Maximal stretch and Lipschitz maps on Riemannian manifolds of negative curvature

Gerhard Knieper; Xian Dai

REVIEW 2 major objections 4 minor 48 references

Maximal stretch and Lipschitz maps on Riemannian manifolds of negative curvature

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For pairs of negatively curved metrics, the set of maximally stretched directions is either the entire unit tangent bundle or a nowhere-dense set, with the dichotomy governed by whether the marked length spectra are proportional.

desk verdict A substantial generalization of Thurston's maximal stretch to variable negative curvature, with a sharp nowhere-dense dichotomy and a good counterexample; sound overall, but two load-bearing regularity transfers need proof. read the letter →

arxiv 2507.02617 v2 pith:Z2UFSC2G submitted 2025-07-03 math.DG math.DSmath.GT

Xian Dai , Gerhard Knieper This is my paper · ORCID

classification math.DGmath.DSmath.GT MSC 37D4037D3553C2453D25

keywords maximalstretchMathersetmarkedlengthspectrumnegativecurvaturegeodesiccurrentsAubrybestLipschitzmapstopologicalentropy

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

Thurston's maximal stretch—the largest ratio of closed-geodesic lengths induced by a pair of metrics—describes how one hyperbolic metric distorts another, and the set where it is attained is a geodesic lamination. This paper asks what survives when the surface is replaced by a closed manifold admitting variable negative curvature and the hyperbolic metrics by the space of all negatively curved metrics. The central structural claim is a dichotomy: the Mather set (the closure of the union of supports of all invariant measures that attain the maximal stretch) is nowhere dense precisely when the marked length spectra of the two metrics—the lengths they assign to the closed geodesic in every free homotopy class—are not proportional, and it equals the whole unit tangent bundle when they are. A companion thermodynamic result says the topological entropy of the geodesic flow restricted to the Mather set equals the full entropy exactly in the proportional case. A sympathetic reader should care because this turns a Teichmüller-theoretic rigidity phenomenon into sharp, checkable statements about arbitrary negatively curved metrics, and it links the stretch to best Lipschitz maps through newly defined weighted least Lipschitz constants.

What carries the argument

The load-bearing object is the infinitesimal time change $a_{g_1,g_2}(v) = g_2(B^{g_2}(\pi(v), v_+^{g_1}), v)$, the derivative at zero of the Busemann-cocycle time change that accompanies the Hölder orbit equivalence $\Psi_{g_1,g_2}: S_{g_1}M \to S_{g_2}M$ pairing orbits with the same endpoints at infinity. Its integral along $\varphi^{g_1}$-orbits is the geodesic stretch cocycle, whose growth rate defines the stretch $I_m(g_1,g_2)$ of an invariant measure; the Mather set is the closure of the union of supports of measures attaining the supremum $S(g_1,g_2)$. On the Aubry set the time change is exactly linear with slope $S(g_1,g_2)$, which is what converts the existence of an open set inside the Mather set into proportionality of the marked length spectra. The entropy statements run through the pressure function $P(r a_{g_1,g_2})$, whose high-parameter limit $S(g_1,g_2) = \lim_{r\to\infty} P(r a_{g_1,g_2})/r$ identifies the maximal stretch as a zero-temperature quantity, and whose variance along equilibrium states measures the entropy gap on the Mather set.

What would settle it

Take a pair of negatively curved metrics on a closed surface with provably non-proportional marked length spectra (for instance the conformal pair of Example 6.15) and test the dichotomy directly: if any open subset of $S_{g_1}M$ were contained in the Mather set $M(g_1,g_2)$, Theorem 3.18 would fail, because an open set of the Anosov flow contains a dense orbit and continuity of the orbit equivalence would then force the length multiplier $\ell_{g_2}([\gamma])/\ell_{g_1}([\gamma])$ to be constant. A computable proxy is the variance identity $h_{\mathrm{top}}(\varphi^{g_1}, M(g_1,g_2)) = h_{\mathrm{top}}(\varphi^{g_1}) - \int_0^\infty r\,\mathrm{Var}(P_{m_r}(a_{g_1,g_2}), m_r)\,dr$: if the left side equalled $h_{\mathrm{top}}(\varphi^{g_1})$ while the spectra are non-proportional, the entropy dichotomy of Theorem 4.9 would collapse.

