{"id":"d039e004-2f45-4ebf-a3c2-09cb8f4f2a27","arxiv_id":"2412.05935","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper reviews and re-exposes existing stability theorems for sharp Sobolev inequalities on manifolds with Ricci lower bounds.","lead":"This survey reviews recent results on how close functions must be to Euclidean bubbles when they almost achieve the sharp Sobolev inequality on Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. It also sketches the proof strategy, combining synthetic RCD geometry, rearrangement inequalities, and concentration compactness.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3's proof sketch turns on the unproved equality rigidity in Theorem 2.6; a gap there would break the limit-space identification.","rationale":"The reader and I identify the same load-bearing point: equality rigidity of the Pólya–Szegő inequality on RCD(0,N) spaces is what turns the a priori RCD(0,d) limit Y into a cone with a radial bubble, and hence gives the approximating sequence (3.5). I do not find a separate internal inconsistency in the survey's derivation: the scaling in Section 3.1 is compatible with Theorem 2.10, the AVR lower bound on Y follows from the volume-cone argument, and the contradiction at the end is valid once Theorem 2.6 is granted. The reason this does not change the verdict is that Theorem 1.3 is a previously published result [139] and the survey explicitly functions as an overview; the rigidity theorem is cited, not newly proved here. Nevertheless, the proof sketch is only as strong as Theorem 2.6, which is presented as the fine/novel tool and is the least independently checked input, so a dedicated verification of its equality case is the single check worth running.","tokens_in":31983,"tokens_out":32256,"duration_ms":340922,"concrete_test":"Independently re-derive [137, Theorem 1.3] for p=2 on a smooth Ricci-flat ALE 4-manifold with AVR>0, e.g. Eguchi–Hanson. The check is to follow the equality case in (2.2) from the isoperimetric inequality to the cone conclusion and verify that the stated hypotheses (RCD(0,N), AVR>0, (u*)' ≠ 0) are sufficient, without hidden regularity or noncriticality assumptions. If the derivation requires extra smoothness or an additional condition, then Theorem 2.6 as stated in Section 2.2 is stronger than what is established, and the proof sketch of Theorem 1.3 loses its basis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.1 obtains the limit space Y from Theorem 2.10 and then identifies Y as a cone and u∞ as a Euclidean bubble by invoking Theorem 2.7, whose rigidity rests entirely on Theorem 2.6, quoted from the preprint [137] without proof. If equality in the Pólya–Szegő inequality (2.2) on an RCD(0,N) space with AVR>0 did not force X to be a cone and u radial under the stated hypothesis (u*)' ≠ 0, then the equality case in the chain leading to (3.5) would not produce the required centered bubble, and the contradiction in the proof of Theorem 1.3 would collapse. This is the least independently supported input in the presentation: the main theorems are otherwise published in [138,139,135], whereas Theorem 2.6 is presented as the fine/novel tool. The survey does not supply a proof or a precise statement of the isoperimetric-rigidity step behind it, so the reader cannot check the key mechanism from this paper alone.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper is an expository survey, centered on the author's recent work, of qualitative and quantitative stability for sharp Sobolev inequalities under Ricci curvature lower bounds. After recalling the Euclidean stability program (Bianchi–Egnell and later quantitative results) and the AB-program, the paper states its main theorems: Theorem 1.3 says that on a noncompact manifold with Ric ≥ 0 and asymptotic volume ratio bounded below, any function whose Sobolev quotient is δ-close to the sharp constant is ε-close, in normalized L2-gradient distance, to a Euclidean bubble; Theorem 1.4 gives the compact analog under Ric ≥ d−1, together with rigidity of the optimal constant and a quantitative diameter estimate (π − diam(M))^d. Sections 2 and 3 develop the three ingredients used in the proof sketch: the RCD calculus and pmGH convergence, a fine Pólya–Szegő inequality with equality rigidity (Theorem 2.6), and a concentration-compactness principle on varying spaces (Theorem 2.10), followed by the argument for Theorem 1.3 and comments on the compact case. Section 4 reports stability in the AB-program and open questions.","tokens_in":32152,"tokens_out":11616,"duration_ms":113828,"significance":"If the theorems are taken as established (the main ones are published in [138,139,135], while [137] is a preprint), the paper gives a valuable unified account of an