The canonical lamination calibrated by a cohomology class

Aidan Backus

REVIEW 3 major objections 5 minor 16 references

The canonical lamination calibrated by a cohomology class

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read Every unit cohomology class yields a canonical lamination

desk verdict Ambitious, mostly coherent construction of the canonical calibrated lamination; the main thing to referee is the unverified use of [Bac24] on the noncompact universal cover. read the letter →

arxiv 2412.00255 v4 pith:RUQXO6UC submitted 2024-11-29 math.DG math.GT

Aidan Backus This is my paper · ORCID

classification math.DGmath.GT MSC 49Q0553C3837F34

keywords laminationsminimalhypersurfacescalibrationsfunctionsofleastgradientstablenormcostableRuelle-Sullivancurrentearthquake

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 proves that on any closed oriented Riemannian manifold of dimension $2$ through $7$, each cohomology class of degree $d-1$ with unit stable dual norm determines a unique largest lamination, a closed union of disjoint minimal hypersurfaces, whose leaves are exactly the hypersurfaces calibrated by every calibration representing the class. This gives a canonical geometric object attached to a purely cohomological datum, analogous to the maximally stretched lamination attached to a best Lipschitz map between hyperbolic surface metrics. The paper then reads geometric information about the stable norm ball, the unit ball of the area-minimizing homology norm, off this lamination: transverse measures on its sublaminations parameterize the dual face of the ball, vertices of maximal flats correspond to closed leaves, and a line segment on the stable unit sphere forces the vanishing of an intersection product. A strict-convexity criterion follows from a statement about the derived series of the fundamental group. The point of caring is that the shape of the stable norm ball is constrained by the topology of $M$ in a way that mirrors the earthquake norm on the tangent space to the space of hyperbolic surface metrics.

What carries the argument

The load-bearing mechanism is the correspondence between functions of least gradient and homologically minimizing laminations. A function $u$ on the universal cover that is equivariant under deck transformations with a homomorphism $\alpha:\pi_1(M)\to\mathbb{R}$ and minimizes total variation has level sets that descend to a measured lamination $\lambda_u$ of minimal hypersurfaces whose Ruelle-Sullivan current equals $du$; this is imported as Theorem 3.7 and holds for $d \leq 7$. Lemma 3.9 selects any $\alpha$ in the dual face $\rho^*$, builds an $\alpha$-equivariant least-gradient function, and converts it into a lamination calibrated by every calibration representing $\rho$, giving the nontriviality input. The technical glue is a package of $L^\infty$ calibration results: the normal trace theorem extends integration of closed $L^\infty$ $(d-1)$-forms to Lipschitz hypersurfaces, the Anzellotti wedge product bounds the pairing between $du$ and a calibration, and the $L^\infty$ Poincar\'e lemma gives continuous potentials so that Stokes' theorem applies; together these let the paper treat calibrations that are only $L^\infty$, not continuous. Curvature bounds for stable minimal hypersurfaces give the compactness that upgrades the set of calibrated hypersurfaces to a Lipschitz lamination.

What would settle it

Take a flat $3$-torus and a unit costable class $\rho$; the theorem predicts the hypersurfaces calibrated by every calibration in $\rho$ are exactly the leaves of one foliation by parallel planes. If one can exhibit two non-parallel minimal hypersurfaces each calibrated by every calibration representing $\rho$, the maximality or uniqueness of the canonical lamination fails.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for every $ ho \in H^{d-1}(M,\mathbb{R})$ with $\|\rho\|_\infty = 1$, there exists a unique largest lamination $\lambda_\rho$ in $M$ such that every calibration $F$ representing $\rho$ calibrates every leaf of $\lambda_\rho$. Here a calibration is a closed $(d-1)$-form of $L^\infty$ comass $1$, and a hypersurface is calibrated if the form restricts to its area form. The lamination is largest in the sense that its leaf set is the intersection of the leaf sets of the individual calibrations' laminations $\lambda_F$, and any hypersurface calibrated by every calibration in the class must be a leaf. The proof constructs for each calibration $F$ the lamination $\lambda_F$ whose leaves are all complete connected $F$-calibrated hypersurfaces, proves these leaves have uniform curvature bounds and are pairwise disjoint, and then takes the intersection over all $F$ in the class. The non-emptiness and calibration-by-every-$F$ property come from a least-gradient function: for any homology class $\alpha$ in the dual face $\rho^*$, an $\alpha$-equivariant function of least gradient produces a measured lamination calibrated by every calibration in $\rho$. As corollaries, the dual face $\rho^*$ is exactly the set of homology classes carried by transverse probability measures on sublaminations of $\lambda_\rho$, every extreme point of $\rho^*$ corresponds to an ergodic measure, rational-direction vertices of maximal flats correspond to closed leaves, and if a maximal flat has too many vertices then the lamination has a spiraling part with no transverse measure.