Watch

Extended reading notes

Core claim

The paper establishes, for any two negatively curved metrics $g_1, g_2$ on a closed manifold, a rigid dichotomy for the Mather set $M(g_1,g_2)$: if $\ell_{g_2}([\gamma]) = C\,\ell_{g_1}([\gamma])$ for all free homotopy classes $[\gamma]$, then every invariant measure is maximally stretched and $M(g_1,g_2) = S_{g_1}M$; otherwise $M(g_1,g_2)$ has empty interior. The engine is a Hölder orbit equivalence between the two geodesic flows, built from Busemann functions, which on the Aubry set (the common exact-equality locus of all weak supersolutions, equal to the zero level set of the Peierls barrier) becomes an exact homothety: the time change is $S(g_1,g_2)\,t$ on the nose. From this follows the entropy characterization: the topological entropy of $\varphi^{g_1}$ on the Mather set equals the full topological entropy if and only if the marked length spectra are proportional, with measures of maximal entropy on the Mather set produced as zero-temperature limits of equilibrium states and the entropy gap written as an integral of a variance of the infinitesimal time change. The final pillar connects stretch to Lipschitz geometry: for any maximally stretched measure $m_0$, $S(g_1,g_2) \leq L_{m_0}(g_1,g_2)$, a map attaining equality sends the geodesics in the support of $m_0$ to the corresponding $g_2$-geodesics, and the projection of the Mather set lies in the stretch locus of every best Lipschitz map whenever $S(g_1,g_2) = L(g_1,g_2)$.

Load-bearing premise

Everything downstream rests on the Hölder regularity of the infinitesimal time change $a_{g_1,g_2}$, asserted via the Hölder structure of the boundary of the universal cover: if $a_{g_1,g_2}$ were merely continuous, the Livsic cohomology step, the existence and uniqueness of equilibrium states, the zero-temperature construction, and the shadowing-based Peierls barrier arguments would lose their justification.

Editorial extensions

If this is right

Non-proportional marked length spectra force the maximally stretched part of the flow to have empty interior, so the stretch is realized only along a topologically small set that plays the role of a geodesic lamination without being one.
On the Aubry set the two geodesic flows are homothetic with factor $S(g_1,g_2)$: every periodic orbit there has $g_2$-length exactly $S(g_1,g_2)$ times its $g_1$-length.
The topological entropy of the flow restricted to the Mather set detects proportionality of the marked length spectra—full entropy exactly in the proportional case—and otherwise equals the full entropy minus an explicit variance-integral deficit.
Measures of maximal entropy on the Mather set exist and arise as zero-temperature limits of equilibrium states, with their metric entropies decreasing monotonically to the limiting value.
Whenever $S(g_1,g_2) = L(g_1,g_2)$, the projection of the Mather set lies inside the stretch locus of all best Lipschitz maps, and under the curvature condition $K^-_{g_1}/K^+_{g_2} < L(g_1,g_2)^2$ the stretch locus is a maximally stretched geodesic lamination.

Reading between the lines

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

I read the nowhere-density of the Mather set in the generic case as the correct replacement for Thurston's lamination conclusion in variable curvature: the paper's own example shows this set can carry positive topological entropy while having empty interior, so 'topologically small' rather than 'zero entropy' is the operative generalization, a distinction that likely transfers to other Anosov and
The variance-integral formula for the entropy on the Mather set is a new numerical invariant of the pair $(g_1,g_2)$ that vanishes exactly when the marked length spectra are proportional; it could serve as a practical numerical detector of length-spectrum rigidity on concrete pairs of metrics.
If, as Question D.4 asks, some maximally stretched measure $m_0$ always satisfies $S(g_1,g_2) = L_{m_0}(g_1,g_2)$, the equality between maximal stretch and least Lipschitz constant would hold after averaging over a measure rather than over the whole manifold, suggesting a measure-theoretic route to $S = L$ under hypotheses far weaker than the curvature condition of Theorem 6.11.
The zero-temperature reading of the maximal stretch suggests the whole construction—Mather set, entropy gap, weighted Lipschitz constants—could be transplanted to any setting with a pressure functional and a length ratio, such as Anosov representations or higher-rank symmetric spaces, a direction the paper itself gestures toward.

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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

Bottom line: this is a serious paper with real new results. It generalizes Thurston's maximal stretch to closed negatively curved manifolds of any dimension, and the central dichotomy (Theorem 3.18) is clean: the Mather set is nowhere dense exactly when the marked length spectra are not proportional, and equals the whole unit tangent bundle when they are. The entropy characterization (Theorem 4.9) and the zero-temperature construction of measures of maximal entropy on the Mather set are genuinely new, and Example 6.15 is a nice counterexample showing positive entropy when the stretch locus is not a lamination. The paper is honest about what is new and what is review; Section 7 is explicitly a review of known results, and the open questions in Appendix D are well posed.