emerging stability theory: it shows how smooth and non-smooth methods combine, and it makes explicit that the bubble families are not extremal in general, which is the essential difficulty. The paper's strengths are its clear statements, the readable outline of the concentration-compactness mechanism, and the honest separation of quoted results from sketched arguments. Its main limitation is that the centerpiece proof is not self-contained at the point where it matters most: the equality rigidity of the Pólya–Szegő inequality is imported from [137] without proof, and a displayed inequality in the proof of Theorem 2.7 is misprinted. These issues do not cast doubt on the published theorems, but they affect the survey's stated goal of a self-contained overview.","major_comments":[{"comment":"The proof of Theorem 1.3 in §3.1 identifies the limit space Y as a Euclidean cone and the limit profile u∞ as a Euclidean bubble by appealing to the rigidity part of Theorem 2.7. The proof of Theorem 2.7 in §2.2 derives that rigidity directly from Theorem 2.6, the equality case of the Pólya–Szegő inequality (2.2). Theorem 2.6 is quoted from the preprint [137] with no proof and no statement of the isoperimetric-rigidity mechanism behind it. Since the abstract and §1.4 advertise a 'self-contained' overview, this is a load-bearing gap in the exposition: a reader cannot check from the paper alone why equality in (2.2) forces X to be an N-cone and u radial under the stated hypothesis. I am not questioning Theorem 1.3, which is published in [139], but the survey should either include a detailed proof or at least an outline of the equality case of Theorem 2.6, or explicitly present it as an external black box and state precisely which result of [137] is being used.","section":"2.2 and 3.1"},{"comment":"In the displayed chain after (2.2), the second inequality is written as S_{N,p} ||(|u|∗)'||_{L^{p*}(m_{0,N})} ≥ || |u|∗ ||_{L^{p*}(m_{0,N})}. This is not the Bliss inequality, which requires the L^p norm of the derivative; as printed the inequality is false for general N,p (it fails by scaling). It should read S_{N,p} ||(|u|∗)'||_{L^p(m_{0,N})} ≥ || |u|∗ ||_{L^{p*}(m_{0,N})}. Because this display is the step where (2.2) is converted into the sharp Sobolev inequality (2.3), the correction is necessary.","section":"2.2, proof of Theorem 2.7"}],"minor_comments":[{"comment":"In the numerator of (1.12), the second term is written with u_{a,b,z0}, but it should be v_{a,b,z0}; the displayed expression should read ||u − v_{a,b,z0}||_{L^{2*}(ν)}.","section":"1.3, Eq. (1.12)"},{"comment":"The dimension parameter N is used without being explicitly set equal to d in the function f(t) := a(1 + bt^2)^{(2−N)/2} and in the exponents 2*; using d consistently would avoid confusion with the abstract N in Theorem 2.10.","section":"3.1"},{"comment":"The exclusion of the case σ = 0 via the maximal diameter theorem is not explained. Since diam(Y_n) = σ_n diam(M_n) ≤ σ_n π, a reader cannot immediately see why σ_n cannot converge to 0; a sentence explaining this point is needed.","section":"3.2"},{"comment":"Several small typos remain, including 'dimensioanl' in §3.1, 'equality equality' in §3.1, 'apriori' in §2.2, and 'Overwiew' in §2.3; these should be corrected in a final pass.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is essentially an account of the author's own results, and the citation density to [135,137,138,139] is high but appropriate for a survey. My recommendation is not based on novelty or correctness concerns: the main theorems are already published, so the requested changes concern the exposition's self-containedness and a misprint in a central display. If the journal is willing to accept surveys that rely on cited preprints for a key rigidity result, the revision could be light; otherwise the missing proof of Theorem 2.6 is the main obstacle."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a useful survey of the author's own stability results for Sobolev inequalities under Ricci lower bounds. There are no new theorems, but the expository packaging is genuinely good: the three-ingredient structure (RCD calculus, fine Pólya–Szegő, concentration compactness on varying spaces) makes otherwise technical arguments legible, and the open-questions section is honest and points at real limitations.\n\nWhat it does well: the proof sketch of Theorem 1.3 is coherent and gives the right idea of why the limit space is a cone and the limit profile is a Euclidean bubble. The paper is careful to separate qualitative stability from quantitative, and it does not oversell. The AB-program discussion is compact but accurate, and the pointers to [135,137,138,139] are appropriate; self-citation is heavy but justified because the paper is explicitly an overview of that work.