{"id":"c7c2d9b1-4027-419a-a0ec-7550616d55b7","arxiv_id":"1908.02535","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New uniform pointwise bounds for harmonic Beltrami differentials yield optimal lower Weil-Petersson curvature bounds and imply that the average total scalar curvature of moduli space is comparable to -g as genus grows.","lead":"Uniform bounds on harmonic Beltrami differentials control the Weil-Petersson curvature of moduli space near the thin part, where surfaces have short closed geodesics. The paper proves the expected -1/systole scaling of Ricci and scalar curvature and shows the average total scalar curvature over large-genus moduli space is comparable to -g.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof gap at intermediate injectivity radii: Proposition 3.3 Part (5) is proved only for ε2'=log(3)/2, while Proposition 1.1 and (4.3) need it up to ε2=sinh^{-1}(1); the gap is fillable via Teo's Lemma 3.1 but unstated.","rationale":"The reader's weakest assumption matches my reading: the local redefinition of ε2 in §3.1 creates a threshold gap between Proposition 3.3 Part (5) and Proposition 1.1. That gap is the most load-bearing issue because it occurs in the exact estimate that the paper says drives the curvature bounds and because (4.3), used in the proof of Theorem 4.2, inherits it. I do not see a fatal flaw: the missing intermediate range can be handled by Teo's Lemma 3.1, since √r C(r) ≤ 0.8091 on [ε2', ε2], so the mathematical conclusions Theorem 1.3 and Theorem 1.7 are likely correct. The paper's numerical monotonicity checks are invoked by plots rather than analytic proof, which slightly weakens rigor but is not the main objection. A revision that disambiguates the two epsilons and adds the one-line argument for ε2' < r ≤ ε2 would resolve the concern. I therefore do not move the reader's conditional verdict; the concern is genuine but repairable.","tokens_in":17102,"tokens_out":27368,"duration_ms":265094,"concrete_test":"Check that M(r)=√r·C(r) ≤ 1 for all r ∈ [ε2', ε2], where ε2'=log(3)/2 and ε2=sinh^{-1}(1), with C(r)=(4π/3(1−sech^6(r/2)))^{-1/2}. If M≤1 on this interval, insert the one-line application of Lemma 3.1 into the proof of Proposition 1.1 and into the derivation of (3.14)/(4.3); this verifies the stated bounds for the full Margulis range. If the check fails, recompute the constants in Theorem 1.3 for the intermediate range.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the unannounced reuse of the symbol ε2. In §3.1 the authors set ε2 = log(3)/2 = sinh^{-1}(1/√3) immediately before Proposition 3.3, and Part (5) of Proposition 3.3 proves ||φ(z)|| ≤ ||φ||_2/√r(z) only for r(z) ≤ ε2' (the smaller constant). Proposition 1.1, however, is stated with the Margulis constant ε2 = sinh^{-1}(1), and its proof in §3.3 invokes Part (5) directly for all inj(z) ≤ ε2. The same gap propagates into (3.13)-(3.14) and then into (4.3), which is the collar bound used in the proof of Theorem 4.2 (= Theorem 1.3), since X3 is defined as collars with inj < ε2 (Margulis). For ε2' < r(z) ≤ ε2 the written proof supplies no bound. This is not a false conclusion: Lemma 3.1 (Teo) covers the missing interval, and √r·C(r) is decreasing with value ≈0.8091 at ε2', so the desired inequality extends by a one-line argument. But as written, the proof of Proposition 1.1 and of the curvature bounds contains a genuine threshold gap; a revision should either rename the two constants or add the intermediate-range argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves uniform pointwise bounds on harmonic Beltrami differentials on finite-area hyperbolic surfaces in terms of the Weil-Petersson norm and the injectivity radius, without dependence on genus or number of cusps (Proposition 1.1). It applies these bounds to show that for surfaces with systole at most 2ε2, the Weil-Petersson Ricci curvature is greater than −4/sys(X), the scalar curvature is greater than −(4/sys(X))(3g−3+n), and uniform bounds hold for directions perpendicular to the span of short geodesic length derivatives. The paper then combines the scalar curvature