Load-bearing premise

Everything rests on an imported theorem that every nonzero equivariant function of least gradient has level sets forming a homologically minimizing lamination of minimal hypersurfaces, which is only proved in dimensions at most $7$; if that correspondence fails for a class admitting unit calibrations, the canonical lamination need not exist.

Editorial extensions

If this is right

Every unit costable cohomology class carves out a canonical lamination, so the dual face $\rho^*$ of the stable unit ball is geometrically realized: its points are exactly the homology classes of transverse probability measures on sublaminations of $\lambda_\rho$.
Vertices of maximal flats of the stable unit sphere coincide with closed leaves of $\lambda_\rho$ exactly when they have rational direction, and irrational vertices are limited by $b_1(M)-1$, so the combinatorial shape of stable flats is constrained by the topology of $M$.
If the stable unit ball is strictly convex, every ergodic calibrated lamination is uniquely ergodic; on manifolds with $b_1(M) \geq 2$, all but countably many boundary homology classes are represented by uniquely ergodic calibrated laminations without closed leaves.
A line segment contained in the stable unit sphere forces the intersection product of its endpoints to vanish, and if the quotient of the first two terms of the derived series of $\pi_1(M)$ is a torsion group, the stable unit ball is strictly convex, so every metric on a torus has a strictly convex stable unit ball and many uniquely ergodic laminations of minimal hypersurfaces.
In dimension $2$, the construction recovers a canonical maximally stretched lamination for homotopy classes of maps to the circle, placing the minimal-hypersurface result in the same family as the stretch lamination for best Lipschitz maps between hyperbolic surface metrics.

Reading between the lines

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

A testable extension is to compute $\lambda_\rho$ numerically in low-dimensional manifolds such as flat tori or hyperbolic $3$-manifolds and check that the extreme points of $\rho^*$ are realized by distinct ergodic sublaminations; this would provide explicit examples of the stable unit ball's non-strict convexity.
The $d \leq 7$ restriction likely tracks the regularity theory for least-gradient level sets rather than any essential feature of calibrations, so a natural project is to extend the construction to $d \geq 8$ for classes whose calibrations are continuous or have special structure, using the paper's $L^\infty$ machinery.
The paper's analogy with the earthquake norm suggests that the canonical lamination $\lambda_\rho$ should be characterized as the maximal set on which every optimal Lipschitz representative of $\rho$ is infinitesimally stretched; in dimension $2$ this is exactly what happens, and proving the analogue in higher dimensions would unify the two theories.
Because the proof identifies $\rho^*$ with the set of transverse measures on sublaminations, the non-strict convexity of the stable norm can be probed by searching for two distinct ergodic calibrated sublaminations of $\lambda_\rho$; this gives a geometric, rather than algebraic, way to detect flats in the stable unit ball.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Two things you should know upfront. First, if Theorem 1.2 is correct, this paper closes the Auer–Bangert program and delivers a genuinely useful object: for every unit costable class ρ there is a largest lamination λ_ρ calibrated by every calibration in the class, and the dual flat ρ* is exactly the set of homology classes of transverse measures on sublaminations of λ_ρ. Second, the proof's bridge from least-gradient functions to laminations—Theorem 3.7—is imported from the author's own [Bac24] and applied to the noncompact universal cover without stating or verifying that the cited theorem's hypotheses hold there. That is the load-bearing step, and it is what a referee should check first.

The construction in Section 4 is real work. The curvature bound via Schoen–Simon, the disjointness argument via the maximum principle, and the compactness argument via Vietoris limits are standard tools but assembled in a clean, readable way. Corollary 1.3 and Theorem 1.4 are plausible and mostly well derived. The earthquake-norm analogy is illuminating and not just decoration; it clearly shaped the statements.

