{"id":"410fc4cc-b7aa-4968-9c1a-158ebd4d0002","arxiv_id":"1908.05410","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Bounded-curvature, bounded-diameter immortal Ricci flows admit only orbifold limits, and type-III flows with sub-logarithmic mu_+ entropy growth are volume non-collapsed with negative Einstein blowdown limits.","lead":"An immortal Ricci flow on a closed manifold, if it keeps curvature and diameter bounded forever, has time slices that sub-converge to a compact Riemannian orbifold. For type-III flows with t^(1/2) diameter growth, an entropy-growth condition t*mu_+'(t) tends to 0 excludes volume collapse, so blowdown limits are compact negative Einstein manifolds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.2 relies on unpublished estimates for conjugate heat densities; without them the limit steady-soliton equation, and hence Theorem 1.1, is unsupported.","rationale":"The reader's weakest_assumption identifies the same point I find decisive. I traced the main line of Theorem 1.1: collapsing time slices, unwrapped neighborhoods (Theorem 1.4), integral convergence (Theorem 1.5), vanishing F' (Lemma 6.1), the resulting steady soliton equation (Proposition 6.2), and the O'Neill maximum principle in Section 6.2. The only step that imports a black box into the Ricci-flow-specific argument is the C^0/C^1 control of the conjugate heat density on collapsing slices. Proposition 6.2 states the needed bounds are obtained by applying [31, Props 5.1 and 5.3] and [47, Thm 3.3], but [31] is a separate unpublished note not included in the submission, so the logical chain cannot be checked by the reader. This is not an accusation of error; the estimates may well be true and applicable. The concern is that they are load-bearing and unverified. For Theorem 1.3, Proposition 6.3 is more self-contained: it derives the analogous bounds from the Euler-Lagrange equation for the mu_+ minimizer, (6.4)-(6.7), using Cheng-Yau gradient estimates. Thus the concern does not undermine both main results equally, which is why I keep the reader's CONDITIONAL verdict rather than moving to REJECT. A complete derivation of the two estimates, or a reduction to a named published theorem, would remove the objection.","tokens_in":51387,"tokens_out":11456,"duration_ms":121822,"concrete_test":"Independently re-derive the two estimates used in Proposition 6.2 from [31, Props 5.1 and 5.3] for the conjugate heat solution u(t) constructed in Section 2.3, under only (1.2): verify sup_t |ln(|M|_{g(t)} u(t))| <= C and sup_t |M|_{g(t)} |grad u(t)| <= C with constants depending only on m, g(0), the curvature bound, and D. Concretely, track the constants in the parabolic Harnack and heat-kernel bounds on the cylinder M x (t_i - 1, t_i + 1] and check whether any estimate uses an a priori lower bound on the volume ratio or injectivity radius. If such a bound is used, add it as an explicit hypothesis in Theorem 1.1; if not, provide the derivation in a revised version so the citation is no longer load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 uses Proposition 6.2 to obtain a gradient steady Ricci soliton limit metric on the unwrapped neighborhoods. This requires feeding rho_i = u(t_i) into the integral-convergence Theorem 1.5, whose hypotheses in Proposition 4.1, (4.1), are sup |ln(rho_i |M|_{g(t_i)})| <= C and sup |grad rho_i| |M|_{g(t_i)} <= C. The paper asserts these from the author's unpublished notes [31, Props 5.1 and 5.3] plus [47, Thm 3.3], under only the uniform curvature-diameter bound (1.2). This is exactly the collapsing regime: no injectivity radius or volume-ratio lower bound is available. If the [31] estimates require any additional hypothesis, such as a non-collapsing condition, a fixed basepoint away from a singular time, or a lower diameter bound, then w_X identically zero does not follow, the integrand identity in Proposition 6.2 breaks, and equation (6.2) has no proof. The subsequent maximum-principle exclusion of corner singularities in Section 6.2 depends entirely on that equation, so Theorem 1.1 would not be established. Proposition 6.3 for Theorem 1.3 derives the corresponding rho_i bounds self-containedly from the mu_+ minimizer, so this concern is concentrated on Theorem 1.1, one of the paper's two advertised central results.