lower bound with Mirzakhani's integral identity to prove that the average total Weil-Petersson scalar curvature over moduli space is comparable to −g as genus grows.","tokens_in":17305,"tokens_out":6975,"duration_ms":68706,"significance":"If the results stand, they advance the quantitative understanding of Weil-Petersson geometry in the thin part of moduli space: the pointwise bound improves Wolpert's asymptotically optimal estimate by making the constant independent of topology, and the curvature lower bounds have the optimal 1/sys(X) growth as the systole tends to zero, complementing Teo's earlier bounds. The application to total scalar curvature, linking an average of curvature to −g, is a clean and notable consequence of the combination with Mirzakhani's theorem. The paper is self-contained and provides explicit constants; however the proof relies on several numeric inequalities justified by inspection of plots rather than analytic proof, and on a threshold-alignment argument that is not written.","major_comments":[{"comment":"The symbol ε2 is used with two different values: the Margulis constant sinh^{-1}(1) in the Introduction, Proposition 1.1 and Theorem 4.2, and ε2' = log(3)/2 immediately before Proposition 3.3. Proposition 3.3 Part (5) is stated and proved for r(z) ≤ ε2', but Proposition 1.1 invokes Part (5) for all z with inj(z) ≤ ε2 = sinh^{-1}(1). For points with ε2' < inj(z) ≤ ε2, the written proof supplies no bound. The gap is fillable: Lemma 3.1 (Teo) gives ||φ(z)|| ≤ C(r)||φ||_2, and √r C(r) is decreasing with value ≈0.8091 at ε2', so the desired ||φ(z)|| ≤ ||φ||_2/√r holds on the intermediate interval; but as written this argument is absent. The same gap propagates into (3.14) and into the bound (4.3) used in Theorem 4.2. The two constants should be renamed, or the intermediate-range argument inserted.","section":"Sections 3.1–3.3 and Eq. (4.3)"},{"comment":"The inequalities 'C(x)√x is monotonically decreasing with C(ε2)√ε2 = .8091', 'H(r) = G(r)√r is monotonically decreasing', and 'm(r) ≤ m0 = .9137 (see figure 1)' are central to the bound (3.13), hence to Proposition 1.1, Corollary 3.5 and Theorem 4.2. As written they are justified by computation or by a plot rather than by a proof. Because these numeric constants are load-bearing, the authors should either give a short analytic argument for the monotonicity and the maximum value, or provide machine-checkable code and precise definitions of the evaluated quantities.","section":"Section 3.1, proof of Proposition 3.3 Part (5)"},{"comment":"The bound m'(r) ≤ 1.2333 'by computation (see figure 2)' is also used to derive the uniform bound ||µ(z)|| ≤ √2 ||µ||_2. This is another load-bearing numerical assertion that rests on visual inspection; it should be backed by an explicit analytic estimate or a verifiable computation.","section":"Section 3.4, Lemma 3.8"}],"minor_comments":[{"comment":"The line 'Recall that C(ε2) = 1.0917' is inconsistent with the value C(ε2) = 0.7439 used in Lemma 3.4 and with the numerical value needed for the Margulis constant; this appears to be a holdover from the smaller ε2. Please correct.","section":"Section 3.4"},{"comment":"The formula '>−−3×3.3394/...' contains a typo; it should read '> −3×3.3394/...'.","section":"Remark 4.3"},{"comment":"The phrase 'The third named author in [18]' should be 'the second author' or 'Y. Wu', since the paper has two authors.","section":"Section 2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper's main idea and results are valuable and the threshold gap is easily repairable; the main barrier to acceptance is the reliance on unproved computational monotonicity and maximum claims in the proof of the key pointwise estimate. I would be comfortable with acceptance after those points are made rigorous or verifiably checked."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. The paper proves uniform-in-topology pointwise bounds for harmonic Beltrami differentials in terms of injectivity radius, and uses them to get the optimal -1/sys lower bounds for WP Ricci and scalar curvature on the thin part, plus the new result that the average total scalar curvature over Mg is comparable to -g. That last item had been open, and the proof is short and clean once you have the curvature bounds.