{"id":"513be887-1a48-49b5-8e30-1873daeb092e","arxiv_id":"2412.12479","paper_version":6,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"If X×S¹ admits a PSC metric whose circle factor is at angle < 45° to the X-slice, then X itself admits a PSC metric, for any closed oriented X of dimension at least two.","lead":"This math paper proves a restricted version of a broad conjecture in geometry: if a space times a circle has a positive-curvature metric, and the circle is not tilted too much relative to the space, then the space itself also has such a metric. The proof introduces an extra dimension to turn a difficult conformal problem into an elliptic equation, and it yields a new restriction on known four-dimensional counterexamples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.3's scaling estimate drops a factor in (22), but the corrected bound is still repairable; the conditional verdict stands.","rationale":"The reader's weakest assumption identifies the same load-bearing step: the smallness of ∂²u/∂t² in Lemma 2.3, specifically the scaling estimate between (21) and (22). My independent check confirms a concrete defect there: the θ-averaged Yamabe step in (22) omits the factor (2π/ϵ)^{2/(n+1)} that arises from the length of the scaled S1 factor. This is a real gap in the written proof. The reason the verdict should not change is that the defect is quantitative, not structural. Recomputing the final estimate with the missing factor gives ∂²u/∂t² bounded by a constant times η plus a constant times (C+1)ϵ^{1/2}; both can be made as small as desired under the freedom to shrink η and ϵ in Proposition 2.1 and Lemma 2.3. I also checked the Gauss-Codazzi and conformal transformation algebra leading to (36)-(39); the signs and coefficients are consistent, with the caveat that (38) appears to contain a typo where the Hessian 4∇V∇V u is printed as the square 4∇V u∇V u, but (39) reverts to the Hessian and the use of equation (24) is then correct. Overall, the central claim is supported by the intended argument, but the scaling proof in Lemma 2.3 needs a careful rewrite before the paper can be accepted as written.","tokens_in":15218,"tokens_out":55932,"duration_ms":474898,"concrete_test":"Independently re-derive the chain from (21) to (22) with the explicit convention θ'=ϵ^{-1}θ, so ∫_{S1}dθ'=2π/ϵ, and apply the relative-Yamabe inequality to \\bar u=u∘π. Track the factor L^{1-2/q}=(2π/ϵ)^{2/(n+1)} exactly, propagate it through (19), (23), and the final conversion from ∂²u/∂(t')² to ∂²u/∂t². If the resulting bound is C0 η + C1(C+1)ϵ^{1/2}, Lemma 2.3 holds with only a minor revision; if any negative power of ϵ survives in the coefficient of η or (C+1), the theorem's PSC conclusion is not established by the present proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equations (21)-(22) of Lemma 2.3 contain a dimensionally inconsistent step. With θ'=ϵ^{-1}θ, the scaled circle has length L=2π/ϵ, so the relative-Yamabe inequality applied to the θ-independent function \\bar u=u∘π gives, after converting to W, an extra factor L^{1-2/q}=(2π/ϵ)^{2/(n+1)} in the bound for ∥u∥_{L^q(W,g')}. This factor is absent from the final equality in (22). This is exactly the fragile comparison the reader flagged, and the treatment of the factor ϵ^{(n-1)/(2(n+1))} between (21) and (22) is not consistent with the measure ∫_{S1}dθ'=2π/ϵ. However, inserting the missing factor and propagating it through (19), (23), and the final conversion back to ∂²u/∂t² yields a bound of the form C0 η + C1(C+1)ϵ^{1/2}, which can still be made smaller than any η' by first fixing η and then shrinking ϵ. Thus Lemma 2.3's conclusion survives the correction; the proof requires a revised scaling argument rather than a new idea. The subsequent positivity estimate in (38)-(39) is algebraically consistent once the printed '4∇V u∇V u' in (38) is read as the Hessian 4∇V∇V u, as in (39).