Ahlfors. An extension of S chwarz's lemma. Trans. Amer. Math. Soc. , 43(3):359--364, 1938

work page 1938
[2]

Anderson

Michael T. Anderson. Complete minimal varieties in hyperbolic space. Invent. Math. , 69:477--494, 1982

work page 1982
[3]

Anderson

Michael T. Anderson . Complete minimal hypersurfaces in hyperbolic n-manifolds . Comment. Math. Helv. , 58:264--290, 1983

work page 1983
[4]

Contraction of convex hypersurfaces in R iemannian spaces

Ben Andrews. Contraction of convex hypersurfaces in R iemannian spaces. J. Differential Geom. , 39(2):407--431, 1994

work page 1994
[5]

Positively curved surfaces in the three-sphere

Ben Andrews . Positively curved surfaces in the three-sphere . In Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures , pages 221--230. Beijing: Higher Education Press; Singapore: World Scientific/distributor, 2002

work page 2002
[6]

Marques, and Andr\'e Neves

Danny Calegari, Fernando C. Marques, and Andr\'e Neves. Minimal surface entropy of negatively curved manifolds. In Perspectives in scalar curvature. V ol. 2 , pages 417--428. World Sci. Publ., Hackensack, NJ, [2023] 2023

work page 2023
[7]

Quasi- F uchsian manifolds close to the F uchsian locus are foliated by constant mean curvature surfaces

Diptaishik Choudhury, Filippo Mazzoli, and Andrea Seppi. Quasi- F uchsian manifolds close to the F uchsian locus are foliated by constant mean curvature surfaces. Math. Ann. , 388(4):3981--4010, 2024

work page 2024
[8]

Geometric structures for maximal representations and pencils

Colin Davalo. Geometric structures for maximal representations and pencils. Arxiv 2407.01254 , 2024

work page arXiv 2024

Show all 37 references

[9]

do Carmo and M

M. do Carmo and M. Dajczer. Rotation hypersurfaces in spaces of constant curvature. Trans. Amer. Math. Soc. , 277(2):685--709, 1983

1983
[10]

On the G auss map of equivariant immersions in hyperbolic space

Christian El Emam and Andrea Seppi. On the G auss map of equivariant immersions in hyperbolic space. J. Topol. , 15(1):238--301, 2022

2022
[11]

Charles L. Epstein. The hyperbolic G auss map and quasiconformal reflections. J. Reine Angew. Math. , 372:96--135, 1986

1986
[12]

Charles L. Epstein. Envelopes of horospheres and weingarten surfaces in hyperbolic 3-space. Arxiv 2401.12115 , 2024

2024 arXiv
[13]

Minimal surfaces and the topology of three-manifolds

Joel Hass. Minimal surfaces and the topology of three-manifolds. In Global theory of minimal surfaces , volume 2 of Clay Math. Proc. , pages 705--724. Amer. Math. Soc., Providence, RI, 2005

2005
[14]

Beyond almost fuchsian space

Zheng Huang and Ben Lowe. Beyond almost fuchsian space. To appear in American Journal of Math., ArXiv:2104.11284 , 2021

2021 arXiv
[15]

Uniqueness and nonuniqueness for the asymptotic plateau problem in hyperbolic space

Zheng Huang, Ben Lowe, and Andrea Seppi. Uniqueness and nonuniqueness for the asymptotic plateau problem in hyperbolic space. Arxiv 2309.00599v2 , 2024

2024 arXiv
[16]

Bifurcation for minimal surface equation in hyperbolic 3-manifolds

Zheng Huang, Marcello Lucia, and Gabriella Tarantello. Bifurcation for minimal surface equation in hyperbolic 3-manifolds. Ann. Inst. Henri Poincar \'e , Anal. Non Lin \'e aire , 38(2):243--279, 2021

2021
[17]

U ber F l\

Heinz Hopf. \" U ber F l\"achen mit einer R elation zwischen den H auptkr\"ummungen. Math. Nachr. , 4:232--249, 1951

1951
[18]

Differential geometry in the large , volume 1000 of Lecture Notes in Mathematics

Heinz Hopf. Differential geometry in the large , volume 1000 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, second edition, 1989. Notes taken by Peter Lax and John W. Gray, With a preface by S. S. Chern, With a preface by K. Voss

1989
[19]

On almost-Fuchsian manifolds

Zheng Huang and Biao Wang . On almost-Fuchsian manifolds . Trans. Am. Math. Soc. , 365(9):4679--4698, 2013

2013
[20]

Counting minimal surfaces in quasi-Fuchsian three-manifolds

Zheng Huang and Biao Wang . Counting minimal surfaces in quasi-Fuchsian three-manifolds . Trans. Am. Math. Soc. , 367(9):6063--6083, 2015

2015
[21]

auser Advanced Texts: Basler Lehrb\

Steven G. Krantz and Harold R. Parks. A primer of real analytic functions . Birkh\"auser Advanced Texts: Basler Lehrb\"ucher. [Birkh\"auser Advanced Texts: Basel Textbooks]. Birkh\"auser Boston, Inc., Boston, MA, second edition, 2002

2002
[22]

Minimal surfaces and particles in 3-manifolds

Kirill Krasnov and Jean-Marc Schlenker. Minimal surfaces and particles in 3-manifolds. Geom. Dedicata , 126:187--254, 2007

2007
[23]

Minimal surfaces and entanglement entropy in anti-de S itter space

Pavel Krtou s and Andrei Zelnikov. Minimal surfaces and entanglement entropy in anti-de S itter space. J. High Energy Phys. , (10):077, front matter+35, 2014

