On Perelman's $W$-entropy and Shannon entropy power for super Ricci flows on metric measure spaces

Xiang-Dong Li

REVIEW 4 major objections 5 minor 61 references

On Perelman's $W$-entropy and Shannon entropy power for super Ricci flows on metric measure spaces

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Perelman's W-entropy monotonicity is extended to super Ricci flows on metric measure spaces.

desk verdict Perelman's W-entropy on synthetic super Ricci flows: the program is right, but the central theorem currently rests on an unstated Bochner identity and needs a major revision before the main claim is supported. read the letter →

arxiv 2505.03202 v1 pith:WZETJ5JV submitted 2025-05-06 math.DG math.MGmath.PR

Xiang-Dong Li This is my paper · ORCID

classification math.DGmath.MGmath.PR MSC 53C2353C2160J6060H30

keywords Li-Yau-Hamilton-PerelmanHarnackinequalityLog-SobolevPerelman'sW-entropyShannonentropypowersuperRicciflowsmetricmeasurespacesvolumenon-localcollapsingproperty

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that Perelman's W-entropy, a functional over solutions of the conjugate heat equation that is monotone on smooth Ricci flows, remains non-increasing on 'super Ricci flows' on metric measure spaces, i.e., spaces with synthetic Ricci curvature lower bounds. The main theorems give explicit dissipation formulas for the W-entropy on closed $(K,n,N)$-super Ricci flows and derive the Shannon entropy power concavity, a Li-Yau-Hamilton-Perelman differential Harnack inequality, an equivalence between volume non-local collapsing and lower boundedness of the W-entropy, and optimal log-Sobolev inequalities. If correct, this extends a tool central to the analysis of Ricci flow singularities to the non-smooth, synthetic setting.

What carries the argument

The central object is the W-entropy functional $W_{N,K}(u,t) = \int_X\big[t|\nabla f|^2 + f - N(1+\frac{Kt}{2})^2\big]u\,d\mu$, where $u = e^{-f}/(4\pi t)^{N/2}$ solves the heat equation of the time-dependent Witten Laplacian $L_t = \Delta_t - \nabla\varphi_t\cdot\nabla$. The argument is carried by the conjugate heat equation $\frac{d}{dt}(e^{-\varphi_t}dm_t)=0$, the definition of a $(K,n,N)$-super Ricci flow via the $N$-dimensional Bakry-Emery Ricci curvature measure, and a distributional Bochner identity $\Gamma_2(L_t)(f,f) = \|\mathrm{Hess}_t f\|^2_{\mathrm{HS}} + \mathrm{Ric}_{N,n}(L_t)(\nabla f,\nabla f)$. These combine to produce the dissipation formulas (4.7) and (4.11).

What would settle it

Construct a closed $(0,n,N)$-super Ricci flow on a metric measure space that satisfies the definition but for which the distributional Bochner identity $\Gamma_2(L_t)(f,f) = \|\mathrm{Hess}_t f\|^2_{\mathrm{HS}} + \mathrm{Ric}_{N,n}(L_t)(\nabla f,\nabla f)$ fails, for example a space with a singular potential $\varphi_t$ outside the assumed differentiability class; then compute $dW_N/dt$ directly and look for a time where it becomes positive, which would refute Theorem 4.4.

Watch

Extended reading notes

Core claim

The central claim is that on every closed $(K,n,N)$-super Ricci flow on metric measure spaces satisfying the conjugate heat equation, the W-entropy $W_{N,K}(u,t)$ is non-increasing in time, with the explicit dissipation estimate $$\frac{d}{dt}W_{N,K}(u) \le -\frac{2t}{N}\int_X u\Big(L\log u + \frac{N}{2t} - \frac{NK}{2}\Big)^2 d\mu$$ (Theorems 4.4 and 4.5). This extends Perelman's entropy monotonicity from smooth Ricci flows to synthetic spaces. The paper also proves the corresponding concavity of the Shannon entropy power, a Li-Yau-Hamilton-Perelman Harnack inequality, and the equivalence between the volume non-collapsing property and lower boundedness of the W-entropy on RCD$(0,N)$ spaces.