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a proof of the S^1-stability conjecture for positive scalar curvature under an additional geometric hypothesis: for a fixed P in S^1, the angle between the unit normal to the slice X x {P} and the circle direction is < pi/4 (Theorem 3.1). The strategy is to extend the metric on X x S^1 to a product metric on W x S^1 with W = X x [-1,1], to solve an elliptic PDE with a carefully chosen inhomogeneous term F concentrated near X x {0}, and then to use two Gauss-Codazzi steps together with conformal changes to show that the induced metric on X has positive scalar curvature. The main technical work is in Section 2: ellipticity of the operator L', existence of a C^{1,alpha}-small solution of the Dirichlet problem, a partial C^2-estimate for the solution in the t-direction, and comparison formulas for Laplacians and conformal factors. Section 3 assembles these estimates into the proof of Theorem 3.1 and derives two corollaries about Yamabe invariants and obstructions for counterexamples.","tokens_in":15534,"tokens_out":5509,"duration_ms":55838,"significance":"If the proof is correct, the paper establishes a genuinely new conditional result: the first PDE-based proof that a PSC metric on X x S^1 satisfying a local angle bound forces a PSC metric on X. The introduction of the auxiliary interval W and the use of the relative Yamabe invariant to control the t-derivative are plausible and potentially reusable ideas. The claimed theorem is not circular: the constants in Proposition 2.1 are chosen to satisfy inequalities, not to force the conclusion, and the main theorem follows from the constructed solution of the elliptic PDE. A notable strength is that the angle condition is explicit and falsifiable, leading to the clean obstruction statement in Corollary 3.1. However, the proof as printed is not fully sound: the scaling argument in Lemma 2.3 contains a missing factor, and several notational inconsistencies make the estimates hard to verify. These issues are local and repairable, so the central strategy is defensible.","major_comments":[{"comment":"The scaling argument with theta' = epsilon^{-1} theta drops a factor. After this change of variables one has \\int_{S^1} d\\theta' = 2\\pi/\\epsilon, so converting the M,g' norm in (20)-(21) back to W,sigma*g' produces an additional factor (2\\pi/\\epsilon)^{1/q} with q = 2(n+1)/(n-1). This factor is not present in the equality at the end of (22), which is therefore dimensionally inconsistent. As a result, the displayed derivation of the bound just below (22) does not follow. This is load-bearing because the estimate (14) is used in (38)-(39) to subtract 4\\eta' from the scalar curvature; if the bound on \\partial^2 u / \\partial t^2 is not established, the positivity conclusion is not justified. The missing factor appears repairable: inserting it and propagating it through (19), (23), and the last line of the proof yields a bound of the form \\bar D \\epsilon^{1/(n+1)} \\eta + \\bar D'(C+1)\\epsilon^{1/2}, which can still be made smaller than \\eta' by fixing \\eta and then shrinking \\epsilon. The lemma needs a corrected scaling proof, not a new idea.","section":"Lemma 2.3, Eqs. (21)-(22)"},{"comment":"The dimension notation is internally inconsistent. At the beginning of Section 2 the paper states \"We assume dim(M) = n-1 \\ge 2\", while Lemma 2.3 and the Sobolev exponent in (20)-(21) require dim M = n+1 when dim X = n-1. The conformal powers and the critical exponent q = 2(n+1)/(n-1) in (21) are those for a manifold of dimension n+1, and the proof of Theorem 3.1 uses the convention that n = dim(X \\times S^1). The statement \"dim(M) = n-1\" is therefore a typo that affects how every Sobolev and conformal formula in the paper is read. This should be corrected globally, since a reader trying to verify the estimates in Section 2 cannot tell whether n denotes dim X, dim(X \\times S^1), or dim M.","section":"Setup and Lemma 2.3, dimension convention"},{"comment":"In the displayed chain (38), the term printed as 4\\nabla_V u_Y \\nabla_V u_Y is not the term that appears in (37) and (39), which require the Hessian 4\\nabla_V\\nabla_V u_Y. If the printed product of gradients were used, the expression would not have the correct homogeneity and the subsequent rearrangement with the K_2 term would not be valid. The preceding equation (37) and the following equation (39) make clear that the