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies long-time limits of immortal Ricci flows on closed m-dimensional manifolds (m >= 3) satisfying the uniform curvature-diameter bound (1.2). Theorem 1.1 asserts that if curvature and diameter stay uniformly bounded in time, then every unbounded sequence of time slices has a subsequence converging to a compact Riemannian orbifold. Theorem 1.3 asserts that for a type-III flow with diam(M, g(t)) = O(t^(1/2)), the condition lim sup_{t→∞} t mu_+'(t) = 0 forces a uniform positive lower bound on the global volume ratio, i.e. non-collapsing; the abstract and introduction add that any blowdown limit is an m-dimensional negative Einstein manifold. The proofs introduce local tools: unwrapped neighborhoods around orbifold points (Theorem 1.4), a collapsing-and-integral-convergence theorem (Theorem 1.5), and a limit central density whose zero locus is the corner singular set (Theorem 1.6). The main results are then derived by showing that the limit metric on the unwrapped neighborhoods satisfies a gradient steady or expanding Ricci soliton equation, and by applying a maximum principle to the central density to exclude corner singularities.","tokens_in":51580,"tokens_out":6236,"duration_ms":60360,"significance":"If correct, the paper would establish a strong structural rigidity statement: long-time limits of controlled immortal Ricci flows are orbifolds, and the entropy condition prevents volume collapsing in a natural type-III class. The local structural theorems (1.4 through 1.6) are deliberately formulated without the global frame-bundle construction, so they may be reusable in other collapsing problems; the short proof of Rong's theorem in Section 7 is a nice illustration. The paper is honest about which parts are standard and which parts are imported, and the overall proof strategy is coherent. However, the main results currently rest on estimates imported from an unpublished preprint, the proof of one advertised conclusion (negative Einstein blowdown) is not actually supplied, and the construction of the central density contains a key estimate that is only sketched. These issues must be resolved before the central claims are fully supported.","major_comments":[{"comment":"The proof of Proposition 6.2 obtains the uniform bounds sup |ln(rho_i |M|)| <= C and sup |grad rho_i| |M| <= C from [31, Props 5.1 and 5.3] and [47, Thm 3.3], but those propositions are not stated, [31] is the author's unpublished preprint, and the setting is exactly the collapsing regime, where no injectivity radius or volume-ratio lower bound is available. If the quoted estimates require any additional hypothesis beyond (1.2), then the equality w_X = 0 in the proof of Proposition 6.2 does not follow, equation (6.2) (the limit gradient steady soliton equation) is unsupported, and the maximum-principle step in Section 6.2 that rules out corner singularities has no basis. This is load-bearing for Theorem 1.1.","section":"Section 6.1, Proposition 6.2"},{"comment":"The abstract and the introduction (page 3) assert that under lim sup_{t→∞} t mu_+'(t) = 0 any blowdown limit is an m-dimensional negative Einstein manifold, citing [10], [26], [41] and [16] en bloc. However, Theorem 1.3 as proved in Section 6.2 only establishes the non-collapsing inequality (1.3); after the contradiction argument the text does not derive the additional soliton rigidity that would identify the blowdown limit as Einstein. Either supply a precise statement and proof of the rigidity step with a named theorem, or weaken the advertised conclusion to the non-collapsing statement (1.3).","section":"Sections 1 and 6.2"},{"comment":"The uniform second fundamental form bound sup |II_{T_i}(z)| <= C_{5.5} is asserted via a sketch combining (5.3) with the flatness of the connection along the T_i orbits, but the actual estimate is not written out and the dependence on the regularity constants is not tracked. This bound is needed to control the sectional curvature of (F M_i/T_i, [\\bar g^1_i]) and hence to define the positive continuous density \\bar chi_Q that underpins Theorem 1.6 and the subsequent maximum-principle argument. Please provide a complete proof with explicit constants.","section":"Section 5.1, inequality (5.5)"}],"minor_comments":[{"comment":"There are many typos and misspellings, e.g. 