The soft spots are local but real. Lemma 5.6's coarea estimate is not justified as written: the volume lower bound is established on a discrete set T and then integrated over full intervals. A referee should ask for a genuine coarea argument or a different proof. Theorem 1.4 states max(0,b_1−1) but the proof gives max(1,b_1−1); the discrepancy only matters when b_1=0, but it is still a mismatch. Lemma 5.4's second clause is mis-stated: as written it says the homology classes are linearly independent and if they span H_{d−1}(M,R) then H_{d−1}(M,R)=0, which cannot be right. Theorem 1.7 is a promissory note to [DU25], and the AB12 lemmas in §5.5 include a 'straightforward generalization' of [BGG69] that is not spelled out. None of these by itself breaks the central claim, but together they make the paper feel less finished than the main theorem deserves.

On the stress-test note: I think the concern lands. The paper does not show that [Bac24, Theorem B] applies on the universal cover, and it does not reconcile the exact form of Theorem 3.7(c) with the cited result. This is not circular—[Bac24] is published and peer-reviewed—so it is a legitimate gap, not a fatal flaw. It may be that a simple equivariant localization argument fills it, but the text needs to say so.

This paper is for people working on stable norms, calibrated geometry, and minimal laminations. It deserves a serious referee. I would send it to review, with a request to verify the [Bac24] bridge first and then clean up the local issues. If the bridge holds, it is a significant paper; if not, it needs substantial revision. Either way, worth the referee time.

Referee Report

3 major / 5 minor

Summary. The paper constructs, for every unit-costable cohomology class ρ in H^{d−1}(M,R) on a closed oriented Riemannian d-manifold with 2≤d≤7, a canonical lamination λ_ρ whose leaves are exactly the complete immersed hypersurfaces calibrated by every calibration representing ρ. The construction first builds, for each calibration F in ρ, a lamination λ_F of F-calibrated minimal hypersurfaces, using α-equivariant functions of least gradient on the universal cover and a theorem (Theorem 3.7) that turns their level sets into homologically minimizing laminations. The canonical lamination is then the largest lamination contained in every λ_F. The paper derives structural consequences for the stable unit ball: the dual flat ρ* is exactly the set of homology classes of transverse probability measures on sublaminations of λ_ρ (Corollary 1.3); maximal flats are polytopes whose rational vertices correspond to closed leaves (Theorem 1.4); strict convexity of the stable ball follows from a condition on the derived series of π_1(M), and a line segment in the boundary of the stable ball forces vanishing intersection product (Theorem 1.6). The final section develops an analogy with the earthquake norm on Teichmüller space.

Significance. If the main theorem is correct, the paper provides a canonical lamination associated to every costable unit class, generalizing the Thurston–Guéritaud–Kassel maximally stretched lamination to calibrated hypersurfaces in higher dimensions. The structural corollaries—the polytope description of flats, the closed-leaf characterization of rational vertices, and the strict-convexity criteria—are concrete, falsifiable statements about the stable norm ball and go substantially beyond what was previously proved. The construction is parameter-free and I do not see a circularity: λ_ρ is built from calibrations and least-gradient functions, and the later theorems are derived from the structure of λ_ρ. However, the proof depends on a load-bearing import ([Bac24, Theorem B] via Theorem 3.7) and on a questionable level-set identity in the proof of Theorem 1.6(2), so the paper is not yet in final form.

major comments (3)