The soft spots are real but not fatal. The Holder continuity of the infinitesimal time change a_{g1,g2} is asserted in Remark 2.8(1) with a one-line appeal to the boundary's Holder structure. The paper even states Busemann functions are C^2 convex, which is stronger than what variable negative curvature generally gives. The missing lemma is load-bearing: Livsic, equilibrium states, and the entropy dichotomy all need a Holder potential. I believe the statement is true, but it needs a proof or a precise reference (e.g., regularity of the gradient of Busemann functions in the boundary parameter). This should be fixed before acceptance.

Second, Theorem 6.11 (S=L and stretch locus is a lamination under a curvature condition) is explicitly delegated to [GK17] with "it is not hard to convince oneself" that the proof transfers to variable curvature. That may be true, but it is exactly the kind of claim that should be written out, since the constant-curvature proof uses tools that are delicate in variable curvature. The appendix example A also leans on [GR24], but that is an external construction, which is acceptable.

The entropy machinery in Section 4 is detailed and internally consistent. The self-citations to [Kni95] and [GKL22] are legitimate building blocks, not padding. If the Holder gap is closed, I see no reason the main theorems fail. I would send this to a serious referee, probably one expert in Anosov thermodynamics and one in Lipschitz maps, and ask for the regularity lemma and a written transfer of [GK17] as conditions. The paper deserves referee time, and with those fixes it would be a strong contribution.

Referee Report

2 major / 4 minor

Summary. This paper develops a theory of maximal stretch for pairs of Riemannian metrics of variable negative curvature on a closed manifold, in the spirit of Thurston's work on Teichmüller spaces. The authors introduce the geodesic stretch of an invariant measure or geodesic current, define the maximal stretch S(g1,g2), the Mather set M(g1,g2), and the Aubry set A(g1,g2), and prove a series of structural results: a dichotomy for the Mather set (nowhere dense unless the marked length spectra are proportional, and equal to the full unit tangent bundle in the proportional case, Theorem 3.18); existence of measures of maximal entropy on the Mather set as zero-temperature limits (Theorem 4.3); an entropy dichotomy characterizing proportionality of marked length spectra (Theorem 4.9); a Hölder orbit equivalence with time change linear on the Aubry set (Theorem 3.14); relations between weighted least Lipschitz constants and geodesic stretch (Section 5); and a comparison between the Mather set and the stretch locus of best Lipschitz maps (Proposition 6.9). The paper also contains an explicit example with positive entropy on the Mather set (Example 6.15), an appendix with an example where S(g1,g2)<L(g1,g2), and a survey section on volume inequalities.

Significance. If the two load-bearing gaps identified below are repaired, this is a substantial contribution. The results provide a genuine extension of Thurston's lamination picture to variable negative curvature, with a sharp marked-length-spectrum dichotomy and a new thermodynamic-formalism connection. The paper is largely self-contained: definitions are precise, many long proofs (Theorems 3.18, 4.3, 4.9, 5.13, 5.14, 6.9 and Example 6.15) are written in detail, and there are no fitted parameters or circular assumptions. The explicit example with positive topological entropy on the Mather set is valuable and clarifies the difference from the Teichmüller-space case. However, the asserted Hölder regularity of the infinitesimal time change and the delegated proof of Theorem 6.11 are central enough that the manuscript should not be accepted in its current form.

major comments (2)