'relavent', 'correpsonding', 'fiberation', 'comapct', 'Thoerem', and 'functioanl'; the manuscript needs a careful proofreading pass.","section":"Throughout"},{"comment":"The commutative diagrams (2.8), (3.8) and (5.8) are not legible in the current PDF because of the ASCII-style formatting; they should be typeset properly.","section":"Figures and diagrams"},{"comment":"The phrase 'ln(det G)−1/2 u∞-harmonic function' is confusing; please clarify the operator and the meaning of this terminology.","section":"Introduction, after (1.6)"},{"comment":"Since [31] supplies central estimates for Proposition 6.2, please state the exact hypotheses and results used from [31] (or include them in an appendix), so that the reader can verify that they apply under only the assumptions of Theorem 1.1.","section":"Section 2.3 and references"},{"comment":"The notation for the averaged density is inconsistent: after introducing rho_{X,i}(x), the proof later refers to 'rho_{X,i}' without the argument in the estimates following (4.2), and the notational dependence on i should be made explicit throughout.","section":"Section 4.1, proof of Proposition 4.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript currently rests at a central point on the author's own unpublished preprint [31], and the referee could not verify that the hypotheses of [31, Props 5.1 and 5.3] cover the collapsing regime used in Proposition 6.2. There is also a mismatch between the abstract's strong claim about negative Einstein blowdown limits and the statement actually proved in Section 6.2. These are fixable in principle, but they require substantial additional material or a weakening of the claims. If the author supplies the missing estimates and a precise proof of the soliton rigidity, the paper would be a solid contribution to the field."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This preprint is worth your time, but read it with the dependencies in mind. The two main theorems are genuinely new statements. Theorem 1.1 — immortal Ricci flow with uniformly bounded curvature and diameter subconverges to a compact Riemannian orbifold — is a strong structural result, and Theorem 1.3 gives a μ_+-based entropy criterion forcing non-collapsing and negative-Einstein blowdowns for type-III flows. If correct, both are significant. The architecture is coherent: collapse → unwrap → integral convergence → limit soliton → maximum principle → orbifold; the F/μ_+ derivative vanishing is the engine, and Sections 3–5 build a local toolbox (Theorems 1.4–1.6) that is largely proven in the text and should be reusable.\n\nNow the soft spots, proportionally. The main one is Proposition 6.2, which for Theorem 1.1 imports uniform control of the conjugate heat densities ρ_i = u(t_i) — specifically |M| ρ_i ∈ [C^{-1}, C] and |M| |∇ρ_i| ≤ C — from the same author's arXiv preprint [31, Props 5.1, 5.3] under only the collapsed regime bound (1.2). This is exactly the setting with no injectivity radius and no volume-ratio lower bound. If those estimates fail or require extra hypotheses, w_X ≡ 0 does not follow, (6.2) has no basis, and Theorem 1.1's corner-singularity exclusion collapses. It is not an obvious error, but a referee must verify [31] or require a self-contained proof. Theorem 1.3's proof, by contrast, is mostly self-contained: Proposition 6.3 derives the analogous bounds from the Euler-Lagrange equation for the μ_+ minimizer. The dependency is concentrated on Theorem 1.1.\n\nTwo smaller items. First, the abstract claims the blowdown limit is negative Einstein, but the proof of Theorem 1.3 only establishes non-collapsing; the negative-Einstein step is asserted via an en-bloc citation to [10, 26, 41, 16], with no named theorem or argument. Second, the second fundamental form bound (5.5) for torus orbits is sketched in a paragraph, not proven; likely fixable, but it is a gap.\n\nWho should read it: people working in collapsing geometry, long-time Ricci flow, and entropy rigidity. It deserves peer review, not a desk reject. If I were an editor I'd send it to a strong geometer and ask them to focus on the [31] estimates and on pinning down the expander rigidity citation.