{"id":"5783ce0f-bd1c-4d38-9950-3106534e5d11","arxiv_id":"1908.03147","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Porous medium flows on manifolds with Ric ≥ -K satisfy a Wasserstein stability bound with a sharp time-dependent exponential factor.","lead":"On curved spaces with Ricci curvature bounded below, the authors prove that porous-medium flows spread apart in the Wasserstein distance at a controlled rate, and they show the rate is sharp on hyperbolic space. Read it for the first quantitative stability estimates for nonlinear diffusion beyond nonnegative curvature.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"As written, Lemma 5.5 defines ψ_{s,t} with the wrong sign of the exponential, so the central equality (5.15) is algebraically false; the exponent must be negative.","rationale":"The reader's verdict is already CONDITIONAL and its rationale identifies the same sign issue; however, the reader's stated weakest assumption points to the external comparison (5.47) used only for optimality. In my read, the most load-bearing obstruction to the central upper-bound claim is the inconsistent definition of ψ_{s,t} in Lemma 5.5, because without the negative exponent the displayed algebra leading to (5.14) fails. This is an internal inconsistency rather than an unsupported citation, and it is directly testable by substitution. The optimality comparison (5.47) is cited to [21, Remark 2.12] and [38] and, even if it were revisited, it would not affect the upper-bound theorem; it only sharpens the exponent. I therefore agree with the CONDITIONAL verdict: accept once the sign is corrected and the few related presentation issues, such as Remark 2.6, are tightened.","tokens_in":97,"tokens_out":22784,"duration_ms":307049,"concrete_test":"Recompute the displayed string after (5.15) in Lemma 5.5 twice: once with ψ_{s,t} = e^{+A}φs(0) as printed and once with ψ_{s,t} = e^{-A}φs(0), keeping A the same. The identity e^A(-1/2 E(ψ) + ⟨ψ⟩) = -1/2 e^{-A}E(φ) + ⟨φ⟩ holds only for the negative exponent, which settles whether the sign is a typo. If confirmed, the fix is local and Theorem 2.4 goes through after correcting Lemma 5.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The upper-bound proof of Theorem 2.4 turns on Lemma 5.5. There, after applying Lemma 5.3 one obtains, with A = 2Kc1Cm[(tM^{m-1})^{2/(2+n(m-1))} ∨ (tM^{m-1}) + εt/(c1Cm)], the bracket -1/2 e^{-A} E_{ρs0}[φs(0)] + ⟨∂sρs0, φs(0)⟩. To factor e^A out of this expression, the new potential must be ψ_{s,t} = e^{-A}φs(0), because E is quadratic: e^A[-1/2 E(e^{-A}φ) + ⟨e^{-A}φ⟩] = -1/2 e^{-A}E(φ) + ⟨φ⟩. The manuscript instead defines ψ_{s,t} := e^{+A}φs(0) in Lemma 5.5, display following (5.15). With that definition, the right side becomes -1/2 e^{3A}E(φ) + e^{2A}⟨φ⟩, which does not equal the preceding bracket. The subsequent use of the Fenchel dual E* is therefore not justified with the printed definition. Since this is the step converting the smoothed Hamiltonian lower bound into the Wasserstein estimate (5.14)-(5.20), the proof of (2.6) as written contains a genuine algebraic gap. The error is almost certainly a sign typo, but it must be corrected before the displayed derivation is sound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Cauchy problem for porous medium-type equations ∂_t ρ = ΔP(ρ) on complete Riemannian manifolds with Ricci curvature bounded below (Ric ≥ -K) and a Sobolev inequality. The main result (Theorem 2.4) establishes a quantitative 2-Wasserstein stability estimate: for two solutions with initial data of mass M, W_2(ρ(t), ρ̂(t)) ≤ exp{K c_1 C_m[(tM^{m-1})^{2/(2+n(m-1))} ∨ (tM^{m-1})]} W_2(µ_0, µ̂_0). The proof combines the Hamiltonian/Eulerian approach of Ambrosio–Mondino–Savaré with a new quantitative L^1–L^∞ smoothing estimate obtained by Moser iteration, and a compact-support property. Theorem 2.5 shows that, in hyperbolic space, the time exponent in the estimate is sharp for small times, and consequently the PME cannot be a gradient flow of a λ-convex functional