Hessian is intended, but the typo should be fixed in a revision.","section":"Eq. (38) in the proof of Theorem 3.1"},{"comment":"The sentence \"The operator L is injective: if Lu = 0, then combining Lem. 2.1, (10) and the maximum principle gives u = 0\" needs more detail. Since L contains the first-order operator \\nabla_V\\nabla_V, the maximum principle for Lu = 0 with u = 0 on \\partial W requires the zeroth-order coefficient R_{g|\\sigma(W)} to be nonnegative and the argument must account for the non-symmetric first-order terms. This is likely correct, but the proof as written is too compressed for a step that underlies the Fredholm alternative and the spectral estimate (12).","section":"Proposition 2.1, injectivity argument"}],"minor_comments":[{"comment":"The typeset title contains a spacing artifact: \"S1-ST ABILITY\" should almost certainly be \"S^1-Stability\".","section":"Title"},{"comment":"Reference [9] in the bibliography contains the malformed URL \"https://https://arxiv.org/abs/2302.05521\"; one \"https://\" should be removed.","section":"References"},{"comment":"The claim that u_W := u+1 is positive for \\eta \\ll 1 follows from the C^{1,\\alpha} bound (8), but the proof does not state the explicit threshold \\eta < 1. Adding this one-line justification would make the remark self-contained.","section":"Remark 2.1(ii)"},{"comment":"The proof of Lemma 2.2 asserts that the L^p norm of F on X \\times [-\\epsilon,\\epsilon] is smaller than \\delta for sufficiently small \\epsilon. This is correct because F is bounded by C+1 and the volume of the support shrinks, but the sentence hides the dependence of the volume on \\epsilon; spelling out the estimate would improve readability.","section":"Lemma 2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper's main theorem is conditional and does not claim to settle the full S^1-stability conjecture, so the known counterexamples in dimension four are not a source of difficulty for the manuscript. The central issue is the scaling estimate in Lemma 2.3, which as printed is not a proof; however, the stress-test analysis shows that the missing factor can be inserted without changing the structure of the argument. I therefore see the paper as repairable within its own scope, and the appropriate decision is major revision rather than rejection. The dimension convention also needs a careful global pass, because the same symbol n is used in several incompatible ways. I would encourage the authors to make the proof of Lemma 2.3 fully explicit, including the computation of the change of variables in the measure on S^1, since that is the step most likely to be checked by future users of the result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a new conditional version of J. Rosenberg's S1-stability conjecture: if X×S1 admits a PSC metric h whose angle between the normal to a slice and ∂θ is < π/4 at one point on the slice, then X admits a PSC metric. The PDE with the operator ∇_V∇_V − Δ is new, and the corollary that every 4D counterexample must have angle ≥ π/4 is a genuinely new necessary condition. The strategy is clever: introduce the auxiliary interval, solve an elliptic PDE with a concentrated inhomogeneous term, then move up and down the diagram with Gauss–Codazzi twice. I think the architecture is sound and the paper is worth serious engagement.\n\nThe main proof is mostly well executed. The ellipticity lemma (2.1) is clean, and the Fredholm/bootstrap argument in Prop. 2.1 is standard. The positivity estimate in (38)–(39) is algebraically consistent once you read the printed '4∇_V u∇_V u' as the Hessian term, which the surrounding text supports. The authors are transparent about what their result does and does not resolve; they do not overclaim.\n\nThe soft spot is Lemma 2.3, the scaling estimate. The stress-test note is right: the transition from (21) to (22) drops a factor of (2π/ϵ)^{2/(n+1)} when scaling θ. The asserted ϵ-independence of the Sobolev constant D2 also needs a more careful statement than the text gives. This is a real gap, but it is repairable: inserting the missing factor and propagating it through the inequalities still yields a bound of the form C0 η + C1(C+1)ϵ^{1/2}, which can be made small by first fixing η and then shrinking ϵ. So the conclusion of Lemma 2.3 survives with a revised scaling argument. Because this lemma is load-bearing for the positivity in (38), the proof is not complete as written, but the flaw is not fatal.