[§3.2 (Theorem 3.7)] Theorem 3.7 is the bridge from least-gradient functions to calibrated laminations: Lemma 3.9 uses it to produce the nontriviality input for the construction of λ_ρ in §4. The proof applies [Bac24, Theorem B] to the noncompact universal cover \tilde M, but the manuscript does not state the hypotheses of that theorem or verify that \tilde M satisfies them. If [Bac24, Theorem B] is proved only for closed manifolds, its hypotheses are not satisfied by \tilde M, and no reduction by deck-group invariance or by passing to a closed quotient is given. Since the nontriviality condition (1) in §4 and therefore Theorem 1.2 collapse without this step, please quote the exact theorem used, show that it applies to \tilde M, or supply the missing argument.
[§5.5 (proof of Theorem 1.6(2))] The proof asserts that for u := (u_α+u_β)/2 one has ∂{u>0} = ∂{u_α>0} ∪ ∂{u_β>0}. This equality is not a consequence of the definitions and is generally false: the support of the average Ruelle–Sullivan current is the union of the two supports, but the topological boundary of a superlevel set of the averaged function need not have both leaves as boundary components. Since the disconnectedness of ∂{u>0} is then fed into Lemma 5.10, the proof of Theorem 1.6(2) is incomplete as written. A corrected argument is needed, for example by choosing the constants or orientations so that the two leaves are boundary components of a single level set, or by working directly with the jump set rather than the superlevel boundary.
[§5.3 (proof of Theorem 1.4)] The step 'The infinite sequence (α_n) is linearly dependent, so M \setminus ⋃N_n must be disconnected; therefore there can be no leaf of λ_ρ which is dense in M' is too compressed. Linear dependence of the homology classes of infinitely many closed leaves does not by itself imply that their union disconnects M, and this implication is used before applying the Morgan–Shelan decomposition. Since Theorem 1.4 is a central structural claim, this topological step should be justified or replaced by a direct argument.

minor comments (5)

[Abstract and §1] There are duplicated words ('the the geometry' in the abstract, 'a an affine map' in §5.1) and the title contains an errant space ('LAMINA TION'); these should be corrected.
[§3.2 (Theorem 3.7)] For self-containedness, the exact statement of [Bac24, Theorem B] that is being invoked should be quoted, especially because Theorem 3.7 is presented as an equivalence but the proof refers to the cited theorem for both directions.
[§5.2 (Lemma 5.6)] In the proof of Lemma 5.6, the sentence 'The image of T in S^1 is a point' is not explained; the intended reason is presumably that the sheets in the ball are lifts of the same leaf and the equivariant function changes by integers under deck transformations, but this should be stated explicitly.
[§5.3 (Theorem 1.4 proof)] The proof writes m ≤ max(1,b_1−1) while the theorem states max(0,b_1−1); the discrepancy should be resolved and the bookkeeping for b_1=0 clarified.
[§5.5] Lemmas 5.9 and 5.10 are attributed to the unpublished manuscript [AB12]; the paper reproduces them with credit, but it would help the reader if the manuscript stated which parts of those lemmas are being reused verbatim and which are modified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the canonical lamination is constructed from calibrations and least-gradient functions, with no fitted input relabeled as a prediction.

full rationale

The derivation is not circular. Theorem 1.2 is proved by fixing a calibration F in the cohomology class ρ, showing in Lemma 4.6 that the set of all F-calibrated complete immersed hypersurfaces forms a lamination λF, and then intersecting these laminations over all calibrations F in ρ via Lemma 2.11. The nontriviality input is Lemma 3.9, which chooses α in the dual flat ρ*, takes an α-equivariant least-gradient function u (Lemma 3.6), and converts u into a measured lamination using Theorem 3.7. Theorem 3.7 is a restatement of the author's earlier theorem [Bac24, Theorem B] rather than a consequence of the paper's target results; it is a prior, parameter-free structural statement about least-gradient functions and level-set laminations, with stated assumptions (d ≤ 7, equivariant BV function of least gradient) that do not include the existence or uniqueness of λρ. The use of the author's own earlier theorem is therefore a citation of independent supporting work, not a circular reduction: there is no fitted parameter renamed as a prediction, no quantity defined in terms of the claimed output, and no equivalence between Theorem 1.2 and its inputs by construction. Corollary 1.3, Theorem 1.4, Corollary 1.5, and Theorem 1.6 are subsequently derived from the structure of the constructed lamination rather than assumed. The skeptic's concern about applying [Bac24, Theorem B] to the noncompact universal cover is a question about whether the cited theorem's hypotheses are satisfied, which is a correctness risk, not a circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

The central claim rests on five imported pillars: the least-gradient-to-lamination correspondence and lamination compactness theorem of the author's [Bac24]; Federer's stable-norm duality; Anzellotti's calculus for L∞ forms; the Morgan-Shalen decomposition with the Arnoux-Levitt lemma; and, for Theorem 1.6(2), two lemmas from the unpublished Auer-Bangert manuscript [AB12]. No free parameters are fitted anywhere. The single invented entity, the canonical lamination λ_ρ, is constructed rather than postulated, and the paper proves falsifiable structural consequences for it.