in the EVI sense. The paper also provides existence, uniqueness, regularity, moment propagation, and continuity properties of the solutions.","tokens_in":44681,"tokens_out":10103,"duration_ms":92646,"significance":"If the results are correct, they constitute a substantial advance: they provide the first quantitative Wasserstein contraction-type estimate for nonlinear diffusion on manifolds beyond the Ric ≥ 0 regime, with explicit constants and a sharp small-time exponent. The proof is largely self-contained, including the smoothing estimate and the compact-support argument, and the treatment of both noncompact and compact manifolds is careful. The optimality result is a valuable contribution, showing that the PME does not admit a gradient-flow/EVI structure on negatively curved manifolds. The paper also gives credit to previous work and clearly delineates the external inputs (e.g., the hyperbolic-space comparison (5.47)).","major_comments":[{"comment":"The definition of ψ_{s,t} has the wrong sign of the exponential, and this makes the displayed equality in (5.15) algebraically false. Setting A = 2K c_1 C_m[(tM^{m-1})^{2/(2+n(m-1))} ∨ (tM^{m-1}) + ε t/(c_1 C_m)], the bracket on the preceding line is -1/2 e^{-A} E_{ρ^s_0}[φ_s(0)] + ⟨d/ds ρ^s_0, φ_s(0)⟩. To factor out e^A and obtain an expression of the form -1/2 E[ψ] + ⟨·, ψ⟩, one must set ψ_{s,t} = e^{-A} φ_s(0), because E is quadratic in its argument. With the printed definition ψ_{s,t} = e^{A} φ_s(0), the resulting integrand becomes e^A[-1/2 e^{2A}E[φ_s(0)] + e^{A}⟨·,φ_s(0)⟩] = -1/2 e^{3A}E[φ_s(0)] + e^{2A}⟨·,φ_s(0)⟩, which does not equal the original bracket and is not bounded by the claimed Fenchel-dual term. Since this step is essential for passing from the smoothed Hamiltonian lower bound to the Wasserstein estimate (5.14)–(5.20), the proof of (2.6) as written contains a genuine algebraic gap. The error appears to be a sign typo, but it must be corrected and the subsequent estimate rechecked before the proof is sound.","section":"§5.2, Lemma 5.5, display following (5.15)"}],"minor_comments":[{"comment":"There are several typos, e.g., 'por ous' and 'W ASSERSTEIN ST ABILITY' in the title; the manuscript would benefit from a careful proofreading pass.","section":"Abstract and title"},{"comment":"The notation 'Cm := Cm−1 2m−2 [2 + n(m − 1)]' is ambiguous; it should be typeset as C^{m-1} 2^{m-2} [2 + n(m-1)] to avoid confusion.","section":"Theorem 2.4 and Lemma 5.3"},{"comment":"The sharpness result relies crucially on the comparison (5.47) with the Euclidean Barenblatt profile, imported from [21, Remark 2.12] and [38]. Since the lower bound (5.50) depends on this comparison, it would be helpful to state explicitly the conditions under which (5.47) is known to hold and whether the constants D and k depend only on n and m.","section":"§5.4, proof of Theorem 2.5"},{"comment":"The discussion of the spaces V'_E and D'_E is rather brief; adding precise references to the relevant statements in [4] would make the compact-case argument easier to verify.","section":"§5.3, compact case"},{"comment":"The notation 'E := exp_x v^⊥' is a slight abuse; it would be clearer to write 'E := exp_x(v^⊥)' or 'E := {exp_x w : w ∈ v^⊥}'.","section":"§5.4, Lemma 5.6"},{"comment":"In inequality (5.46), the constant '3κ' appears without explanation; it would be clearer to introduce a new constant (e.g., κ') at that point.","section":"§5.4, proof of Lemma 5.7"}],"recommendation":"major_revision","confidential_remarks":"The sign error in Lemma 5.5 is the only load-bearing technical issue I found; it is almost certainly a typo and is easily corrected. After that correction, the main stability proof appears coherent. The optimality theorem depends on the external comparison (5.47), so I recommend the authors verify that this comparison indeed holds for the full range of parameters and times used in the proof. Overall, the paper is a solid and substantial contribution, and I expect it to be acceptable after the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper proves a quantitative W2 stability bound for porous medium-type equations on complete manifolds with Ric ≥ −K, with an explicit time-dependent exponent, and shows sharpness on hyperbolic space. That is a real advance over the known Ric ≥ 0 contraction results of Sturm and Otto–Westdickenberg, and the combination of L1–L∞ smoothing with the Hamiltonian approach is well suited to the problem.