\n\nI also noted a minor issue: the maximum-principle argument for injectivity in Prop. 2.1 with R>0 is terse, but the positivity of R and standard maximum principle make it plausible; not a serious concern.\n\nCitation pattern is fine; reliance on [9] is motivational and does not carry the result. The paper is oriented, uses the extra dimension as a real tool, and the math is honest. I would accept this for peer review and recommend the referee focus on Lemma 2.3 for a rigorous scaling argument. The paper deserves revision, not desk rejection.","headline":"A genuinely new conditional proof of the S1-stability direction via a codimension-two PDE, with a repairable technical gap in the scaling estimate.","tokens_in":16014,"tokens_out":654,"would_cite":true,"duration_ms":7545,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","58J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a PSC metric on X × S1 descends to a PSC metric on X whenever the circle direction makes an angle strictly below π/4 with the slice normal at some point of S1.","keywords":["positive scalar curvature","S1-stability conjecture","elliptic PDE","conformal geometry","Gauss-Codazzi equation","Yamabe invariant","codimension two"],"falsifier":"Search for a counterexample inside the theorem's scope: a closed oriented 4-manifold $X$ with no PSC metric, together with a PSC metric $h$ on $X \\times \\mathbb{S}^1$ for which $\\angle_h(\\mu, \\partial_\\theta) < \\pi/4$ at some slice — Theorem 3.1 says no such pair can exist, so finding one would refute it. More locally, one can test the proof's fragile step directly: for an explicit metric such as the round metric on $\\mathbb{S}^n \\times \\mathbb{S}^1$, run the scaling in Lemma 2.3 ($t' = \\epsilon^{-1}t$, $\\theta' = \\epsilon^{-1}\\theta$) and compute whether the Sobolev constants really stay bounded as $\\epsilon \\to 0$; a blow-up would invalidate the estimate (14) and with it the curvature bound (38)–(39).","tokens_in":15029,"feed_emoji":"📐","tokens_out":22416,"duration_ms":155104,"temperature":0.7,"pith_summary":"This paper addresses the $\\mathbb{S}^1$-stability conjecture, which asserts that a closed oriented manifold $X$ carries a positive scalar curvature (PSC) metric if and only if the product $X \\times \\mathbb{S}^1$ does, a statement known to fail in dimension four. What the paper tries to establish is the non-trivial direction under a geometric restriction: if a PSC metric $h$ on $X \\times \\mathbb{S}^1$ has, at some slice $X \\times \\{P\\}$, the circle direction $\\partial_\\theta$ making an angle strictly less than $\\pi/4$ with the unit normal to the slice, then $X$ itself admits a PSC metric. The proof works in every dimension and bypasses spin geometry and minimal surfaces entirely; it thickens the product to $X \\times [-1,1] \\times \\mathbb{S}^1$, solves an elliptic PDE whose ellipticity is exactly the angle condition, and uses a conformal change plus two Gauss-Codazzi reductions to show that the restricted metric on $X$ has positive scalar curvature. If the proof holds, the failure of the conjecture is a geometric phenomenon — the circle direction tilting at least $45^\\circ$ away from the slice normal somewhere on every slice — rather than a purely topological one.","feed_headline":"A 45-degree angle condition makes the S1-stability conjecture true","feed_subtitle":"If the circle direction stays within 45° of the slice normal, the base inherits a positive scalar curvature metric","key_machinery":"The load-bearing object is the elliptic operator $L' := \\nabla_V\\nabla_V - \\Delta_{\\sigma^*g}$ acting on functions on $W = X \\times [-1,1]$, where $V$ is the component of the slice normal $\\mu$ tangent to $X$ after writing $\\mu = a\\partial_\\theta + V$ with $a = h(\\mu, \\partial_\\theta)^{-1}$. Ellipticity is decided by the principal symbol, whose positivity