\n\nSoft spots, in order of importance. First, the key rigidity input in the noncompact proof, Theorem 2.6, is quoted from the preprint [137], and no proof is supplied. The equality case of the Pólya–Szegő inequality is what forces the limit space to be a cone and the limit function radial; without it, the contradiction in Theorem 1.3 collapses. Since the paper advertises a 'self-contained overview,' this is a genuine gap, though not a fatal flaw for a survey: the result is stated precisely and is available in [137]. Second, the concentration compactness theorem, Theorem 2.10, is also only sketched, with the technical core deferred to [139]; again acceptable for a survey, but worth flagging if a reader wants to verify the chain from this paper alone. Third, there is a small typo after Theorem 1.3: 'Theorem 1.1 is stated only for p=2' should almost certainly read 'Theorem 1.3.' Also, in Theorem 1.4(iii) the first norm uses v_{a,b,z0} while the second uses u_{a,b,z0}; harmless but should be fixed.\n\nOn balance, the central claims are sound and already published; the survey is a fair and useful entry point. It deserves a serious referee, mostly to check the claims of self-containedness and catch the small errors.\n\nRecommendation: send it to review. A referee should ask for a few clarifying sentences about Theorem 2.6 and its precise role, but the paper is in good shape.","headline":"A useful survey of the author's own stability results, with a coherent proof sketch; the main soft spot is that the key Pólya–Szegő rigidity is quoted from an unpublished preprint rather than proved.","tokens_in":32679,"tokens_out":3639,"would_cite":false,"duration_ms":36323,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E35","53C21","53C23"],"pacs":[],"model":"deepseek-v4-flash","headline":"On manifolds with Ricci lower bounds, the sharp Sobolev inequality is stable: functions that almost attain it are close to explicit Euclidean or spherical bubble profiles, even when exact extremizers do not exist.","keywords":["Sobolev inequality","stability","Ricci curvature lower bounds","RCD spaces","Pólya-Szegő inequality","concentration compactness","Euclidean volume growth","extremal functions"],"falsifier":"Find one RCD(0,N) space with positive asymptotic volume ratio and a nonzero function attaining equality in the sharp Sobolev inequality (2.3) whose space is not a Euclidean cone and whose extremal is not radial; Theorem 2.7 would then be false. Since the proof of Theorem 1.3 uses exactly this rigidity on the limit space produced by concentration compactness, such an example would invalidate the stability claim.","tokens_in":31757,"feed_emoji":"📐","tokens_out":16879,"duration_ms":147600,"temperature":0.7,"pith_summary":"This review paper consolidates a stability program for sharp Sobolev inequalities under Ricci curvature lower bounds. The central claim is that almost attaining a sharp Sobolev constant forces a function to be close to the explicit bubble family, even on manifolds where no exact extremal function exists. On noncompact manifolds with nonnegative Ricci curvature and Euclidean volume growth, a near-extremizer is close in relative gradient $L^2$ distance to a Euclidean bubble. On closed manifolds with Ricci curvature at least $d-1$, the analogous statement holds with spherical bubbles, and the optimal Sobolev constant is nearly the sphere's value exactly when the diameter is nearly $\\pi$. The paper surveys the techniques behind these results and gives a self-contained overview of the proof in the noncompact case.","feed_headline":"Near-sharp Sobolev functions are bubbles on Ricci-bounded manifolds","feed_subtitle":"On manifolds with Ricci lower bounds, near equality in the sharp inequality forces the explicit bubble shape up to small error.","key_machinery":"Three tools carry the argument. The first is the synthetic RCD(0,N) class, meaning metric measure spaces with Ricci curvature bounded below by 0 and dimension bounded above by N, together with pointed measured Gromov-Hausdorff convergence: Mosco-convergence of Cheeger energies makes Sobolev constants stable under limits of spaces (Lemma 2.5). The second is a fine Pólya-Szegő rearrangement inequality on RCD(0,N) spaces with positive asymptotic volume ratio (Theorem 2.6): the gradient $L^p$ norm of a function controls the gradient of its decreasing rearrangement, and equality forces the space to be a Euclidean cone and the function to be radial about the tip. Combined with the one-dimensional Bliss inequality, this yields the sharp Sobolev inequality and its full equality case (Theorem 2.7). The third is a concentration-compactness principle for functions on varying spaces (Theorem 2.10) that rules