assumptions (7)

domain assumption Equivariant functions of least gradient exist for every homology class, and their level sets form homologically minimizing laminations with Ruelle-Sullivan current du (Theorem 3.7, from [Bac24, Theorem B]).
Load-bearing bridge from calibrations to laminations: Lemma 3.9 and the proof of Theorem 1.2 convert an α-equivariant least-gradient function into a measured lamination calibrated by every calibration in ρ. Imported from the author's published [Bac24], not re-proved here, and it carries the d ≤ 7 restriction.
domain assumption Disjoint minimal hypersurfaces with uniformly bounded second fundamental forms and closed union form a Lipschitz lamination ([Bac24, Theorem A]); stable minimal hypersurfaces in dimension d ≤ 7 satisfy Schoen-Simon curvature estimates [SS81].
Used in Lemmas 4.3 and 4.6 to show the set of F-calibrated hypersurfaces is the leaf set of the lamination λ_F. This is the source of the dimension restriction 2 ≤ d ≤ 7.
standard math Federer's duality: the costable norm equals the minimal comass, attained by an L∞ closed form (Theorem 2.6, [Fed74]).
Guarantees calibrations in a unit costable class exist, so the family of laminations intersected in Theorem 1.2 is nonempty; also used in Lemma 3.8 and Section 3.3.
standard math Anzellotti's trace and wedge-product theorems, the L∞ Poincaré lemma, and the normal trace theorem (Theorems 2.1, 2.2, 2.4, from [Anz83] and [CM10]).
Technical foundation allowing L∞ calibrations to be integrated along Lipschitz hypersurfaces; needed for the very definition of calibrated laminations and for Lemma 3.3.
standard math Morgan-Shalen decomposition of measured oriented laminations (Theorem 2.12, [MS88]) and the Arnoux-Levitt linear-independence lemma (Lemma 5.4, [AL86]).
Used in Theorem 5.5 (extreme points of M(λ) are exactly the ergodic measures) and in Theorem 1.4 to bound the number of irrational-direction vertices of a maximal flat.
ad hoc to paper For an equivariant least-gradient function u, the superlevel set {u^{ab} > t} on the universal abelian cover is connected (Lemma 5.9), and a disconnected boundary ∂{u^{ab} > t} missing a set of spanning curves gives a nontrivial class in H^1(M^{ab}, R) (Lemma 5.10).
Technical core of Theorem 1.6(2) (strict convexity of the stable ball). Attributed to the unpublished Auer-Bangert manuscript [AB12]; the author reproduces the proofs 'with full credit', but the original is not publicly checkable.
domain assumption Every p-harmonic representative F_p of ρ converges along a subsequence to a least-gradient function u whose transverse measure lies in M(λ_ρ) (Theorem 1.7, attributed to [DU25]).
Stated in Section 1 with proof omitted and credited to in-preparation work; not used in the proofs of Theorems 1.2-1.6, but presented as a result of the paper.

invented entities (1)

canonical calibrated lamination λ_ρ independent evidence
purpose: Unique largest lamination whose leaves are calibrated by every calibration representing the unit costable class ρ; the central object used to derive Corollary 1.3 and Theorems 1.4-1.6.
Not an ad hoc postulate: λ_ρ is explicitly constructed in Section 4 as the intersection (Lemma 2.11) of the leaf sets of the λ_F, and its structural properties (dual flat ρ*, polytope vertices, strict convexity consequences) are proven, giving checkable consequences a counterexample could refute.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The canonical lamination calibrated by a cohomology class." pith.science (2026). https://pith.science/paper/RUQXO6UC