is equivalent to $|V|^2_h < 1$, i.e. to $h(\\partial_\\theta, \\partial_\\theta)/h(\\mu, \\partial_\\theta)^2 < 2$ — the same inequality as the angle hypothesis $\\angle_h(\\mu, \\partial_\\theta) < \\pi/4$. The operator feeds the PDE $4\\nabla_V\\nabla_V u - 4\\Delta_{\\sigma^*g}u + R_g|_W u = F$ with Dirichlet boundary data, whose solution $u$, chosen with arbitrarily small $C^{1,\\alpha}$ norm via a source term $F$ concentrated in a thin slab around $X \\times \\{0\\}$, produces the conformal factor $u_M^{4/(n-2)}$ on $M$. Two applications of the Gauss-Codazzi equation — first from $M$ to the hypersurface $X \\times \\mathbb{S}^1$, then from there to $X$ — convert the positivity of the scalar curvature of $\\tilde{g}$ into positivity of $\\tau^*\\iota^*\\tilde{g}$, with comparison lemmas transferring Laplacian and gradient information between $M$, $W$, and the slices.","core_discovery":"On the paper's own terms, the central claim is Theorem 3.1: for an oriented closed manifold $X$ with $\\dim X = n-1 \\geq 2$, if $X \\times \\mathbb{S}^1$ carries a PSC metric $h$ satisfying $\\angle_h(\\mu, \\partial_\\theta) < \\pi/4$ on $X \\times \\{P\\}$ for some $P \\in \\mathbb{S}^1$ — where $\\mu$ is the unit normal to the slice chosen so that $h(\\mu, \\partial_\\theta) > 0$ — then $X$ admits a PSC metric. The proof constructs a metric $\\tilde{g} = u_M^{4/(n-2)} g$ on $M = X \\times [-1,1] \\times \\mathbb{S}^1$, with $g = h \\oplus dt^2$, where $u_M$ is the pullback of $u + 1$ and $u$ solves the elliptic Dirichlet problem $4\\nabla_V\\nabla_V u - 4\\Delta_{\\sigma^*g} u + R_g|_W u = F$ on $W = X \\times [-1,1]$. The angle hypothesis is precisely the ellipticity condition $h(\\partial_\\theta, \\partial_\\theta)/h(\\mu, \\partial_\\theta)^2 < 2$ of the operator $L' = \\nabla_V\\nabla_V - \\Delta_{\\sigma^*g}$, and the inhomogeneous term $F$, concentrated near $X \\times \\{0\\}$, makes the solution's $C^{1,\\alpha}$ norm arbitrarily small while remaining dominant when the scalar curvature of $\\tau^*\\iota^*\\tilde{g}$ is estimated through two applications of Gauss-Codazzi. The paper states as Corollary 3.1 that any counterexample to the conjecture must have angle at least $\\pi/4$ on every slice, and as Corollary 3.2 that under the angle condition a positive Yamabe invariant on the product forces one on the base.","pith_inferences":["The equality of the geometric threshold ($\\pi/4$) and the ellipticity threshold ($|V|^2 < 1$) suggests the angle condition is not a technical artifact: the condition that makes curvature descend is the same condition that makes the auxiliary PDE solvable, and probing that coincidence in related descent problems could be productive.","The trick of adding a dimension to turn a non-elliptic operator into an elliptic one (the analogous operator on $X \\times \\mathbb{S}^1$ alone is $\\nabla_\\mu\\nabla_\\mu - \\Delta_h$, which is never elliptic) could plausibly transfer to other problems where curvature or index information must pass between a manifold and a submanifold or quotient.","A concrete check: take a known four-dimensional counterexample and the PSC product metric on it that is known to exist, compute the angle field $\\angle_h(\\mu, \\partial_\\theta)$ slice by slice, and locate where the $\\pi/4$ threshold is crossed; Corollary 3.1 predicts a crossing on every slice, and the angular profile would show how sharp the theorem's hypothesis is."],"forward_implications":["The $\\mathbb{S}^1$-stability conjecture becomes a theorem for every metric satisfying the angle condition: a PSC metric on $X \\times \\mathbb{S}^1$ forces one on $X$ in all dimensions, with no spin, minimal-surface, or dimension hypotheses.","Conversely, every known or possible counterexample — such as the odd-degree hypersurface in $\\mathbb{CP}^3$ and the connected sums $M \\# k\\mathbb{CP}^2$ discussed in the paper — must have angle $\\angle_h(\\mu, \\partial_\\theta) \\geq \\pi/4$ somewhere on every slice $X \\times \\{P\\}$ (Corollary 3.1).","Under the angle condition the Yamabe invariant propagates from product to base: $\\lambda_h(X \\times \\mathbb{S}^1) > 0$ implies $\\lambda_{\\tau^*h}(X) > 