[Remark 2.8(1); Sections 2.2 and 4] The assertion that the infinitesimal time change a_{g1,g2}(v)=g2(B^{g2}(π(v),v_+^{g1}),v) is Hölder continuous is load-bearing and is not proved. The Hölder structure of the boundary gives Hölder continuity of the Busemann function b_ξ in the boundary point ξ, but a_{g1,g2} involves the spatial gradient B^{g2}(x,ξ)=grad_x b_ξ, and Hölder regularity of this gradient in ξ does not follow from the cited boundary regularity. This regularity underpins the Livsic theorem application in Corollary 2.10, the uniqueness and analyticity of equilibrium states in Proposition 4.5, and hence the entropy dichotomy Theorem 4.9 and the zero-temperature construction Theorem 4.3. If only continuity is available, the Livsic theorem for Hölder functions cannot be invoked and the stated conclusions do not follow. Please supply a proof, or a precise reference, that (x,ξ)mapsto grad_x b_ξ is uniformly Hölder in ξ, or replace the affected arguments by ones that require only the regularity that is actually established.
[Section 6.4, Theorem 6.11] Theorem 6.11 is advertised as a generalization of [GK17] to variable negative curvature, but the proof consists of saying 'it is not hard to convince oneself' that the arguments of [GK17] transfer, followed by a scaling argument. The local analysis in [GK17, Lemma 5.2] uses the Kirszbraum–Valentine extension theorem and Toponogov comparison in a way that is not automatic for variable curvature; the curvature condition (23) alone may not suffice without checking the relevant comparison estimates pointwise. Since Theorem 6.11 is one of the main advertised results (an open condition under which S(g1,g2)=L(g1,g2) and the stretch locus is a geodesic lamination), this transfer must be written out, or the theorem must be restated with explicit hypotheses and a complete proof.

minor comments (4)

[Corollary 4.6] The name 'Bowen–Magulis' should be 'Bowen–Margulis'.
[Section 2.1, after Equation (1)] The statement that Busemann functions are C^2 convex functions is stronger than the standard regularity for variable negative curvature; please replace it by the regularity actually used or add a reference for the C^{1,α} regularity of Busemann functions.
[Proposition 7.2 proof] The phrase 'from Equation (24)' appears before Equation (24) is introduced; please renumber the displayed equations or rephrase the cross-reference.
[Title page and throughout] There are several typographical issues, including 'NEGA TIVE CUR V A TURE' on the title page, 'Arzel` a-Ascoli' with a misplaced accent, and 'porposition' instead of 'proposition'; these should be corrected in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Mather-set and entropy dichotomies are derived from the definitions via orbit-equivalence and thermodynamic arguments, not assumed as inputs.

full rationale

The paper's central dichotomy (Theorem 3.18) is not circular. S(g1,g2) is defined as a supremum over currents of the length-ratio functional (Def. 3.1 and Def. 3.7), and the theorem is proved by showing that an open Mather set would force the time change of the orbit equivalence in Theorem 3.14 to be exactly S(g1,g2)t on a dense orbit, hence on all of S_{g1}M, which yields proportional marked lengths; the proportional case follows by direct density of periodic orbits. This is a genuine argument from the definitions, not an assumption of the conclusion. The thermodynamic results (Thm 4.3 and Cor 4.9) are derived from the variational principle, Livsic's theorem via Cor 2.10, and standard pressure and variance formulas (Prop 4.5); zero-temperature limits are shown to be maximally stretched and then maximal-entropy on the Mather set. The self-citations [Kni95] and [GKL22] appear as Theorem 3.17 and are used to organize the entropy-ratio dichotomy in Cor 4.6 and Prop 4.7, but that theorem is a stated parameter-free published result with assumptions (g1,g2 in R^-(M)) that do not include the target Mather-set statement; under the given rules it is independent support and does not constitute circularity. The only flagged gap is Remark 2.8(1), which asserts Holder continuity of a_{g1,g2} without proof; this is a regularity and rigor concern that underpins the Livsic and equilibrium-state steps, but it is not a case of a result being equivalent to its input by construction. No fitted quantities, renamed fits, or imported uniqueness claims are used.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central theorems are parameter-free: S(g1,g2), the Mather set and the Aubry set are defined by supremum and intersection operations with no fitted constants. The listed free parameters appear only in the constructed examples (6.15, A, B). The listed axioms are standard background results or explicit external theorems the paper cites; the self-cited Theorem 3.17 [Kni95, GKL22] is highlighted because it is load-bearing for the entropy dichotomy.

free parameters (3)

a = a>1
Conformal factor in Example 6.15: phi=a on the subsurface M1 and 0<phi<a elsewhere, producing S=sqrt(a) and a non-lamination stretch locus.
epsilon_1, epsilon_2 = small positive, epsilon_1<epsilon_2 and epsilon_2-epsilon_1 small
Perturbation parameters in Appendix A inherited from [GR24], used to force S(g1,g2)<L(g1,g2).
s = 0<s<1, small
Scaling parameter in Lemma B.1 and Example B.2 defining the conformal perturbation g_s=kappa_s g_0 with stretch locus equal to a simple closed geodesic.