\n\nThe paper does a lot well. The smoothing estimate is proved self-containedly via Moser iteration, with careful control of constants as m ↓ 1. The use of the Ambrosio–Mondino–Savaré machinery is appropriate, and the optimality construction in hyperbolic space is clever, turning an Ollivier-type geodesic expansion plus Barenblatt comparison into a matching lower bound. The main theorem is new and the sharpness example is convincing.\n\nThe soft spot is exactly where the reader put it: Lemma 5.5. The displayed definition of ψ_{s,t} has the wrong sign in the exponential. With ψ = e^{+A}φ, the equality after (5.15) does not hold; you need ψ = e^{-A}φ for the E_\rho factor to cancel. I checked the algebra and the stress-test note is right. This is almost certainly a sign typo, but it sits in the load-bearing step that converts the smoothed Hamiltonian lower bound into the Wasserstein estimate, so as printed the proof of Theorem 2.4 has a genuine gap. It should be an easy fix.\n\nTwo smaller things are worth mentioning. First, the optimality proof relies on the comparison (5.47) between the hyperbolic-space PME solution and the Euclidean Barenblatt profile from [21] and [38]. That is an external result, but a credible one; the upper bound (2.6) does not depend on it. Second, Remark 2.6 states the non-gradient-flow conclusion for “general negatively-curved manifolds” when the proof only covers hyperbolic space. The logic is acceptable—if a universal gradient-flow structure existed it would hold on hyperbolic space—but the phrasing could be tightened.\n\nOverall: this is a solid paper with one correctable typo. I would send it to a serious referee. If the sign in Lemma 5.5 is fixed and the remark is reworded, I would expect it to be accepted.","headline":"A genuinely new Wasserstein stability estimate under negative Ricci bounds, but a sign typo in Lemma 5.5 currently breaks the main proof—fixable, and the paper deserves peer review.","tokens_in":45229,"tokens_out":3637,"would_cite":true,"duration_ms":36343,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K65","49Q22","53C21","58J35","35B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that on manifolds with Ricci curvature bounded below by $-K$, the Wasserstein distance between two porous medium-type solutions stays controlled by an explicit exponential factor, and that this estimate is sharp in…","keywords":["porous medium equation","Wasserstein distance","Ricci curvature bounds","stability estimates","hyperbolic space","smoothing effect","nonlinear diffusion","optimal transport"],"falsifier":"Compute, for a concrete case such as $n=3$, $m=2$, $M=1$ in $H^3_K$, the $W_2$ distance between solutions starting from two Dirac masses separated by a small $\\delta$, at times spanning several scales. The paper's lower bound predicts logarithmic growth with exponent $2/(2+n(m-1))=2/5$ in $t$; observing a different exponent, or a prefactor not proportional to $K$ as $t\\downarrow 0$, would refute the sharpness claim. For the upper bound, a direct check is to verify that the ratio $W_2(\\rho(t),\\hat\\rho(t))/W_2(\\mu_0,\\hat\\mu_0)$ never exceeds the stated exponential.","tokens_in":44160,"feed_emoji":"🌊","tokens_out":8847,"duration_ms":86924,"temperature":0.7,"pith_summary":"On a complete Riemannian manifold whose Ricci curvature is bounded below by $-K$, the paper proves a quantitative 2-Wasserstein stability estimate for porous medium-type equations: the distance between two solutions at time $t$ is controlled by the distance between their initial data times an explicit exponential