@misc{pith2026241200255,
  author       = {Pith},
  title        = {Pith review of: The canonical lamination calibrated by a cohomology class},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RUQXO6UC}},
  note         = {Machine review of arXiv:2412.00255}
}

abstract

Let $M$ be a closed oriented Riemannian manifold of dimension $2 \leq d \leq 7$, and let $\rho \in H^{d - 1}(M, \mathbb R)$ have unit norm. We construct a lamination $\lambda_\rho$ whose leaves are exactly the minimal hypersurfaces which are calibrated by every calibration in $\rho$. The geometry of $\lambda_\rho$ is closely related to the the geometry of the unit ball of the stable norm on $H_{d - 1}(M, \mathbb R)$, and so we deduce several results constraining the geometry of the stable norm ball in terms of the topology of $M$. These results establish a close analogy between the stable norm on $H_{d - 1}(M, \mathbb R)$ and the earthquake norm on the tangent space to Teichm\"uller space.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 10 canonical work pages

[1]

Minimising currents and the s table norm in codimension one

[AB01] F. Auer and V. Bangert. “Minimising currents and the s table norm in codimension one”. Comptes Ren- dus de l’Acad´ emie des Sciences - Series I - Mathematics 333.12 (2001), 1095–1100. issn: 0764-4442. doi: https://doi.org/10.1016/S0764-4442(01)02188-7 . url: https://www.sciencedirect.com/science/article/pii/S0764444 [AB12] F. Auer and V. Bangert. “...

work page doi:10.1016/s0764-4442(01)02188-7 2001
[6]

Transverse mea sures and best Lipschitz and least gradient maps

arXiv: 2410.08296 [math.DG] . url: https://arxiv.org/abs/2410.08296. [DU24b] G. Daskalopoulos and K. Uhlenbeck. “Transverse mea sures and best Lipschitz and least gradient maps”. Journal of Diﬀerential Geometry 127.3 (2024), 969 –1018. doi: 10.4310/jdg/1721071495. url: https://doi.org/10.4310/jdg/1 [DU25] G. Daskalopoulos and K. Uhlenbeck. Best Lipschitz ...

arXiv 2024
[10]

Lp regularity of least gradient functions

isbn: 9783642617980. url: https://books.google.com/books?id=l9L6CAAAQBAJ. [G´ o20] W. G´ orny. “ Lp regularity of least gradient functions”. Proc. Amer. Math. Soc. 148.7 (2020), 3009–3019. issn: 0002-9939,1088-6826. doi: 10.1090/proc/15031. url: https://doi.org/10.1090/proc/15031. [HL82] R. Harvey and H. B. Lawson. “Calibrated geometries”. Acta Mathematic...

arXiv 2020
[11]

The earthquake metric on Teichm{\"u}ller space

arXiv: 2404.19515 [math.GT] . url: https://arxiv.org/abs/2404.19515. [Kec12] A. Kechris. Classical Descriptive Set Theory . Graduate Texts in Mathematics. Springer New York,

work page Pith review arXiv
[12]

The Nielsen Realization Problem

isbn: 9781461241904. url: https://books.google.com/books?id=WR3SBwAAQBAJ. [Ker83] S. P. Kerckhoﬀ. “The Nielsen Realization Problem”. Annals of Mathematics 117.2 (1983), 235–265. issn: 0003486X. url: http://www.jstor.org/stable/2007076 (visited on 05/27/2024). [Liu23] Z. Liu. Homologically area-minimizing surfaces that cannot be calibrated

work page arXiv 1983
[309]

Calibrations and laminations

issn: 1559-002X. doi: 10.1007/s12220-024-01758-8 . url: https://doi.org/10.1007/s122 [BC17] V. Bangert and X. Cui. “Calibrations and laminations ”. Mathematical Proceedings of the Cambridge Philosophical Society 162.1 (2017), 151–171. doi: 10.1017/S0305004116000475. [BD17] J. F. Brock and N. M. Dunﬁeld. “Norms on the cohomology of hyperbolic 3-manifolds”....

work page doi:10.1007/s12220-024-01758-8 2017
[1984]

Maximally stretched la minations on geometrically ﬁnite hyperbolic mani- folds

isbn: 9780817631536. url: https://books.google.com/books?id=dNgsmArDoeQC. [GK17] F. Gu´ eritaud and F. Kassel. “Maximally stretched la minations on geometrically ﬁnite hyperbolic mani- folds”. Geometry and Topology 21.2 (2017), 693–840. doi: 10.2140/gt.2017.21.693. url: https://doi.org/10.2140%2Fgt.201 [Gro07] M. Gromov. Metric Structures for Riemannian a...

