{"id":"f21f4759-1b17-48c3-82a3-0858c7f31087","arxiv_id":"2608.10324","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For closed manifolds with positive Yamabe invariant below the sphere value, stability optimizers exist whenever the stability constant lies strictly below a local and a one-bubble threshold, and the bubble-threshold inequality is verified in most geometric settings.","lead":"This paper proves that, under two threshold conditions, the best constant in the quantitative stability inequality for the Yamabe problem is attained by an actual function on closed manifolds with positive Yamabe invariant. It identifies a one-bubble compactness threshold and a mass-dominance condition that push the stability constant strictly below the concentration threshold.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's compactness proof assumes the weak limit has unit Lp norm, which is exactly the strong-convergence conclusion, leaving the mixed weak-limit-plus-bubble case untreated.","rationale":"The reader's verdict identifies a real normalization inconsistency in the mass-dominance condition: the introduction's (M_x0) uses a denominator |S^n|^{(n−2)/n}, while Proposition 5.2 derives a coefficient with denominator ∥U1∥_{L^p(R^n)}^2, differing by the dimension-dependent factor [n(n−2)]^{(n−2)/2}/2^{n−2}. This affects Theorem 1.4 and should be fixed. However, the more load-bearing gap is in the proof of Theorem 1.2 itself, since that theorem underpins all existence claims. In Claim 2 of Section 3, the proof assumes ∥u∥_p=1 before proving strong convergence. This is not a harmless normalization: for a normalized minimizing sequence, Brezis–Lieb shows 1 = ∥u∥_p^p + ∥v_k∥_p^p + o(1), so ∥u∥_p=1 is equivalent to the desired vanishing of the remainder. The formula (3.2) and the calculus lemma are built on this unproven identity. If 0<∥u∥_p<1, the correct constraint is different and the argument does not exclude a mixed sequence consisting of a fixed nonzero part plus a concentrating bubble. The introduction's dichotomy—converge to an optimizer or concentrate to zero—omits this mixed alternative. Since Theorem 1.2 is the core of the paper, this gap is the single most load-bearing concern. The paper may still be correct; a standard concentration-compactness splitting lemma could close the gap, so the appropriate verdict remains conditional rather than reject. I therefore keep the reader's conditional verdict while flagging a different and more central obstruction.","tokens_in":26409,"tokens_out":22775,"duration_ms":209512,"concrete_test":"Re-derive the asymptotic inequality (3.2) under the correct Brezis–Lieb splitting with s := ∥u∥_p ∈ (0,1), using 1 = (s^p + ∥v_k∥_p^p)^{2/p} and X_k = E[v_k]/Y_0. The reduced function becomes Ψ_s(X) = (a+X)^{(γ−2)/2}(a+X−b)/(a+X−c)^{γ/2} on the constrained set X ≥ (1−s^p)^{2/p}. Check whether Ψ_s can have an interior minimum for some s<1; if it can, Lemma 3.1 no longer forces X_k→0 and a mixed minimizing sequence is not excluded. For a concrete instance, take n=3 product-cylinder data from the companion paper and numerically evaluate liminf Sγ along sequences u + w_k where w_k is a concentrating Aubin–Talenti bubble, to see whether values below cγ are attainable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.2, Claim 2 (Section 3) starts: 'Assume again that u_k = u+v_k with ∥u∥_{Lp(M)} = 1'. Claim 1 establishes only that the weak limit is nonzero; it does not prove ∥u∥_p=1. This normalization is not harmless. For the normalized sequence u_k, Brezis–Lieb gives 1 = ∥u_k∥_p^p = ∥u∥_p^p + ∥v_k∥_p^p + o(1), so ∥u∥_p=1 already forces ∥v_k∥_p→0, and Proposition 2.7 then yields E[v_k]→0, i.e., the desired strong convergence. Thus the assumption is equivalent to the conclusion it is meant to establish. If instead 0<∥u∥_p<1, the mixed alternative—a nonzero weak limit together with a concentrating bubble—must be excluded, but equation (3.2) is derived using (1+∥v_k∥_p^p)^{2/p} instead of the correct (∥u∥_p^p+∥v_k∥_p^p)^{2/p}=1. The auxiliary function Ψ and Lemma 3.1 are then applied to a different reduced problem, and no argument such as a splitting lower bound liminf Sγ(u_k) ≥ min(c_loc, c_bub) is supplied for the mixed case. Consequently, the