\n\nRecommendation: engage with it, conditionally. If the dependencies check out, this is a solid, useful paper.","headline":"Serious preprint; Theorem 1.1's proof hinges on the author's own unpublished heat-density estimates, and the negative-Einstein claim in Theorem 1.3 is cited en bloc rather than proven.","tokens_in":52209,"tokens_out":11034,"would_cite":true,"duration_ms":98053,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53E20","53C23"],"pacs":[],"model":"deepseek-v4-flash","headline":"Immortal Ricci flows with bounded curvature and diameter sub-converge to compact Riemannian orbifolds, and type-III flows with $t\\mu_+'(t)\\to 0$ are non-collapsing with negative Einstein blowdown limits.","keywords":["immortal Ricci flow","type-III Ricci flow","collapsing geometry","orbifold limits","entropy functionals","negative Einstein manifold","non-collapsing","Gromov-Hausdorff convergence"],"falsifier":"Construct an immortal Ricci flow on a closed $m$-manifold ($m\\ge3$) satisfying the uniform curvature-diameter bound, with bounded curvature and diameter, such that some sequence $(M,g(t_i))$ converges in the Gromov-Hausdorff sense to a space containing a corner singularity—for instance a limit interval where the generic fiber is a 2-torus and over some points the fiber is a circle; such a limit would refute Theorem 1.1. Alternatively, to test Theorem 1.3, compute $t\\mu_+'(t)$ along a type-III flow with $\\operatorname{diam}(M,g(t))=O(t^{1/2})$ whose global volume ratio tends to zero; if $t\\mu_+'(t)\\to0$, the non-collapsing conclusion is contradicted.","tokens_in":51082,"feed_emoji":"📐","tokens_out":12149,"duration_ms":109427,"temperature":0.7,"pith_summary":"The paper asks what happens to an immortal Ricci flow—a solution defined for all positive time on a closed manifold—as time goes to infinity, under the uniform curvature-diameter bound $\\operatorname{diam}(M,g(t))^2\\sup_M|\\operatorname{Rm}|\\le D^2$. It establishes that any unbounded sequence of time slices sub-converges, in the Gromov-Hausdorff sense, to a compact Riemannian orbifold: volume collapsing can occur, but only down to orbifold singularities, never to the corner singularities that appear in arbitrary collapsing sequences. For type-III flows whose diameter grows at most like $t^{1/2}$, it proves a second result: if the derivative of the $\\mu_+$-entropy satisfies $\\limsup_{t\\to\\infty} t\\mu_+'(t)=0$, the global volume ratio cannot degenerate, and every blowdown limit is an $m$-dimensional negative Einstein manifold. A sympathetic reader would care because these are structural statements about long-time limits of immortal Ricci flows that do not assume static or soliton data from the start.","feed_headline":"Immortal Ricci flows: limits are orbifolds, not corners","feed_subtitle":"When curvature and diameter stay bounded, long-time limits are compact orbifolds; entropy decay stops volume collapse.","key_machinery":"The load-bearing object is the unwrapped neighborhood $W_{x_0}$ around an orbifold point $x_0$ of the collapsing limit. Passing to the fiberwise universal cover removes the nilpotent lattice actions, gives a uniform injectivity radius, and produces a limit metric $\\tilde g_\\infty$ on $W_{x_0}$ together with a Riemannian submersion $p:(W_{x_0},\\tilde g_\\infty)\\to(V_{x_0},\\hat g_X)$ whose fibers are simply connected nilpotent Lie groups; the commuting family of Killing fields tangent to the fibers, the limit central distribution $\\mathcal C$, has volume density $\\sqrt{\\det H}$ that locally represents $\\chi_C$. The asymptotic vanishing of the functional derivatives yields the gradient steady or expanding soliton equation for $\\tilde g_\\infty$, and the submersion curvature identities convert it into an elliptic equation for $\\log\\det H$ whose structure supports the maximum-principle argument: the maximum of $\\chi_C$ lies in the orbifold part, and the equation forces $\\chi_C$ to be constant, eliminating corner singularities.","core_discovery":"The central discovery is that Ricci flow evolution itself is strong enough to rule out the worst collapsing singularity type. When a sequence of time slices $(M,g(t_i))$ collapses to a lower-dimensional limit $(X,d)$, the a priori limit is