{"id":"b31a0bd7-1d99-438a-a4f8-8fddb65aa356","arxiv_id":"2507.02617","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Closed negatively curved manifolds have a Mather set of maximally stretched orbits that is nowhere dense unless the two metrics have proportional marked length spectra, in which case it is the whole unit tangent bundle.","lead":"The authors generalize Thurston's maximal stretch, a tool for comparing two hyperbolic geometries on a surface, to closed manifolds of any dimension with negative curvature. They prove when the most-stretched orbits form a thin set, when they fill the whole manifold, and relate this to best Lipschitz maps and thermodynamic formalism.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Hölder regularity of a_{g1,g2} is asserted without proof; without it, the Livsic and thermodynamic steps behind Theorem 4.9 lose their footing.","rationale":"The reader's weakest assumption is the right one. The central dichotomy (Thm 3.18) and the entropy characterization (Thm 4.9) are the core claims. Thm 3.18's proof is internally consistent given the Aubry/Mather inclusions and continuity of G; the potential issue there is only the union-vs-closure wording, which is fixable. The genuinely load-bearing unproven point is the Hölder regularity of a, used in every analytic step: strong supersolution existence (Prop 3.10 via [LT05]), Livsic (Cor 2.10), equilibrium states and pressure analyticity (Prop 4.5), and the variance-zero characterization (Cor 4.6). Without it, the entropy dichotomy fails; with it, the paper's proofs are plausible and represent a substantial advance. The external transfers ([GK17] in Thm 6.11, [GR24] in App A) are secondary to the central claim and were already flagged by the reader. Therefore I agree with the reader's conditional verdict; no change needed.","tokens_in":65570,"tokens_out":31649,"duration_ms":367205,"concrete_test":"Verify from [Bou95, Section 2.6] and [ST21, Prop 2.13] whether Busemann functions on closed negatively curved manifolds depend Hölder continuously on the boundary point in the C^1 topology (equivalently, whether the strong stable foliation is C^{1,α}). If yes, add a lemma proving a_{g1,g2} is Hölder; the paper's arguments then go through. If only C^0 continuity holds, test Cor 2.10 against a continuous function with zero periodic averages that is not a Livsic coboundary; this would show Theorem 4.9 needs a new proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Remark 2.8(1) asserts Hölder continuity of a_{g1,g2}(v)=g2(B^{g2}(π(v),v_+^{g1}),v) by appealing to the Hölder structure of the boundary. But the boundary Hölder structure directly gives Hölder continuity of the Busemann function b_ξ(x0,x) in ξ, not of its spatial gradient B(x,ξ)=grad_x b_ξ. The paper even states (Section 2.1) that Busemann functions are C^2 convex functions, which is stronger than what is generally available for variable negative curvature; the correct regularity is typically C^{1,α} with Hölder dependence of the gradient on the boundary point. This missing lemma is load-bearing: [LT05] (Prop 3.10) needs a Hölder potential for a strong supersolution; Cor 2.10 uses the Livsic theorem for Hölder functions; Section 4 (Prop 4.5, Cor 4.6, Cor 4.9) uses uniqueness/analyticity of equilibrium states and the variance formula. If a were only continuous, a continuous coboundary obstruction could break Livsic, and the dichotomy h_top(φ,M)=h_top ⇔ proportional spectra would not be established. The text provides no proof or reference for the C^1 regularity of B in ξ.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":65810,"tokens_out":7541,"duration_ms":89061,"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":[{"comment":"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":"Remark 2.8(1); Sections 2.2 and 4"},{"comment":"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.","section":"Section 6.4, Theorem 6.11"}],"minor_comments":[{"comment":"The name 'Bowen–Magulis' should be 'Bowen–Margulis'.","section":"Corollary 4.6"},{"comment":"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.","section":"Section 2.1, after Equation (1)"},{"comment":"The phrase 'from Equation (24)' appears before Equation (24) is introduced; please renumber the displayed equations or rephrase the cross-reference.","section":"Proposition 7.2 proof"},{"comment":"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.","section":"Title page and throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a differential geometry or dynamical systems journal and contains several novel and interesting results. The main risk is that the two gaps identified in the major comments—Hölder regularity of the infinitesimal time change and the transfer of [GK17] to variable curvature—are fixable but require substantial additional work. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nSecond, 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.\n\nThe 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.","headline":"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.","tokens_in":66388,"tokens_out":3130,"would_cite":true,"duration_ms":35954,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D40","37D35","53C24","53D25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["maximal stretch","Mather set","marked length spectrum","negative curvature","geodesic currents","Aubry set","best Lipschitz maps","topological entropy"],"falsifier":"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.","tokens_in":65326,"feed_emoji":"📐","tokens_out":16366,"duration_ms":148032,"temperature":0.7,"pith_summary":"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.","feed_headline":"Maximal stretch is everywhere or a nowhere-dense set","feed_subtitle":"The Mather stretch set is the whole tangent bundle exactly when the two length spectra are proportional.","key_machinery":"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.","core_discovery":"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 $S(g_1,g_2) = L(g_1,g_2)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the maximal stretch for hyperbolic surfaces, proves it equals the least Lipschitz constant and that the attained set is a geodesic lamination; the model theory this paper generalizes.","marker":"[Thu98]"},{"why":"Supplies the entropy inequality and existence of ergodic maximally stretched measures used in Theorem 3.17 and Lemma 3.16, plus the asymmetric metric structure for entropy-normalized metrics.","marker":"[GKL22]"},{"why":"Provides the Hopf parametrization and the Hölder orbit equivalence between geodesic flows that underpin Lemma 2.5, Proposition 2.13 and the measure-current correspondence.","marker":"[ST21]"},{"why":"Gives existence of strong supersolutions for Anosov flows, used to build the Aubry set and to obtain the homothety statement of Theorem 3.14.","marker":"[LT05]"},{"why":"Supplies the analytic pressure formalism and variance formulas used in Proposition 4.5 and Corollary 4.6 to derive the entropy dichotomy for the Mather set.","marker":"[PP90]"},{"why":"Density of periodic orbit measures is used to realize the maximal stretch as a supremum over closed geodesics and to show every invariant measure is maximally stretched in the proportional case.","marker":"[Sig72]"},{"why":"Contributes the stretch-locus theory and the curvature condition $K^-_{g_1}/K^+_{g_2} < L(g_1,g_2)^2$ that guarantee the stretch locus is a maximally stretched geodesic lamination (Theorem 6.11).","marker":"[GK17]"},{"why":"Provides the perturbed surface metrics used in Appendix A to exhibit $S(g_1,g_2) < L(g_1,g_2)$.","marker":"[GR24]"}],"fun_headline_variants":["Dichotomy for Mather set in negative curvature","Maximal stretch set: full or nowhere-dense","Rigid stretch dichotomy on curved manifolds","Length spectra decide stretch set's size","Mather set: all or nothing for stretch"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Dichotomy for Mather set in negative curvature","Maximal stretch set: full or nowhere-dense","Rigid stretch dichotomy on curved manifolds","Length spectra decide stretch set's size","Mather set: all or nothing for stretch"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000207,"raw_usage":{"total_tokens":1475,"prompt_tokens":1095,"completion_tokens":380,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":711,"completion_tokens_details":{"reasoning_tokens":309}},"tokens_in":711,"tokens_out":380,"duration_ms":4135,"temperature":1.0,"reasoning_tokens":309,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:27:19.941080+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}