central compactness theorem is not proved as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the quantitative stability of the Yamabe inequality on closed Riemannian manifolds (M,g) with n≥3 and 0 < Y(M,[g]) < Y(S^n,[g_0]). It defines the stability constant c_γ(M,g) as the infimum of a stability quotient S_γ over functions that are not Yamabe optimizers, and asks whether this infimum is attained. The main result (Theorem 1.2) states that if c_γ is strictly below a local threshold c_loc and a bubble threshold c_bub, then every minimizing sequence converges strongly in H^1 and an optimizer for c_γ exists. The bubble threshold is computed in Corollary 2.8 as c_bub = 1 − Y(M,[g])/Y(S^n,[g_0]). Theorems 1.3 and 1.4 provide sufficient geometric conditions for c_γ < c_bub: for n≥6 non-locally-conformally-flat manifolds via Aubin's test functions, and for 3≤n≤5 or locally conformally flat manifolds via Schoen's Green function modified test functions, under a new 'mass dominance condition' (M_x0). The proof combines a concentration-compactness argument adapted from [15] with explicit asymptotic expansions of the stability quotient along test functions.","tokens_in":26657,"tokens_out":10394,"duration_ms":87511,"significance":"If correct, the paper makes a valuable contribution to the quantitative stability theory of sharp functional inequalities on manifolds. The identification of a one-bubble threshold in the non-spherical case, as opposed to the two-bubble threshold on the sphere treated in [15], is a genuinely new insight and clarifies the structure of the compactness argument. The paper contains a number of clean and useful technical pieces: the compactness proof for the optimizer set O_1 (Lemma 2.1), the distance identity in Lemma 2.3, the explicit computation of c_bub in Corollary 2.8, and the calculus lemma (Lemma 3.1) ruling out local minima of the auxiliary function Ψ. The expansion of the stability mass m along test functions and the detailed energy computations in Propositions B and C are careful and largely self-contained in the appendices. However, the two major issues detailed below currently prevent the main existence theorems from being fully established, so the significance is conditional on a substantial revision.","major_comments":[{"comment":"The proof of Claim 2 assumes 'u_k = u+v_k with ||u||_{L^p(M)} = 1', but Claim 1 only establishes that the weak limit u is nonzero; it does not prove that u carries the full L^p mass. This assumption is equivalent to the desired conclusion: if ||u||_{L^p(M)} = 1, then Brezis–Lieb gives ||v_k||_{L^p(M)} → 0, and the Yamabe inequality then yields E[v_k] → 0, so strong convergence already holds. In the complementary case 0 < ||u||_{L^p(M)} < 1, the formula in equation (3.2) uses (1+||v_k||_p^p)^{2/p} in the numerator instead of the correct (||u||_p^p+||v_k||_p^p)^{2/p}, and the subsequent reduction to the one-variable function Ψ does not apply. No argument is supplied to exclude this mixed weak-limit-plus-bubble scenario, such as a splitting lower bound of the form liminf_k S_γ(u_k) ≥ min(c_loc, c_bub). Consequently, Theorem 1.2 is not proved as written.","section":"Section 3, Claim 2 (Proof of Theorem 1.2)"},{"comment":"The definition of the rescaled stability mass coefficient m_∞(x0) is inconsistent between the introduction and Proposition 5.2. In the introduction, the mass dominance condition (M_x0) is stated with m_∞(x0) = [α_n^2 (n−2)^2 ω_{n−1}^2 / |S^n|^{(n−2)/n}] sup_{h∈O_1} (∫_M h^{p−1} G_{x0} dV_g)^2, whereas Proposition 5.2, Eq. (5.6), derives m_∞(x0) = [α_n (n−2) ω_{n−1}]^2 / ||U_1||^2_{L^p(R^n)} sup_{h∈O_1} (∫_M h^{p−1} G_{x0} dV_g)^2. These two expressions differ by a dimension-dependent constant; for n=3 the ratio of the denominators is √3/2. Since the proof of Theorem 1.4 uses the Proposition 5.2 version while the statement of Theorem 1.4 and the introduction use the other version, the condition verified in the proof is not the condition stated in the theorem. This ambiguity affects the validity of the theorem's hypothesis and must be resolved.","section":"Introduction (M_x0) and Proposition 5.2, Eq. (5.6)"},{"comment":"In the proof of Claim 2, the text states 'By Proposition 2.7, one obtains ||v_k||_p^p ≤ Y_0^{-p/2} E[v_k]^{p/2} + o(1)' for a sequence v_k ⇀ 0 in H^1 that is not normalized to have L^p norm one. Proposition 2.7 is stated for sequences with ||v_k||_{L^p(M)} = 1. A scaling argument can likely bridge this gap, but as written the application is not justified, and since this inequality is used to derive the lower bound on Ψ(X_k), the step should be made explicit.","section":"Section 3, use of Proposition 2.7"}],"minor_comments":[{"comment":"The convergence in the definition of the local threshold c_loc (1.9) is not specified; it should say whether the limit is taken in H^1(M), C^2(M), or another topology. The proof of Claim 3 suggests H^1 convergence is intended.","section":"Section 1, definition of c_loc (1.9)"},{"comment":"The paper never verifies the strict inequality c_γ(M,g) < c_loc_γ(M,g) for any manifold; the existence results in Corollary 1.5 are therefore conditional on this hypothesis, with verification deferred to the companion paper [1]. This is acceptable for a theorem of sufficient conditions, but the abstract's wording 'we prove the existence of stability optimizers' could be misread as unconditional, and a sentence emphasizing the conditional nature would help.","section":"Abstract and Corollary 1.5"},{"comment":"The proof of Proposition C is presented as a sketch: Claim 3 states that the transition annulus contributes −β_n ||U_1||^2_{L^p(R^n)} m_{x0} µ^{n−2} + o(µ^{n−2}) and refers to Schoen's integration-by-parts argument, but the details of the surface integral and the precise coefficient β_n are not shown. Since the mass dominance condition hinges on this coefficient, a fuller derivation would improve the paper.","section":"Appendix A, Proposition C proof"},{"comment":"In Lemma 2.5, the statement says u_k ⇀ 0 in L^p(M), but the proof uses convergence in H^1(M); this is harmless because H^1 compactly embeds into L^2 and weakly in L^p, but the statement could be tightened.","section":"Section 2, Lemma 2.5"}],"recommendation":"major_revision","confidential_remarks":"The paper is well-written and the ideas are promising, but the proof of Theorem 1.2 has a serious gap in the treatment of the mixed weak-limit-plus-bubble case, and the definitional inconsistency for m_∞(x0) between the introduction and Proposition 5.2 affects the statement of Theorem 1.4. The applicability of the main results also depends entirely on the companion paper [1] for the unverified condition c_γ < c_loc_γ. Given these issues, major revision is appropriate. I would encourage the authors to add a profile-decomposition argument to handle the mixed case and to unify the two definitions of m_∞(x0)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nYou should know two things about arXiv:2608.10324. The main abstract result, Theorem 1.2, is likely correct in spirit but the proof as written has a genuine gap in the mixed weak-limit-plus-bubble case. And the definition of the rescaled stability mass m_infinity is inconsistent between the introduction and Proposition 5.2, so the mass-dominance condition in Theorem 1.4 is not the one actually proved.\n\nWhat's genuinely new: the one-bubble compactness threshold c_bub = 1 - Y_M/Y_0, which is different from the sphere two-bubble threshold in [15]; the treatment of degenerate stability exponents gamma >= 2 via a calculus lemma; and the mass-dominance condition that resolves the competition between the Green function mass and the stability mass in the LCF and low-dimension cases. The paper is clearly written, the c_bub computation is clean, and the Aubin/Schoen test function expansions in Sections 4 and 5 are standard but carefully done. This is a serious contribution to quantitative Yamabe theory.\n\nThe soft spots, in order of severity.\n\nFirst, the gap in Claim 2 of Theorem 1.2, which the stress-test note flags. I read that note and think it is right. The proof assumes the weak limit u satisfies ||u||_p = 1. Claim 1 only establishes u is nonzero. If 0 < ||u||_p < 1, then the remainder v_k carries positive L^p mass, and the formula (3.2) using (1+||v_k||_p^p)^{2/p} is not the correct expansion. The auxiliary function Psi is built for the normalized case. No argument is given that excludes the mixed case that a nonzero weak limit coexists with a concentrating bubble. A splitting lower bound like liminf S_gamma(u_k) >= min(c_loc, c_bub) would do it, but that argument is absent. I think this is repairable — the two-threshold strategy should handle the mixed case with a two-parameter optimization — but as written the proof of the central theorem is incomplete.