2014
[24]

Asymptotic counting of minimal surfaces and of surface groups in hyperbolic 3-manifolds [according to C alegari, M arques and N eves]

Fran c ois Labourie. Asymptotic counting of minimal surfaces and of surface groups in hyperbolic 3-manifolds [according to C alegari, M arques and N eves]. Ast\'erisque , (430):Exp. No. 1179, 425--457, 2021

2021
[25]

Blaine Lawson, Jr

H. Blaine Lawson, Jr. Complete minimal surfaces in S 3 . Ann. of Math. (2) , 92:335--374, 1970

1970
[26]

Morrey, Jr

Charles B. Morrey, Jr. On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I . A nalyticity in the interior. Amer. J. Math. , 80:198--218, 1958

1958
[27]

I. G. Petrowsky. Sur l'analyticit\'e des solutions des syst\`emes d'\'equations diff\'erentielles. Rec. Math. N.S. [Mat. Sbornik] , 5/47:3--70, 1939

1939
[28]

Hyam Rubinstein

J. Hyam Rubinstein . Minimal surfaces in geometric 3-manifolds . In Global theory of minimal surfaces. Proceedings of the Clay Mathematics Institute 2001 summer school, Berkeley, CA, USA, June 25--July 27, 2001 , pages 725--746. Providence, RI: American Mathematical Society (A...

2001
[29]

Domains of discontinuity for almost-Fuchsian groups

Andrew Sanders . Domains of discontinuity for almost-Fuchsian groups . Trans. Am. Math. Soc. , 369(2):1291--1308, 2017

2017
[30]

Minimal discs in hyperbolic space bounded by a quasicircle at infinity

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

2016
[31]

The M orse- S ard theorem for real-analytic functions

Ji r \' Sou c ek and Vladim \' r Sou c ek. The M orse- S ard theorem for real-analytic functions. In Theory of nonlinear operators ( P roc. S ummer S chool, B abylon, 1971) , pages 121--125. Academia [Publishing House of the Czechoslovak Academy of Sciences], Prague, 1973

1971
[32]

Sacks and K

J. Sacks and K. Uhlenbeck. Minimal immersions of closed R iemann surfaces. Trans. Amer. Math. Soc. , 271(2):639--652, 1982

1982
[33]

Sullivan

D. Sullivan. Combinatorial invariants of analytic spaces. In Proceedings of L iverpool S ingularities--- S ymposium, I (1969/70) , volume Vol. 192 of Lecture Notes in Math. , pages 165--168. Springer, Berlin-New York, 1971

1969
[34]

Schoen and Shing Tung Yau

R. Schoen and Shing Tung Yau. Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature. Ann. of Math. (2) , 110(1):127--142, 1979

1979
[35]

Minimal surfaces in germs of hyperbolic 3-manifolds

Clifford Henry Taubes. Minimal surfaces in germs of hyperbolic 3-manifolds. In Proceedings of the C asson F est , volume 7 of Geom. Topol. Monogr. , pages 69--100. Geom. Topol. Publ., Coventry, 2004

2004
[36]

uller theory in R iemannian geometry . Lectures in Mathematics ETH Z\

Anthony J. Tromba. Teichm\"uller theory in R iemannian geometry . Lectures in Mathematics ETH Z\"urich. Birkh\"auser Verlag, Basel, 1992. Lecture notes prepared by Jochen Denzler

1992
[37]

Uhlenbeck

Karen K. Uhlenbeck. Closed minimal surfaces in hyperbolic 3 -manifolds. In Seminar on minimal submanifolds , volume 103 of Ann. of Math. Stud. , pages 147--168. Princeton Univ. Press, Princeton, NJ, 1983

1983

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Lars V. Ahlfors. An extension of S chwarz's lemma. Trans. Amer. Math. Soc. , 43(3):359--364, 1938

work page 1938

[2] [2]

Anderson

Michael T. Anderson. Complete minimal varieties in hyperbolic space. Invent. Math. , 69:477--494, 1982

work page 1982

[3] [3]

Anderson

Michael T. Anderson . Complete minimal hypersurfaces in hyperbolic n-manifolds . Comment. Math. Helv. , 58:264--290, 1983

work page 1983

[4] [4]

Contraction of convex hypersurfaces in R iemannian spaces

Ben Andrews. Contraction of convex hypersurfaces in R iemannian spaces. J. Differential Geom. , 39(2):407--431, 1994

work page 1994

[5] [5]

Positively curved surfaces in the three-sphere

Ben Andrews . Positively curved surfaces in the three-sphere . In Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures , pages 221--230. Beijing: Higher Education Press; Singapore: World Scientific/distributor, 2002

work page 2002

[6] [6]

Marques, and Andr\'e Neves

Danny Calegari, Fernando C. Marques, and Andr\'e Neves. Minimal surface entropy of negatively curved manifolds. In Perspectives in scalar curvature. V ol. 2 , pages 417--428. World Sci. Publ., Hackensack, NJ, [2023] 2023

work page 2023

[7] [7]

Quasi- F uchsian manifolds close to the F uchsian locus are foliated by constant mean curvature surfaces

Diptaishik Choudhury, Filippo Mazzoli, and Andrea Seppi. Quasi- F uchsian manifolds close to the F uchsian locus are foliated by constant mean curvature surfaces. Math. Ann. , 388(4):3981--4010, 2024

work page 2024

[8] [8]

Geometric structures for maximal representations and pencils

Colin Davalo. Geometric structures for maximal representations and pencils. Arxiv 2407.01254 , 2024

work page arXiv 2024