an orbifold with corners; the corner set $\\tilde S$ is characterized as the zero locus of a limit central density $\\chi_C$. The paper shows that along an immortal Ricci flow the asymptotic vanishing of the $F$-entropy derivative forces the limit metric on unwrapped neighborhoods to satisfy the gradient steady soliton equation, hence a submersion-curvature elliptic equation for $\\log\\det H$ with nonnegative right-hand side; the maximum principle then forces $\\chi_C$ to be constant, so $\\tilde S$ is empty. In the type-III case, the analogous use of the $W_+$-entropy and the hypothesis $t\\mu_+'(t)\\to0$ produces a gradient expanding soliton equation with an extra positive term $\\dim Z$; the maximum principle forces $\\dim Z=0$, contradicting collapse of the global volume ratio and yielding the non-collapsing conclusion.","pith_inferences":["If the same maximum-principle and elliptic-equation mechanism extends to other geometric flows possessing monotone entropy functionals, one would expect analogous orbifold-only compactness for their long-time limits.","A testable weakening of Theorem 1.3 would replace $\\limsup t\\mu_+'(t)=0$ by an integrability condition such as $\\int^\\infty t\\mu_+'(t)\\,dt<\\infty$; the proof only needs a sequence of times along which the limit integrand vanishes, so the dichotomy may persist under milder decay.","In dimension three, combining this orbifold-limit statement with known long-time behavior could further restrict which geometric model pieces an immortal Ricci flow can collapse to, with orbifold points corresponding to quotients of the pieces.","The local tools (unwrapped neighborhoods, integral convergence, central density) are formulated for arbitrary collapsing sequences with bounded curvature and diameter, so they may be useful for measured Gromov-Hausdorff limits of other families with no Ricci flow structure present."],"forward_implications":["Under the assumptions of Theorem 1.1, every unbounded sequence of time slices has a subsequence converging to a compact Riemannian orbifold, so the long-time limit singular set can only consist of orbifold points.","If a time slice is sufficiently collapsed along such a flow, the underlying manifold is an infranil fiber bundle over a compact lower-dimensional orbifold, equivalently a manifold carrying a pure polarized $F$-structure.","For type-III flows with diameter growth $O(t^{1/2})$ and $t\\mu_+'(t)\\to0$, the global volume ratio $|M|_{g(t)}\\operatorname{diam}(M,g(t))^{-m}$ has a positive lower bound, so the flow does not asymptotically collapse.","Any blowdown limit of such a type-III flow is an $m$-dimensional negative Einstein manifold.","The theorem gives a Ricci-flow counterpart of a classical volume non-collapsing theorem for negatively Ricci-curved manifolds with bounded curvature and diameter."],"supporting_citations":[{"why":"Supplies the uniform conjugate heat density and gradient bounds used in Proposition 6.2 to pass from vanishing $F'$ to the limit soliton equation.","marker":"[31]"},{"why":"Describes the local structure of collapsing limits as quotients by nilpotent groups and identifies corner singularities; it is the basis for the orbifold-versus-corner dichotomy.","marker":"[19]"},{"why":"Constructs the nilpotent structure and invariant metrics used to unwrap fibers and to form the limit central distribution.","marker":"[11]"},{"why":"Introduces measured Gromov-Hausdorff convergence and the limit density $\\chi_X$; it underlies the integral convergence theorems.","marker":"[17]"},{"why":"Extends the fiber-bundle and density constructions over orbifold points; it supports Corollary 1.2 and the local representation of $\\chi_C$.","marker":"[20]"},{"why":"Defines the $W_+$- and $\\mu_+$-functionals and their derivative formula; it supplies the entropy-decay hypothesis in Theorem 1.3.","marker":"[16]"},{"why":"Provides the maximum-principle elliptic-equation strategy for eliminating corner singularities, adapted here to Ricci flow limits.","marker":"[39]"},{"why":"Used together with the compactness and entropy references to conclude from non-collapsing that blowdown limits are negative Einstein manifolds.","marker":"[10]"},{"why":"A compactness theorem for Ricci flows used to obtain the local