out vanishing and dichotomy and yields a strong $L^{2*}$ limit. In the proof of Theorem 1.3, each manifold is rescaled so that a fixed fraction of the mass of $|u|^{2*}$ sits in the unit ball; the rescaled spaces and functions pass to a limit, the rigidity step identifies the limit profile, and a Euclidean bubble is scaled back to the original manifold.","core_discovery":"The central claim is that sharp Sobolev inequalities are stable on Riemannian manifolds with Ricci lower bounds. On a noncompact $d$-dimensional manifold with $\\mathrm{Ric}_g \\ge 0$ and asymptotic volume ratio above $V$, if a nonzero function $u$ nearly saturates $\\|u\\|_{L^{2*}(M)} \\le \\mathrm{AVR}(M)^{-1/d} S_{d,2} \\|\\nabla u\\|_{L^2(M)}$, then there are $a \\in \\mathbb{R}$, $b>0$, and $z_0 \\in M$ such that the relative gradient distance to the Euclidean bubble $u_{a,b,z_0}(x)=a(1+b\\,d_g(x,z_0)^2)^{(2-d)/2}$ is at most $\\varepsilon$ (Theorem 1.3). On a closed manifold with $\\mathrm{Ric}_g \\ge d-1$, the analogous claim holds with spherical bubbles, and the optimal Sobolev constant $A_{\\mathrm{opt}}(M)$ equals the sphere's value only when $M$ is isometric to the round sphere, with the quantitative bound $A(S^d)-A_{\\mathrm{opt}}(M) \\ge C_d(\\pi-\\mathrm{diam}(M))^d$ (Theorem 1.4). The paper presents a self-contained overview of the proof, which argues by contradiction: a near-extremizing sequence on varying spaces is rescaled and passed to a limit RCD space, concentration compactness produces a nonzero limit function attaining equality, and the rigidity of the sharp inequality identifies the limit as a Euclidean cone or spherical suspension with a bubble profile, which is then pulled back to the original manifolds.","pith_inferences":["A quantitative version of the noncompact stability theorem, with an explicit $\\delta(\\varepsilon,d,V)$, is not supplied by the contradiction proof; the paper itself lists this as an open problem, and the compactness method suggests that a new local analysis around bubbles would be needed.","Because the rigidity input is formulated on RCD spaces, the same proof scheme should transfer the stability conclusion from smooth manifolds to the non-smooth limit spaces themselves, not only to approximating manifolds.","One could test the theorem numerically on a Ricci-flat asymptotically locally Euclidean four-manifold, where the asymptotic volume ratio lies in $(0,1)$ and no extremizer exists, by computing the Sobolev quotient of rescaled bubble profiles and checking whether the deficit vanishes only along the explicit bubble family.","The quantitative diameter bound and the functional stability together suggest an almost-Obata statement: on closed manifolds with $\\mathrm{Ric}_g \\ge d-1$, a nearly maximal Sobolev constant should force the manifold to be close to the round sphere in a geometric sense, not just in diameter."],"forward_implications":["A near-extremizer on a noncompact manifold with nonnegative Ricci curvature and Euclidean volume growth cannot split into separate concentration pockets or escape to infinity: it must look like a single Euclidean bubble, up to gradient error $\\varepsilon$.","On manifolds with nonnegative Ricci curvature and asymptotic volume ratio in $(0,1)$, equality in the sharp Sobolev inequality is impossible, so the stability statement genuinely covers a regime where no exact extremizer exists.","On closed manifolds with $\\mathrm{Ric}_g \\ge d-1$, the deficit $A(S^d)-A_{\\mathrm{opt}}(M)$ is bounded below by a dimensional constant times $(\\pi-\\mathrm{diam}(M))^d$, so the optimal Sobolev constant is nearly spherical exactly when the diameter is nearly $\\pi$.","Almost-equality in the sphere-comparison Sobolev inequality on such closed manifolds forces the function to be close to a spherical bubble in the mixed $W^{1,2}$ plus $L^{2*}$ sense, as stated in Theorem 1.4(iii).","The same rearrangement machinery yields quantitative diameter-stability for the $p$-spectral gap, subcritical Sobolev constants, and the logarithmic Sobolev inequality under the same Ricci lower bound, as collected in Theorem 4.4."],"supporting_citations":[{"why":"This work derives the sharp Sobolev inequality (1.8) on spaces with nonnegative Ricci curvature and Euclidean volume growth, the inequality whose stability is the subject of the paper.","marker":"[26]"},{"why":"This work gives the one-dimensional sharp inequality used in the rearrangement proof of the sharp Sobolev inequality (Theorem 2.7).","marker":"[37]"},{"why":"This work supplies pointed measured convergence and Mosco-convergence of Cheeger energies, the calculus for passing to limit spaces along the contradiction proof.","marker":"[102]"},{"why":"This work characterizes when cones and suspensions over