factor. The exponent is $K c_1 C_m[(tM^{m-1})^{2/(2+n(m-1))} \\lor tM^{m-1}]$, so short times are governed by a nonlinear power and long times by a linear term. This extends the known contraction property for $K=0$ to arbitrary negative curvature bounds, for $m>1$ and general nonlinearities satisfying the McCann-type condition. A matching lower bound in hyperbolic space shows the short-time power is sharp and rules out a Wasserstein gradient-flow description of the porous medium equation on negatively curved manifolds.","feed_headline":"Porous-medium flows get a sharp Wasserstein stability bound","feed_subtitle":"On Ricci-negatively curved manifolds, two solutions drift apart at most exponentially in a nonlinear time power.","key_machinery":"The load-bearing mechanism is the Hamiltonian functional $E_{\\rho(t)}[\\varphi(t)] = \\int_{M^n} |\\nabla \\varphi(t)|^2 \\rho(t)\\, dV$, evaluated on a solution $\\rho$ and on a solution $\\varphi$ of the backward adjoint equation $\\partial_t \\varphi = -P'_\\varepsilon(\\rho)\\Delta\\varphi$. Along the flow it obeys the differential inequality $\\frac12 \\frac{d}{dt} E_{\\rho(t)}[\\varphi(t)] \\ge -K \\int \\Gamma(\\varphi(t)) P_\\varepsilon(\\rho(t))\\, dV$; the paper closes this inequality using the quantitative $L^1$–$L^\\infty$ smoothing estimate from Proposition 4.3, which bounds the sup norm by a power of $t^{-1}$ times the initial mass. The lower-bound proof uses a different geometric ingredient: the small-time comparison of the hyperbolic-space Barenblatt solution with the Euclidean Barenblatt profile, combined with a distance expansion for nearby spheres.","core_discovery":"The paper's central claim is that solutions of $\\partial_t \\rho = \\Delta P(\\rho)$ started from measures of mass $M$ on a complete manifold with $\\mathrm{Ric} \\ge -K$ and a Sobolev inequality satisfy the stability estimate (2.6), with the explicit exponential factor above. The same theorem gives the quantitative smoothing bound (2.5), $\\|\\rho(t)\\|_{L^\\infty} \\le C(t^{-n/(2+n(m-1))} M^{2/(2+n(m-1))} + M)$. The optimality theorem states that in hyperbolic space $H^n_K$, for two close Dirac initial data, the distance ratio is at least $1 + K\\kappa(tM^{m-1})^{2/(2+n(m-1))}$ for small times; hence the exponent cannot be improved, and the flow is not a gradient flow of a $\\lambda$-convex functional in the Wasserstein metric when $K>0$.","pith_inferences":["One extension left implicit: the smoothing-plus-Hamiltonian mechanism suggests the same kind of bound should hold for the fast-diffusion range $m\\in(0,1)$ up to the extinction time, but the compact-support and smoothing estimates would need a different treatment.","The optimality theorem is proved only in constant-curvature hyperbolic space; a natural test is whether the same small-time exponent is sharp on every Cartan-Hadamard manifold, where the Barenblatt comparison is also available.","The proof's reliance on PDE smoothing suggests that adapting the estimate to metric-measure spaces would require a quantitative regularization estimate in that setting; the paper singles this out as an obstruction."],"forward_implications":["For the model case $P(\\rho)=\\rho^m$, the bound gives an explicit quantitative replacement for the contraction that was previously available only when $\\mathrm{Ric}\\ge 0$.","Letting $m\\downarrow 1$ recovers the linear heat-flow estimate $W_2(\\rho(t),\\hat\\rho(t)) \\le e^{Kt} W_2(\\mu_0,\\hat\\mu_0)$.","The hyperbolic-space lower bound implies the porous medium equation is not, for $K>0$, a gradient flow of a $\\lambda$-convex energy with respect to $W_2$ in the evolutionary variational inequality sense.","Under a Euclidean Sobolev inequality in place of the lower-dimensional one, the long-time linear term disappears and the exponent reduces to just the power $(tM^{m-1})^{2/(2+n(m-1))}$.","The estimates cover a whole class of nonlinearities $P$ satisfying the growth conditions (H4)–(H5), not only the pure power law."],"supporting_citations":[{"why":"Provides