doi:10.2140/gt.2017.21.693 2017
[1996]

Stable Norms of Surfaces: Local Struct ure of the Unit Ball at Rational Directions

url: https://theses.hal.science/tel-00589624. [Mas97] D. Massart. “Stable Norms of Surfaces: Local Struct ure of the Unit Ball at Rational Directions”. Geomet- ric & Functional Analysis GAF A7 (1997), 996–1010. url: https://api.semanticscholar.org/CorpusID:119954418. 26 REFERENCES [MRL14] J. M. Maz´ on, J. D. Rossi, and S. S. de Le´ on. “Functio ns of Lea...

arXiv 1997

Show all 16 references

[1998]

The Fenchel-Nielsen Deformation

doi: 10.48550/ARXIV.MATH/9801039. url: https://arxiv.org/abs/math/9801039. [Wol82] S. Wolpert. “The Fenchel-Nielsen Deformation”. Annals of Mathematics 115.3 (1982), 501–528. issn: 0003486X, 19398980. url: http://www.jstor.org/stable/2007011 (visited on 10/28/2024). Department...

1982 doi
[2010]

Real Flat Chains, Cochains and Variati onal Problems

isbn: 9780857290212. url: https://books.google.com/books?id=PiDET2fS7H4C. [Fed74] H. Federer. “Real Flat Chains, Cochains and Variati onal Problems”. Indiana University Mathematics Journal 24.4 (1974), 351–407. issn: 00222518, 19435258. url: http://www.jstor.org/stable/2489082...

1974
[2011]

A Visit with the ∞ -Laplace Equation

isbn: 9780821853238. url: https://books.google.com/books?id=k4ODAwAAQBAJ. [Cra08] M. G. Crandall. “A Visit with the ∞ -Laplace Equation”. Calculus of Variations and Nonlinear Par- tial Diﬀerential Equations: With a historical overview by El vira Mascolo . Ed. by B. Dacorogna a...

2008 doi
[2012]

Sur l’unique ergodicit´ e de s 1-formes ferm´ ees singuli` eres

[AL86] P. Arnoux and G. Levitt. “Sur l’unique ergodicit´ e de s 1-formes ferm´ ees singuli` eres.”Inventiones mathe- maticae 84 (1986), 141–156. url: http://eudml.org/doc/143337. [Anz83] G. Anzellotti. “Pairings between measures and boun ded functions and compensated compactne...

1986 doi
[2015]

C 1,α -regularity for inﬁnity harmonic functions on two dimensio ns

isbn: 9781482242393. url: https://books.google.com/books?id=e3R3CAAAQBAJ. [ES08] L. Evans and O. Savin. “ C 1,α -regularity for inﬁnity harmonic functions on two dimensio ns”. Calculus of Variations 32 (2008), 325–347. [EW10] M. Einsiedler and T. Ward. Ergodic Theory: with a v...

2008
[2022]

Currents, ﬂows and diﬀeom orphisms

arXiv: 2206.01371 [math.GT] . url: https://arxiv.org/abs/2206.01371. [RS75] D. Ruelle and D. Sullivan. “Currents, ﬂows and diﬀeom orphisms”. Topology 14.4 (1975), 319–327. issn: 0040-9383. doi: https://doi.org/10.1016/0040-9383(75)90016-6 . url: https://www.sciencedirect.com/s...

1975
[2023]

Normes stables des surfaces

arXiv: 2310.19860 [math.DG] . [Mar] F. C. Marques. Surfaces minimales: Th´ eorie variationelle et applications. [Mas96] D. Massart. “Normes stables des surfaces”. Theses. Ecole normale sup´ erieure de lyon - ENS LYON, June

arXiv
[2024]

Geodesics with bounded in tersection number on surfaces are sparsely dis- tributed

arXiv: 2403.07702 [math.DG] . [BS85] J. S. Birman and C. Series. “Geodesics with bounded in tersection number on surfaces are sparsely dis- tributed”. Topology 24.2 (1985), 217–225. issn: 0040-9383. doi: https://doi.org/10.1016/0040-9383(85)90056-4 . url: https://www.sciencedi...