convergence at the rescaled time slices.","marker":"[26]"},{"why":"The $F$-entropy and its monotonicity give the asymptotic vanishing of $F'$ in Lemma 6.1.","marker":"[41]"}],"fun_headline_variants":["Ricci flow: orbifold limits, no corner collapses","Type-III Ricci blowdowns are negative Einstein","Curvature and diameter bounds give orbifold limits","Entropy decay kills corner collapse in Ricci flow","Long-time Ricci flow: orbifold or negative Einstein"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on uniform bounds for the volume-normalized conjugate heat densities $\\rho_i=u(t_i)$ on the collapsing time slices—$|M|_{g(t_i)}u(t_i)$ between $C^{-1}$ and $C$ and $|M|_{g(t_i)}|\\nabla\\rho_i|\\le C$—estimates imported from the paper's own unpublished companion preprint [31, Props. 5.1 and 5.3]; if those fail, the limit integrand $w_X$ need not vanish and the gradient soliton equation on the unwrapped neighborhoods does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Ricci flow: orbifold limits, no corner collapses","Type-III Ricci blowdowns are negative Einstein","Curvature and diameter bounds give orbifold limits","Entropy decay kills corner collapse in Ricci flow","Long-time Ricci flow: orbifold or negative Einstein"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000932,"raw_usage":{"total_tokens":3955,"prompt_tokens":879,"completion_tokens":3076,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":2998}},"tokens_in":495,"tokens_out":3076,"duration_ms":22417,"temperature":1.0,"reasoning_tokens":2998,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:17:22.701973+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an immortal Ricci flow on a closed $m$-manifold ($m\\ge3$) satisfying the uniform curvature-diameter bound, with bounded curvature and diameter, such that some sequence $(M,g(t_i))$ converges in the Gromov-Hausdorff sense to a space containing a corner singularity—for instance a limit interval where the generic fiber is a 2-torus and over some points the fiber is a circle; such a limit would refute Theorem 1.1. Alternatively, to test Theorem 1.3, compute $t\\mu_+'(t)$ along a type-III flow with $\\operatorname{diam}(M,g(t))=O(t^{1/2})$ whose global volume ratio tends to zero; if $t\\mu_+'(t)\\to0$, the non-collapsing conclusion is contradicted.","supporting_citations":[{"cited_title":"Improving Subseasonal Forecasting in the Western U.S. with Machine Learning","cited_arxiv_id":"1809.07394","evidence_quote":"Supplies the uniform conjugate heat density and gradient bounds used in Proposition 6.2 to pass from vanishing $F'$ to the limit soliton equation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Describes the local structure of collapsing limits as quotients by nilpotent groups and identifies corner singularities; it is the basis for the orbifold-versus-corner dichotomy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the nilpotent structure and invariant metrics used to unwrap fibers and to form the limit central distribution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces measured Gromov-Hausdorff convergence and the limit density $\\chi_X$; it underlies the integral convergence theorems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the fiber-bundle and density constructions over orbifold points; it supports Corollary 1.2 and the local representation of $\\chi_C$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the $W_+$- and $\\mu_+$-functionals and their derivative formula; it supplies the entropy-decay hypothesis in Theorem 1.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the maximum-principle elliptic-equation strategy for eliminating corner singularities, adapted here to Ricci flow limits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Used together with the compactness and entropy references to conclude from non-collapsing that blowdown limits are negative Einstein manifolds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"A compactness theorem for Ricci flows used to obtain the local convergence at the rescaled time slices."},{"cited_title":"Preprint, arXiv: math / 0211159","cited_arxiv_id":null,"evidence_quote":"The $F$-entropy and its monotonicity give the asymptotic vanishing of $F'$ in Lemma 6.1."}],"review_version":1}