metric measure spaces satisfy the RCD condition, so the limit objects produced by rigidity belong to the same synthetic class.","marker":"[112]"},{"why":"This work supplies the locally compact concentration-compactness principle whose vanishing-dichotomy-compactness trichotomy is generalized to varying base spaces.","marker":"[121]"},{"why":"This work supplies the limit-case concentration-compactness principle that provides the template for extracting strong limits from near-extremal Sobolev sequences.","marker":"[122]"},{"why":"This work supplies the fine Pólya-Szegő rigidity theorem (Theorem 2.6) whose equality cases identify limit spaces as Euclidean cones and limit functions as radial bubbles.","marker":"[137]"},{"why":"This work establishes rigidity and almost-rigidity of Sobolev inequalities on compact spaces with lower Ricci bounds, feeding into Theorem 1.4 and the stability of Sobolev constants under convergence.","marker":"[138]"},{"why":"This work is the source of the main noncompact stability theorem (Theorem 1.3) and of the generalized concentration-compactness principle (Theorem 2.10) on varying RCD spaces.","marker":"[139]"}],"fun_headline_variants":["Stable Sobolev: near-extremizers are bubbles on Ricci-bounded spaces","Bubbles forced: near-equality in Sobolev implies explicit shape under Ricci lower bounds","Ricci lower bounds enforce bubble profile for near-optimal Sobolev functions","Sobolev stability on Ricci-bounded manifolds: near-extremizers are bubbles"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that equality in the Pólya-Szegő rearrangement inequality on an RCD(0,N) space with positive asymptotic volume ratio forces the space to be a Euclidean cone and the function to be radial about the tip; if that rigidity failed, the limit identification at the heart of the stability proof would no longer go through.","fun_headline_variants_meta":{"raw":{"variants":["Stable Sobolev: near-extremizers are bubbles on Ricci-bounded spaces","Bubbles forced: near-equality in Sobolev implies explicit shape under Ricci lower bounds","Ricci lower bounds enforce bubble profile for near-optimal Sobolev functions","Sobolev stability on Ricci-bounded manifolds: near-extremizers are bubbles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000286,"raw_usage":{"total_tokens":1704,"prompt_tokens":992,"completion_tokens":712,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":608,"completion_tokens_details":{"reasoning_tokens":620}},"tokens_in":608,"tokens_out":712,"duration_ms":7281,"temperature":1.0,"reasoning_tokens":620,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:10:46.409258+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find one RCD(0,N) space with positive asymptotic volume ratio and a nonzero function attaining equality in the sharp Sobolev inequality (2.3) whose space is not a Euclidean cone and whose extremal is not radial; Theorem 2.7 would then be false. Since the proof of Theorem 1.3 uses exactly this rigidity on the limit space produced by concentration compactness, such an example would invalidate the stability claim.","supporting_citations":[{"cited_title":"Gigli, A","cited_arxiv_id":null,"evidence_quote":"This work supplies pointed measured convergence and Mosco-convergence of Cheeger energies, the calculus for passing to limit spaces along the contradiction proof."},{"cited_title":"Ketterer, Cones over metric measure spaces and the maximal diameter theorem , J","cited_arxiv_id":null,"evidence_quote":"This work characterizes when cones and suspensions over metric measure spaces satisfy the RCD condition, so the limit objects produced by rigidity belong to the same synthetic class."},{"cited_title":"Lions, The concentration-compactness principle in the calculus of variations","cited_arxiv_id":null,"evidence_quote":"This work supplies the locally compact concentration-compactness principle whose vanishing-dichotomy-compactness trichotomy is generalized to varying base spaces."},{"cited_title":"The limit case","cited_arxiv_id":null,"evidence_quote":"This work supplies the limit-case concentration-compactness principle that provides the template for extracting strong limits from near-extremal Sobolev sequences."},{"cited_title":"Nobili and I","cited_arxiv_id":null,"evidence_quote":"This work establishes rigidity and almost-rigidity of Sobolev inequalities on compact spaces with lower Ricci bounds, feeding into Theorem 1.4 and the stability of Sobolev constants under convergence."},{"cited_title":"Math., 440 (2024), p","cited_arxiv_id":null,"evidence_quote":"This work is the source of the main noncompact stability theorem (Theorem 1.3) and of the generalized concentration-compactness principle (Theorem 2.10) on varying RCD spaces."}],"review_version":1}