1985 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Minimising currents and the s table norm in codimension one

[AB01] F. Auer and V. Bangert. “Minimising currents and the s table norm in codimension one”. Comptes Ren- dus de l’Acad´ emie des Sciences - Series I - Mathematics 333.12 (2001), 1095–1100. issn: 0764-4442. doi: https://doi.org/10.1016/S0764-4442(01)02188-7 . url: https://www.sciencedirect.com/science/article/pii/S0764444 [AB12] F. Auer and V. Bangert. “...

work page doi:10.1016/s0764-4442(01)02188-7 2001

[2] [6]

Transverse mea sures and best Lipschitz and least gradient maps

arXiv: 2410.08296 [math.DG] . url: https://arxiv.org/abs/2410.08296. [DU24b] G. Daskalopoulos and K. Uhlenbeck. “Transverse mea sures and best Lipschitz and least gradient maps”. Journal of Diﬀerential Geometry 127.3 (2024), 969 –1018. doi: 10.4310/jdg/1721071495. url: https://doi.org/10.4310/jdg/1 [DU25] G. Daskalopoulos and K. Uhlenbeck. Best Lipschitz ...

arXiv 2024

[3] [10]

Lp regularity of least gradient functions

isbn: 9783642617980. url: https://books.google.com/books?id=l9L6CAAAQBAJ. [G´ o20] W. G´ orny. “ Lp regularity of least gradient functions”. Proc. Amer. Math. Soc. 148.7 (2020), 3009–3019. issn: 0002-9939,1088-6826. doi: 10.1090/proc/15031. url: https://doi.org/10.1090/proc/15031. [HL82] R. Harvey and H. B. Lawson. “Calibrated geometries”. Acta Mathematic...

arXiv 2020

[4] [11]

The earthquake metric on Teichm{\"u}ller space

arXiv: 2404.19515 [math.GT] . url: https://arxiv.org/abs/2404.19515. [Kec12] A. Kechris. Classical Descriptive Set Theory . Graduate Texts in Mathematics. Springer New York,

work page Pith review arXiv

[5] [12]

The Nielsen Realization Problem

isbn: 9781461241904. url: https://books.google.com/books?id=WR3SBwAAQBAJ. [Ker83] S. P. Kerckhoﬀ. “The Nielsen Realization Problem”. Annals of Mathematics 117.2 (1983), 235–265. issn: 0003486X. url: http://www.jstor.org/stable/2007076 (visited on 05/27/2024). [Liu23] Z. Liu. Homologically area-minimizing surfaces that cannot be calibrated

work page arXiv 1983

[6] [309]

Calibrations and laminations

issn: 1559-002X. doi: 10.1007/s12220-024-01758-8 . url: https://doi.org/10.1007/s122 [BC17] V. Bangert and X. Cui. “Calibrations and laminations ”. Mathematical Proceedings of the Cambridge Philosophical Society 162.1 (2017), 151–171. doi: 10.1017/S0305004116000475. [BD17] J. F. Brock and N. M. Dunﬁeld. “Norms on the cohomology of hyperbolic 3-manifolds”....

work page doi:10.1007/s12220-024-01758-8 2017

[7] [1984]

Maximally stretched la minations on geometrically ﬁnite hyperbolic mani- folds

isbn: 9780817631536. url: https://books.google.com/books?id=dNgsmArDoeQC. [GK17] F. Gu´ eritaud and F. Kassel. “Maximally stretched la minations on geometrically ﬁnite hyperbolic mani- folds”. Geometry and Topology 21.2 (2017), 693–840. doi: 10.2140/gt.2017.21.693. url: https://doi.org/10.2140%2Fgt.201 [Gro07] M. Gromov. Metric Structures for Riemannian a...

doi:10.2140/gt.2017.21.693 2017

[8] [1996]

Stable Norms of Surfaces: Local Struct ure of the Unit Ball at Rational Directions

url: https://theses.hal.science/tel-00589624. [Mas97] D. Massart. “Stable Norms of Surfaces: Local Struct ure of the Unit Ball at Rational Directions”. Geomet- ric & Functional Analysis GAF A7 (1997), 996–1010. url: https://api.semanticscholar.org/CorpusID:119954418. 26 REFERENCES [MRL14] J. M. Maz´ on, J. D. Rossi, and S. S. de Le´ on. “Functio ns of Lea...

arXiv 1997