0$ (Corollary 3.2).","The obstruction to the conjecture is located geometrically — in how far the circle direction tilts from the slice normal — rather than in a topological invariant, for the class of metrics the theorem covers."],"supporting_citations":[{"why":"States the S1-stability conjecture and its four-dimensional hypersurface counterexample; the conjecture and its failure cases are the problem the paper addresses.","marker":"[8]"},{"why":"Supplies the other known four-dimensional counterexamples (connected sums of Kähler surfaces with copies of CP2) that Corollary 3.1 must exclude via the angle bound.","marker":"[6]"},{"why":"Defines the relative Yamabe invariant used in inequality (20), whose positivity is essential to the smallness estimate in Lemma 2.3.","marker":"[2]"},{"why":"Provides the conformal Laplacian strategy that motivates the conformal transformation and the elliptic PDE (7) built in Section 2.","marker":"[9]"},{"why":"Gives the elliptic boundary regularity estimate (11) with Dirichlet boundary conditions used to control the solution u.","marker":"[1]"},{"why":"Supplies the bootstrapping theorem that promotes the weak solution of (7) to a smooth solution.","marker":"[4]"}],"fun_headline_variants":["S1-stability proven under 45° angle bound","Angle <45° on X×S1 forces PSC on X","Circle tilt <45° proves S1-stability","PSC on base from 45°-tilted S1 metric"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the technical estimate in Lemma 2.3: the second derivative of the constructed solution in the extra dimension $t$ can be made arbitrarily small near the middle slice, which rests on a scaling argument whose Sobolev constants are claimed to stay bounded as $\\epsilon \\to 0$; if that bound fails, the positivity of the final scalar curvature is no longer forced.","fun_headline_variants_meta":{"raw":{"variants":["S1-stability proven under 45° angle bound","Angle <45° on X×S1 forces PSC on X","Circle tilt <45° proves S1-stability","PSC on base from 45°-tilted S1 metric"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000542,"raw_usage":{"total_tokens":2662,"prompt_tokens":1074,"completion_tokens":1588,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":690,"completion_tokens_details":{"reasoning_tokens":1515}},"tokens_in":690,"tokens_out":1588,"duration_ms":14395,"temperature":1.0,"reasoning_tokens":1515,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:03:28.663570+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a counterexample inside the theorem's scope: a closed oriented 4-manifold $X$ with no PSC metric, together with a PSC metric $h$ on $X \\times \\mathbb{S}^1$ for which $\\angle_h(\\mu, \\partial_\\theta) < \\pi/4$ at some slice — Theorem 3.1 says no such pair can exist, so finding one would refute it. More locally, one can test the proof's fragile step directly: for an explicit metric such as the round metric on $\\mathbb{S}^n \\times \\mathbb{S}^1$, run the scaling in Lemma 2.3 ($t' = \\epsilon^{-1}t$, $\\theta' = \\epsilon^{-1}\\theta$) and compute whether the Sobolev constants really stay bounded as $\\epsilon \\to 0$; a blow-up would invalidate the estimate (14) and with it the curvature bound (38)–(39).","supporting_citations":[{"cited_title":"Rosenberg","cited_arxiv_id":null,"evidence_quote":"States the S1-stability conjecture and its four-dimensional hypersurface counterexample; the conjecture and its failure cases are the problem the paper addresses."},{"cited_title":"Positive scalar curvature and exotic structures on simply connected four manifolds","cited_arxiv_id":"2501.01113","evidence_quote":"Supplies the other known four-dimensional counterexamples (connected sums of Kähler surfaces with copies of CP2) that Corollary 3.1 must exclude via the angle bound."},{"cited_title":"Akutagawa and B","cited_arxiv_id":null,"evidence_quote":"Defines the relative Yamabe invariant used in inequality (20), whose positivity is essential to the smallness estimate in Lemma 2.3."},{"cited_title":"Cherrier","cited_arxiv_id":null,"evidence_quote":"Supplies the bootstrapping theorem that promotes the weak solution of (7) to a smooth solution."}],"review_version":1}