assumptions (8)

standard math Geodesic flow of a closed negatively curved manifold is topologically transitive and periodic orbits are dense (Sigmund).
Used in Theorem 3.18 to pass from a dense orbit in an open subset of the Mather set to all of S_{g1}M, and in Lemma 3.16 and Proposition 2.20 to approximate invariant measures by periodic orbits.
standard math Equilibrium states exist and are unique for Holder potentials on Anosov flows, and the entropy map is upper semicontinuous (Bowen-Ruelle).
Used throughout Section 4, e.g. Lemma 4.1 (vanishing temperature limit of pressure) and Theorem 4.3 (maximal entropy measures on the Mather set).
standard math Livsic theorem: a Holder cocycle with zero integrals on all periodic orbits is a coboundary.
Used in Corollary 2.10 to connect marked length spectrum equality with time-preserving conjugacy.
domain assumption [LT05, Theorem 1] existence of a Holder sub-action for an Anosov flow when the potential has nonpositive integrals over all periodic orbits.
Used in Proposition 3.10 to obtain a strong supersolution for a_{g1,g2}-S(g1,g2).
domain assumption Theorem 3.17 from [Kni95] and [GKL22]: h_top(g2)/h_top(g1) times S(g1,g2) is at least 1, with equality iff marked length spectra are proportional.
Quoted in Section 3.3.2 and used in Corollary 4.6; it is prior work of the second author and is not reproved here.
domain assumption [GK17, Theorem 5.1 and Lemma 5.2] for hyperbolic spaces: under curvature comparison L>1 and 0>K^- >= K^+, the stretch locus is a maximally stretched geodesic lamination.
Used in Theorem 6.11, whose proof reduces the variable curvature case to [GK17] via scaling.
domain assumption [GR24] existence of a C^2-small perturbation producing a geodesic with one self-intersection and prescribed length differences epsilon_1, epsilon_2.
Basis of Appendix A's example with S(g1,g2)<L(g1,g2); the paper proves the nonexistence of Lip<1 maps using this construction.
standard math Marked length spectrum rigidity for closed negatively curved surfaces (Croke, Otal).
Used in Remark 1.2 and Proposition 7.2 to conclude proportionality or isometry from equality of lengths.

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Pith. "Pith review of Maximal stretch and Lipschitz maps on Riemannian manifolds of negative curvature." pith.science (2026). https://pith.science/paper/Z2UFSC2G

@misc{pith2026250702617,
  author       = {Pith},
  title        = {Pith review of: Maximal stretch and Lipschitz maps on Riemannian manifolds of negative curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z2UFSC2G}},
  note         = {Machine review of arXiv:2507.02617}
}

abstract

In his seminal work on Teichm\"uller spaces (\cite{Th98}), Thurston introduced the maximal stretch for a pair of hyperbolic metrics on a closed surface of genus $\geq 2$ and showed that the logarithm of this quantity induces an asymmetric metric in the Teichm\"uller space. He also showed that the subset of the surface on which the maximal stretch is attained is a geodesic lamination. In this paper, we define the maximal stretch analogously for closed manifolds equipped with Riemannian metrics of variable negative curvature and investigate the structure of the related Mather set on the unit tangent bundle. In contrast to the Teichm\"uller space, the Mather set may not be lifts of geodesic laminations in this broader setting. However, in our paper, we will discuss similar features shared by the Mather set with geodesic laminations. We also connect the study of the Mather set with the theory of best Lipschitz maps.

Figures

Figures reproduced from arXiv: 2507.02617 by the authors.

**Figure 1.** A figure for Property (A). The orbit O is a nonisolated leaf and Pv is a traversal to O. which contradicts the isolated orbit condition. Therefore O must be the orbit of a closed g1-geodesic. On the other hand, if Λ ̸= O, then it leads to a contradiction with the fact that Λ is minimal. Next we assume all orbits of Λ are not isolated. We take one orbit O. Since O is not isolated, there exists a point v in O that sa… view at source ↗

**Figure 2.** (M, g1) is a closed hyperbolic surface of genus 2. The yellow part M1 is a submanifold of (M, g1) bounded by a separating simple closed g1-geodesic γ g1 0 . The generators of the fundamental group π1(M1) is represented by two simple closed g1-geodesics γ g1 1 and γ g1 2 in the figure. We also know that the identity map satisfies Lip(id, g1, g2) = max v∈S g1M ||v||g2 ||v||g1 = max v∈S g1M φ(π(v)) = √ a. As a conseque… view at source ↗

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