the nonnegative-curvature contraction result that the present estimate extends.","marker":"[35]"},{"why":"Introduces the Eulerian calculus for Wasserstein contraction that underlies the proof.","marker":"[32]"},{"why":"Supplies the Hamiltonian, adjoint-equation, and variational-solution framework used to derive the estimate.","marker":"[4]"},{"why":"Provides the Moser-iteration smoothing scheme adapted to the nonlinearity.","marker":"[16]"},{"why":"Gives the hyperbolic-space Barenblatt solution and its scaling used in the lower bound.","marker":"[38]"},{"why":"Provides the comparison of hyperbolic porous-medium solutions with the Euclidean Barenblatt profile stated in (5.47).","marker":"[21]"},{"why":"Supplies the small-radius distance expansion in hyperbolic space used to build the Wasserstein lower bound.","marker":"[30]"}],"fun_headline_variants":["Sharp Wasserstein stability for porous medium flows on curved manifolds","Optimal Wasserstein exponent for porous media on Ricci-negative manifolds","Porous medium stability: sharp bounds and non-gradient-flow proof","Stability of porous medium equations on manifolds with Ricci lower bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sharpness result depends on the external fact that, in hyperbolic space, the porous-medium solution from a point mass stays below the Euclidean Barenblatt profile with the Euclidean time scale; the main stability upper bound does not rely on this comparison.","fun_headline_variants_meta":{"raw":{"variants":["Sharp Wasserstein stability for porous medium flows on curved manifolds","Optimal Wasserstein exponent for porous media on Ricci-negative manifolds","Porous medium stability: sharp bounds and non-gradient-flow proof","Stability of porous medium equations on manifolds with Ricci lower bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1372,"prompt_tokens":843,"completion_tokens":529,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":459,"completion_tokens_details":{"reasoning_tokens":456}},"tokens_in":459,"tokens_out":529,"duration_ms":6062,"temperature":1.0,"reasoning_tokens":456,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:24:32.308051+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a concrete case such as $n=3$, $m=2$, $M=1$ in $H^3_K$, the $W_2$ distance between solutions starting from two Dirac masses separated by a small $\\delta$, at times spanning several scales. The paper's lower bound predicts logarithmic growth with exponent $2/(2+n(m-1))=2/5$ in $t$; observing a different exponent, or a prefactor not proportional to $K$ as $t\\downarrow 0$, would refute the sharpness claim. For the upper bound, a direct check is to verify that the ratio $W_2(\\rho(t),\\hat\\rho(t))/W_2(\\mu_0,\\hat\\mu_0)$ never exceeds the stated exponential.","supporting_citations":[{"cited_title":"Sturm, Convex functionals of probability measures and nonlinear d iffusions on manifolds , J","cited_arxiv_id":null,"evidence_quote":"Provides the nonnegative-curvature contraction result that the present estimate extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Eulerian calculus for Wasserstein contraction that underlies the proof."},{"cited_title":"Fotache, M","cited_arxiv_id":null,"evidence_quote":"Provides the Moser-iteration smoothing scheme adapted to the nonlinearity."},{"cited_title":"Vázquez, Fundamental solution and long time behavior of the porous me dium equation in hyperbolic space , J","cited_arxiv_id":null,"evidence_quote":"Gives the hyperbolic-space Barenblatt solution and its scaling used in the lower bound."},{"cited_title":"Grillo, M","cited_arxiv_id":null,"evidence_quote":"Provides the comparison of hyperbolic porous-medium solutions with the Euclidean Barenblatt profile stated in (5.47)."},{"cited_title":"Ollivier, Ricci curvature of Markov chains on metric spaces , J","cited_arxiv_id":null,"evidence_quote":"Supplies the small-radius distance expansion in hyperbolic space used to build the Wasserstein lower bound."}],"review_version":1}