\n\nThe main technical work is in Section 3. The authors refine Wolpert's collar analysis with explicit constants, improving both Wolpert's surface-dependent threshold and Teo's non-optimal growth rate. The constants are explicit, the argument is mostly self-contained, and the external references used as inputs (Teo's Lemma 3.1, Tromba/Wolpert curvature formulas, Mirzakhani's volume estimate) are appropriate. The one self-citation, to Wolf-Wu, is used only in remarks to show the thin-part assumption cannot be removed. That is honest.\n\nThe soft spots are real but manageable.\n\n- The ε2 notation collision is the most important. The Margulis constant is ε2 = sinh^{-1}(1). But in Section 3.1 the authors redefine ε2 = log(3)/2, a smaller number, and Proposition 3.3 Part (5) proves the bound ||φ(z)|| ≤ ||φ||2/√r(z) only for that smaller threshold. Proposition 1.1 then invokes Part (5) for all inj(z) ≤ ε2 (Margulis). The intermediate range is not covered as written. It is easily covered: Teo's bound plus monotonicity of √r C(r) gives the desired inequality up to the larger constant (√ε2' C(ε2') ≈ 0.809 < 1). But the text does not say this, so the written proof has a genuine threshold gap.\n\n- There is also an internal contradiction about H(r) = G(r)√r. The proof of Part (5) says H is monotonically decreasing; Remark 3.6 says it is monotonically increasing. One of those is a typo, and the actual behavior (increasing from 1/√π) is consistent with the later claim, but as written it is inconsistent.\n\n- A few numerical bounds, such as m(r) ≤ 0.9137 and m'(r) ≤ 1.2333, are justified by plots rather than analytic proofs. That is a defensible choice in a paper like this, but a referee should ask for the verification data or an analytic argument.\n\n- Minor: Lemma 3.8 says C(ε2)=1.0917, which is actually the value at the smaller ε2; the paper refers to a \"third named author\" though there are two; Remark 4.3's bound looks fine.\n\nNone of this undermines the central conclusions. The theorems appear correct, and the main gap is fixed in a few lines. I would send this out for review, with a request to resolve the ε2 conflict and clean up the monotonicity statement. For anyone working on Weil-Petersson geometry, this is a useful paper to have.","headline":"Solid, genuinely new bounds on Weil-Petersson curvature near the boundary of moduli space; the main theorems look right, but the written proof has a fixable threshold gap caused by the reuse of the symbol ε2 for two different constants.","tokens_in":17954,"tokens_out":4641,"would_cite":true,"duration_ms":42065,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["30F60","53C21","32G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any finite-area hyperbolic surface with systole at most $2\\epsilon_2$, the Weil-Petersson Ricci curvature at $X$ exceeds $-4/\\operatorname{sys}(X)$.","keywords":["uniform bounds","harmonic Beltrami differentials","Weil-Petersson curvature","total scalar curvature","moduli space","systole","injectivity radius","hyperbolic surfaces"],"falsifier":"Evaluate $m(r)=\\min(\\sqrt{r}\\,G(r),\\sqrt{r}\\,C(r))$ on the interval $(\\log(3)/2,\\,\\sinh^{-1}(1))$. Proposition 3.3(5) requires the resulting constant to stay at or below $1$ so that $\\|\\varphi(z)\\|\\le \\|\\varphi\\|_2/\\sqrt{r(z)}$; if any $r$ in that interval gives a value above $1$, or if an explicit quadratic differential attains the larger value, the pointwise bound and hence Theorems 1.3 and 1.7 fail. The proof displays the graph only on $(0,\\log(3)/2]$, so a direct computation on the missing interval settles the issue.","tokens_in":16785,"feed_emoji":"📐","tokens_out":16599,"duration_ms":158007,"temperature":0.7,"pith_summary":"This paper proves uniform, topology-independent bounds on how large a harmonic Beltrami differential can be at a point in terms of its Weil-Petersson norm and the local injectivity radius. The key estimate is $|\\mu(z)|^2 \\le \\|\\mu\\|_{WP}^2/\\operatorname{inj}(z) \\le 2\\|\\mu\\|_{WP}^2/\\operatorname{sys}(X)$ wherever the injectivity radius is at most the Margulis constant $\\epsilon_2$. From this it derives lower bounds on Weil-Petersson Ricci and scalar curvature: $\\operatorname{Ric}_{WP}(\\mu)>-4/\\operatorname{sys}(X)$ and $\\operatorname{Sca}_{WP}(X)>-(4/\\operatorname{sys}(X))(3g-3+n)$ for surfaces with short systole. These bounds have the optimal $-1/\\operatorname{sys}$ growth as systole tends to zero. Combined with a known estimate on the moduli-space integral of $1/\\operatorname{sys}$, the paper concludes that the average total Weil-Petersson scalar curvature over moduli space is uniformly comparable to $-g$ as genus grows.","feed_headline":"Ricci curvature bounded below by −4/systole on short-systole surfaces","feed_subtitle":"A uniform bound on harmonic Beltrami differentials gives curvature lower bounds with optimal systole growth.","key_machinery":"The carrying mechanism is a refinement of collar estimates for holomorphic quadratic differentials. Around each short geodesic the lifted differential is split into a Laurent series $\\varphi=\\varphi_-+\\varphi_0+\\varphi_+$; the constant term is controlled by the $L^2$ mass of the collar, while the nonconstant parts satisfy maximum-principle bounds with exponential decay in the injectivity radius. The refined statement, Proposition 3.3, yields a function $G(r)$ whose $\\sqrt{r}\\,G(r)$ stays below about $0.914$ for $r\\le\\epsilon_2$; together with a known pointwise bound for larger radii this gives Proposition 1.1. A separate orthonormal-frame lemma converts the pointwise bound into a uniform upper bound on $\\sum_i |\\mu_i(z)|^2$ over an orthonormal basis, and the curvature formula recalled in Section 2, with the positive self-adjoint operator $D=-2(\\Delta-2)^{-1}$, turns that sum into the Ricci and scalar curvature inequalities.","core_discovery":"The central claim is that the $L^\\infty$ norm of a harmonic Beltrami differential is controlled by its $L^2$ (Weil-Petersson) norm through the local injectivity radius, with constants that do not depend on genus or number of punctures. The sharp form is Proposition 1.1: if $\\operatorname{sys}(X)\\le 2\\epsilon_2$, then every $\\mu$ satisfies $|\\mu(z)|^2\\le \\|\\mu\\|_{WP}^2/\\operatorname{inj}(z)\\le 2\\|\\mu\\|_{WP}^2/\\operatorname{sys}(X)$ at all points with $\\operatorname{inj}(z)\\le\\epsilon_2$. The main application is Theorem 1.3: under the same systole assumption, $\\operatorname{Ric}_{WP}(\\mu)>-4/\\operatorname{sys}(X)$ for unit vectors, and $\\operatorname{Sca}_{WP}(X)>-(4/\\operatorname{sys}(X))(3g-3+n)$. Because this rate matches the known reciprocal-systole curvature blowup near the boundary of moduli space, the bound is optimal in its growth order. The paper also shows Theorem 1.7: the average total Weil-Petersson scalar curvature over moduli space is uniformly comparable to $-g$ as genus goes to infinity.","pith_inferences":["Editorial inference: the threshold mismatch is likely repairable by applying the known pointwise ball estimate directly on the intermediate interval $[\\log(3)/2,\\sinh^{-1}(1)]$; if that repair works, the stated theorems stand, but the printed proof needs the extra line to be complete.","Editorial inference: the same Laurent-splitting collar analysis could be adapted to hyperbolic surfaces with geodesic boundary or cone singularities, where collar widths depend on boundary length, giving sup-norm bounds and curvature consequences for the corresponding moduli spaces; the paper does not pursue this.","Editorial inference: the average-curvature result is an average over moduli space with Weil-Petersson measure; it does not say a typical random surface has scalar curvature comparable to $-g$ pointwise, since the thin part may dominate the average. A finer question left open is whether the comparability has a sharp leading constant.","Editorial inference: numerical constants in the proof (for example $3.3394$ in place of $4$ in the Ricci bound) suggest the universal constants are not optimal; a direct optimization