\n\nSecond, the m_infinity definition is not consistent. The introduction defines the coefficient with |S^n|^{(n-2)/n} in the denominator; Proposition 5.2 derives ||U_1||^2_{L^p(R^n)}. These differ by a dimension-dependent factor. Since Theorem 1.4 is stated with the introduction's formula and proved with the Proposition's, the theorem as stated is not what is proved. This is a straightforward fix, but it matters: readers checking (M_x0) against the introduction will be checking a different inequality.\n\nThird, the paper's claim of a 'complete characterization' overstates things, because c_gamma < c_loc_gamma is not verified for any manifold and is deferred to a companion paper. The results are sufficient conditions, not a full characterization.\n\nWho is this for? Specialists in geometric analysis and quantitative Sobolev inequalities. It deserves a serious referee, and I would send it to peer review rather than desk reject. With the compactness gap closed and the mass notation made consistent, this would be a strong paper.","headline":"Nice ideas, real proof gap: Theorem 1.2 misses the mixed weak-limit/bubble case, and the mass coefficient is defined two incompatible ways.","tokens_in":27253,"tokens_out":8799,"would_cite":false,"duration_ms":70658,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","46E35","58J05","53C21"],"pacs":[],"model":"deepseek-v4-flash","headline":"On closed manifolds, Yamabe stability constants are attained when they stay below a one-bubble threshold.","keywords":["Yamabe problem","Sobolev inequality","stability constant","Bianchi–Egnell inequality","Aubin–Schoen test functions","mass dominance condition","Yamabe stability optimizer","bubble threshold"],"falsifier":"Compute the three constants on the product cylinders $S^1(\\tau)\\times S^{n-1}$, where the optimizers are explicit near $\\tau\\le\\tau^*$; if at some parameter the best stability constant equals the bubble threshold $1-Y(M,[g])/Y(S^n,[g_0])$ while the local inequality is strict, the compactness conclusion would fail exactly there and the threshold picture would need revision.","tokens_in":26109,"feed_emoji":"📐","tokens_out":12968,"duration_ms":117293,"temperature":0.7,"pith_summary":"The paper aims to show that the best constant in the quantitative stability inequality for the Yamabe functional is attained on closed Riemannian manifolds, not only on the round sphere. The main theorem gives two threshold conditions: the stability constant must lie strictly below the threshold set by sequences approaching a Yamabe minimizer, and strictly below the threshold set by sequences that concentrate into a single bubble. The bubble threshold is computed explicitly as $1 - Y(M,[g])/Y(S^n,[g_0])$, and unlike the sphere case, where the threshold comes from two bubbles, one bubble already measures the possible loss because the set of Yamabe minimizers is compact away from the sphere. The paper then proves that the stability constant can be pushed strictly below the bubble threshold by Aubin's Weyl-curvature test functions when $n\\ge 6$ and the manifold is not locally conformally flat, and by Schoen's Green-function test functions under a new mass dominance condition in the remaining dimensions and locally conformally flat case. The upshot is that existence of a stability optimizer is reduced to checking explicit constants attached to the conformal geometry.","feed_headline":"One-bubble threshold decides Yamabe stability optimizers","feed_subtitle":"The existence threshold is a single bubble, not two; minimizing sequences converge and the stability constant is attained.","key_machinery":"Three objects carry the argument. First, the stability quotient $S_\\gamma(u)=E[u]^{(\\gamma-2)/2}(E[u]-Y(M,[g])\\|u\\|_{L^p}^2)/\\mathrm{dist}(u,O)^\\gamma$, whose infimum is $c_\\gamma$. Second, the stability mass $m(u)=\\sup_{h\\in O_1}(\\int_M