of $H(r)$ and the orthonormal-frame bound could produce tighter curvature constants without changing the argument."],"forward_implications":["The pointwise bound and its constants are independent of $g$ and $n$, so the curvature lower bounds hold uniformly across all finite-area moduli spaces with $3g+n>5$; earlier results of this shape either depended on genus or required the systole to be smaller than a type-dependent constant.","The lower bound $\\operatorname{Ric}_{WP}(\\mu)>-4/\\operatorname{sys}(X)$ has optimal growth: near a short geodesic the holomorphic sectional curvature is known to behave like $-3/(\\pi\\ell_\\alpha)$, so no bound with a slower blowup than $-1/\\operatorname{sys}$ is possible.","The scalar bound of order $-(3g-3+n)/\\operatorname{sys}(X)$ on the thin part upgrades, through integration, to a two-sided comparability $\\int_{\\mathcal M_g}\\operatorname{Sca}_{WP}\\,dX \\asymp -g\\,\\operatorname{Vol}_{WP}(\\mathcal M_g)$ as $g\\to\\infty$.","The short-systole assumption cannot be removed: there exist large-genus surfaces whose injectivity radius grows like $\\log g$, and for them uniform negative upper bounds on sectional curvature block the reciprocal-systole lower bound.","A direct corollary is a constant lower bound on sectional curvature when one direction fixes the lengths of all short geodesics: for $\\mu\\ne 0$ in that perpendicular subspace and any $v$, $K_{WP}(\\mu,v)>-4$."],"supporting_citations":[{"why":"Supplies the pointwise estimate relating the sup norm of a quadratic differential to its $L^2$ norm via the function $C(r)$, quoted as Lemma 3.1 and used throughout the thin-part analysis.","marker":"[10]"},{"why":"Gives the asymptotic $\\|\\varphi(z)\\| \\le (1+\\epsilon)\\sqrt{2/\\pi}\\,\\|\\varphi\\|_2/\\sqrt{\\operatorname{sys}(X)}$ for sufficiently short systoles, the target that this paper makes uniform in genus, and supplies the curvature behavior showing the $-1/\\operatorname{sys}$ rate is optimal.","marker":"[17]"},{"why":"Provides the curvature tensor formula and positivity properties of the operator $D$ that convert pointwise bounds into Ricci and scalar curvature inequalities, and the scalar curvature upper bound used in the average comparison.","marker":"[14]"},{"why":"Supplies the negative curvature and the scalar curvature upper bound on all of moduli space used for the upper side of the total scalar curvature estimate.","marker":"[11]"},{"why":"Establishes the asymptotic comparability of $\\int_{\\mathcal M_g} 1/\\operatorname{sys}(X)\\,dX$ with the Weil-Petersson volume, the input that turns the short-systole curvature bound into the $-g$ average.","marker":"[8]"},{"why":"Provides uniform curvature lower bounds on thick parts and large-systole negative curvature obstructions that show the short-systole assumption in the theorems cannot be dropped.","marker":"[13]"},{"why":"Supplies the Collar Lemma and the injectivity-radius formulas on collar and cusp neighborhoods used to set up the thin-part decomposition.","marker":"[3]"},{"why":"Gives the vanishing of the constant Laurent coefficient for directions perpendicular to short geodesic length functions, used in the proof of the bound for such directions.","marker":"[12]"},{"why":"Supplies the length-function derivative formula that identifies the constant term of the Laurent expansion with the pairing against the gradient of a short geodesic length, used in Lemma 3.8.","marker":"[6]"}],"fun_headline_variants":["Harmonic Beltrami differentials: uniform L∞ bound via systole","Weil-Petersson Ricci curvature: sharp systole blowup","Optimal systole growth for WP curvature bounds","Average WP scalar curvature ~ −g for large genus","Uniform harmonic Beltrami bound yields sharp WP Ricci blowup"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the pointwise inequality $|\\mu(z)|^2\\le \\|\\mu\\|_{WP}^2/\\operatorname{inj}(z)$ for every point of injectivity radius at most $\\epsilon_2$. The written proof of the collar estimate Proposition 3.3(5) is stated only up to a smaller threshold $\\log(3)/2$, while Proposition 1.1 uses the same symbol $\\epsilon_2$ up to $\\sinh^{-1}(1)$; the proof does not explicitly cover the intermediate injectivity radii, so the curvature conclusions rest on a step the text does not fully supply.","fun_headline_variants_meta":{"raw":{"variants":["Harmonic Beltrami differentials: uniform L∞ bound via systole","Weil-Petersson Ricci curvature: sharp systole blowup","Optimal systole growth for WP curvature bounds","Average WP scalar curvature ~ −g for large genus","Uniform harmonic Beltrami bound yields sharp WP Ricci blowup"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000885,"raw_usage":{"total_tokens":3833,"prompt_tokens":970,"completion_tokens":2863,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":2780}},"tokens_in":586,"tokens_out":2863,"duration_ms":20161,"temperature":1.0,"reasoning_tokens":2780,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:41:07.722911+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $m(r)=\\min(\\sqrt{r}\\,G(r),\\sqrt{r}\\,C(r))$ on the interval $(\\log(3)/2,\\,\\sinh^{-1}(1))$. Proposition 3.3(5) requires the resulting constant to stay at or below $1$ so that $\\|\\varphi(z)\\|\\le \\|\\varphi\\|_2/\\sqrt{r(z)}$; if any $r$ in that interval gives a value above $1$, or if an explicit quadratic differential attains the larger value, the pointwise bound and hence Theorems 1.3 and 1.7 fail. The proof displays the graph only on $(0,\\log(3)/2]$, so a direct computation on the missing interval settles the issue.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the pointwise estimate relating the sup norm of a quadratic differential to its $L^2$ norm via the function $C(r)$, quoted as Lemma 3.1 and used throughout the thin-part analysis."},{"cited_title":"Wolpert, Geodesic-length functions and the Weil-Petersson curvature tensor, J","cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic $\\|\\varphi(z)\\| \\le (1+\\epsilon)\\sqrt{2/\\pi}\\,\\|\\varphi\\|_2/\\sqrt{\\operatorname{sys}(X)}$ for sufficiently short systoles, the target that this paper makes uniform in genus, and supplies the curvature behavior showing the $-1/\\operatorname{sys}$ rate is optimal."},{"cited_title":"Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent","cited_arxiv_id":null,"evidence_quote":"Provides the curvature tensor formula and positivity properties of the operator $D$ that convert pointwise bounds into Ricci and scalar curvature inequalities, and the scalar curvature upper bound used in the average comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the negative curvature and the scalar curvature upper bound on all of moduli space used for the upper side of the total scalar curvature estimate."},{"cited_title":"Mirzakhani, Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus","cited_arxiv_id":null,"evidence_quote":"Establishes the asymptotic comparability of $\\int_{\\mathcal M_g} 1/\\operatorname{sys}(X)\\,dX$ with the Weil-Petersson volume, the input that turns the short-systole curvature bound into the $-g$ average."},{"cited_title":"Wolf and Y","cited_arxiv_id":null,"evidence_quote":"Provides uniform curvature lower bounds on thick parts and large-systole negative curvature obstructions that show the short-systole assumption in the theorems cannot be dropped."},{"cited_title":"Buser, Geometry and spectra of compact Riemann surfaces, Modern Birkh¨ auser Classics","cited_arxiv_id":null,"evidence_quote":"Supplies the Collar Lemma and the injectivity-radius formulas on collar and cusp neighborhoods used to set up the thin-part decomposition."},{"cited_title":"Wolf, The Weil-Petersson Hessian of length on Teichm¨ uller space,J","cited_arxiv_id":null,"evidence_quote":"Gives the vanishing of the constant Laurent coefficient for directions perpendicular to short geodesic length functions, used in the proof of the bound for such directions."},{"cited_title":"Imayoshi and M","cited_arxiv_id":null,"evidence_quote":"Supplies the length-function derivative formula that identifies the constant term of the Laurent expansion with the pairing against the gradient of a short geodesic length, used in Lemma 3.8."}],"review_version":1}