h^{p-1}u\\,dV_g)^2$, which converts the distance to the optimizer set into $E[u]-Y(M,[g])m(u)$ and measures how much of $u$ is seen by the $L^p$-normalized Yamabe optimizers. Third, the two thresholds $c_\\gamma^{\\mathrm{loc}}$ and $c_\\gamma^{\\mathrm{bub}}$ encode convergence to an optimizer versus concentration into a bubble. A one-variable calculus lemma shows that the auxiliary function $\\Psi(X)$ obtained by separating a minimizing sequence into a nontrivial weak limit and a weakly vanishing remainder has no local minimum; its infimum can occur only at the boundary values $S_\\gamma(u)$ or $c_\\gamma^{\\mathrm{bub}}$. The strict inequalities in Theorem 1.2 rule out both boundaries and force the remainder to vanish. To verify the bubble inequality, Aubin–Talenti bubbles centered at a point of nonzero Weyl curvature produce an energy correction of order $\\mu^4$ that dominates the stability mass of order $\\mu^{n-2}$ for $n\\ge6$, while Schoen's Green-function modified bubbles have both corrections of order $\\mu^{n-2}$, and the sign is decided by the mass dominance inequality $\\beta_n m_{x_0}>(\\gamma/2)(Y(S^n,[g_0])-Y(M,[g]))m_\\infty(x_0)$.","core_discovery":"The central claim is Theorem 1.2. For a closed $n$-dimensional manifold with $0<Y(M,[g])<Y(S^n,[g_0])$ and stability exponent $\\gamma\\ge2$, let $c_\\gamma$ be the best constant in the stability inequality, let $c_\\gamma^{\\mathrm{loc}}$ be the infimum of stability quotients over sequences converging to a Yamabe optimizer, and let $c_\\gamma^{\\mathrm{bub}}$ be the infimum over $L^p$-normalized sequences converging weakly to zero. If $c_\\gamma<c_\\gamma^{\\mathrm{loc}}$ and $c_\\gamma<c_\\gamma^{\\mathrm{bub}}$, then every minimizing sequence for $c_\\gamma$ converges strongly in $H^1$ up to a subsequence, and a stability optimizer exists. The paper computes $c_\\gamma^{\\mathrm{bub}}=1-Y(M,[g])/Y(S^n,[g_0])$ for all $\\gamma\\ge2$, proves the strict inequality $c_\\gamma<c_\\gamma^{\\mathrm{bub}}$ for non-locally-conformally-flat manifolds of dimension $n\\ge6$ using the negative Weyl-curvature correction in Aubin's test functions, and proves it for $3\\le n\\le5$ or locally conformally flat manifolds whenever a mass dominance condition holds at some point. The comparison is controlled by the stability mass $m(u)=\\sup_{h\\in O_1}(\\int_M h^{p-1}u\\,dV_g)^2$, which enters the identity $\\mathrm{dist}(u,O)^2=E[u]-Y(M,[g])m(u)$ and competes with the energy deficit in the Taylor expansion of the stability quotient.","pith_inferences":["The same two-threshold mechanism should transfer to any sharp inequality whose optimizer set is compact and whose stability quotient has a distance-to-optimizers representation; the sphere is the special case where the optimizer set is noncompact and produces an additional bubble.","Because the right-hand side of the mass dominance condition grows linearly in $\\gamma$, larger stability exponents require a larger Green-function mass; this suggests that manifolds with degenerate stability ($\\gamma>2$) should be the first place where existence of an optimizer fails as a geometric parameter is varied.","A dimension-dependent normalization separates the introduction's version of the mass coefficient from the one used in Proposition 5.2, so before applying Theorem 1.4 to a concrete manifold one should identify which version of $m_\\infty(x_0)$ is being verified; this is an editorial caution, not a claim of the paper."],"forward_implications":["Whenever the two strict inequalities of Theorem 1.2 hold, the best stability constant $c_\\gamma(M,g)$ is attained by a function that is not itself a Yamabe optimizer, and the minimizing sequence converges strongly in $H^1$.","In dimensions $n\\ge6$ with a point of nonvanishing Weyl tensor, only the local inequality $c_\\gamma<c_\\gamma^{\\mathrm{loc}}$ has to be checked, since Theorem 1.3 supplies the bubble inequality automatically.","In dimensions $3\\le n\\le5$ and on locally conformally flat manifolds, a single point whose Green-function mass dominates the rescaled stability mass is sufficient to remove the bubble obstruction.","The bubble threshold $1-Y(M,[g])/Y(S^n,[g_0])$ is independent of $\\gamma$, whereas on the round sphere the threshold $2-2^{2/p}$ arises from two bubbles; the difference is exactly the compactness of the optimizer set.","Combined with the known sphere result, Theorem 1.2 completes the positive-Yamabe existence picture for stability optimizers."],"supporting_citations":[{"why":"proves existence of a stability optimizer for the Euclidean Sobolev inequality and supplies the two-threshold concentration-compactness strategy adapted here.","marker":"[15]"},{"why":"establishes the quantitative Yamabe stability inequality (1.6) with exponent gamma, the starting point for defining the stability constant.","marker":"[10]"},{"why":"introduces the Bianchi–Egnell stability estimate for the Sobolev inequality, the prototype for the stability quotient studied here.","marker":"[5]"},{"why":"provides the Aubin test functions and the negative Weyl-curvature energy correction used to prove the bubble inequality in dimension at least six.","marker":"[2]"},{"why":"supplies the compactness of Lp-normalized Yamabe optimizers, conformal normal coordinates, and the classical energy expansions used repeatedly.","marker":"[16]"},{"why":"contributes Schoen's Green-function modified test functions and the mass expansion underlying the mass dominance condition.","marker":"[17]"},{"why":"gives examples where the stability exponent must be taken larger than two, motivating the proof for general gamma.","marker":"[11]"}],"fun_headline_variants":["One-bubble threshold yields Yamabe stability optimizers","Stability optimizers exist under one-bubble condition","Yamabe stability constant attained: one bubble rules","Single-bubble threshold guarantees stability optimizers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the strict inequality between the best stability constant and the local threshold set by sequences approaching a Yamabe optimizer; the paper proves conditions that ensure the bubble half of the comparison but leaves the local half unverified for any concrete manifold, deferring it to a companion paper.","fun_headline_variants_meta":{"raw":{"variants":["One-bubble threshold yields Yamabe stability optimizers","Stability optimizers exist under one-bubble condition","Yamabe stability constant attained: one bubble rules","Single-bubble threshold guarantees stability optimizers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00036,"raw_usage":{"total_tokens":1997,"prompt_tokens":1045,"completion_tokens":952,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":891}},"tokens_in":661,"tokens_out":952,"duration_ms":8141,"temperature":1.0,"reasoning_tokens":891,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:12:53.466511+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the three constants on the product cylinders $S^1(\\tau)\\times S^{n-1}$, where the optimizers are explicit near $\\tau\\le\\tau^*$; if at some parameter the best stability constant equals the bubble threshold $1-Y(M,[g])/Y(S^n,[g_0])$ while the local inequality is strict, the compactness conclusion would fail exactly there and the threshold picture would need revision.","supporting_citations":[{"cited_title":"Engelstein, R","cited_arxiv_id":null,"evidence_quote":"establishes the quantitative Yamabe stability inequality (1.6) with exponent gamma, the starting point for defining the stability constant."},{"cited_title":"Bianchi and H","cited_arxiv_id":null,"evidence_quote":"introduces the Bianchi–Egnell stability estimate for the Sobolev inequality, the prototype for the stability quotient studied here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the Aubin test functions and the negative Weyl-curvature energy correction used to prove the bubble inequality in dimension at least six."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the compactness of Lp-normalized Yamabe optimizers, conformal normal coordinates, and the classical energy expansions used repeatedly."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"contributes Schoen's Green-function modified test functions and the mass expansion underlying the mass dominance condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives examples where the stability exponent must be taken larger than two, motivating the proof for general gamma."}],"review_version":1}