Load-bearing premise

The proofs of Theorems 4.4 and 4.5 require a distributional Bochner formula for the time-dependent Witten Laplacian on closed RCD spaces, which the paper invokes as 'the Riemannian Bochner formula (??)' without stating or citing it; if that identity is not valid on the spaces covered, the W-entropy monotonicity is unsupported.

Editorial extensions

If this is right

The W-entropy monotonicity provides a Lyapunov function for the conjugate heat flow on synthetic spaces, so Perelman-style non-collapsing arguments can be ported to RCD spaces.
The Shannon entropy power $e^{2H/N}$ is concave (or $(-2K)$-concave) along the heat flow on closed $(0,n,N)$ (or $(K,n,N)$) super Ricci flows, giving a sharp Fisher information bound $I(u(t)) \le N/(2t)$.
The Li-Yau-Hamilton-Perelman Harnack inequality holds for the fundamental solution on such spaces, yielding differential Harnack estimates.
On RCD$(0,N)$ spaces, volume non-local collapsing is equivalent to lower boundedness of the W-entropy, and the limit of the W-entropy equals the logarithm of the volume ratio constant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the missing Bochner identity is supplied, the same dissipation formulas imply analogous monotonicity for $(K,\infty)$-super Ricci flows, where the finite-$N$ terms disappear.
The equivalence between non-collapsing and lower bounded W-entropy suggests a quantitative stability statement: a lower bound on $W$ with a given constant $A$ implies explicit volume growth with constant $e^{-A}$, as in (8.11).
The entropy power concavity on synthetic spaces could be used to prove information-theoretic inequalities, such as Costa's entropy power inequality, for heat semigroups on RCD spaces.
The regularity issue noted in Remark 9.5 for the extremal of the log-Sobolev functional may block the Euler-Lagrange characterization on non-smooth spaces; a nonsmooth counterexample would pinpoint the limit of Theorem 9.4.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. The paper is a reasonable next step: it sets up (K,n,N)-super Ricci flows on mm spaces and writes down the Perelman W-entropy in that setting, with the static RCD case already treated by Kuwada–Li and Li–Zhang. The time-dependent statements are new as formulations, and the applications (LYHP Harnack, non-collapsing vs lower-bounded W-entropy, log-Sobolev) are natural. That is real work, and the literature is handled honestly.

The problem is the proof of the central theorem. Theorems 4.4 and 4.5 are derived in §5.3 by invoking a "Riemannian Bochner formula (??)" that is never stated, cited, or proved. This is not a cosmetic gap. The identity needed is stronger than Eq. (3.17); it has to decompose Γ2(L)(log u, log u) into a squared Hessian term plus Ric_{N,n} plus a (L−Tr ∇²)²/(N−n) term. On a smooth space this is algebraic, but on a general closed RCD space it requires log u to have a Hessian of finite Hilbert–Schmidt norm and the measure-valued Γ2 to have no singular part. The hypotheses u∈W^{1,2}∩D(L)∩L∞ with Lu∈L∞ do not put f=−log u in the test-function class where such identities are known. Until that is supplied, the monotonicity claim "on every closed RCD space" is unsupported.

Other soft spots are smaller but real. Theorem 8.1's proof mentions an estimate ∫ d²u ≤ C4(N) and refers to equation (31), which does not exist. The proofs of Theorems 6.2, 6.3, and 9.4 are omitted. These are probably fixable, but they are not in the paper.

The paper deserves a serious referee rather than a desk reject: the definitions are sensible, the missing Bochner identity looks repairable, and the program is coherent. But in its current form the central claim is not proven. I would recommend major revision, not acceptance. If I were the editor I would send it out, and if asked I'd advise rejecting the present version and inviting a resubmission that states and proves the Bochner identity—or narrows the hypotheses to a class where it holds—and fills the gaps.

Referee Report

4 major / 5 minor

Summary. The paper proposes to extend Perelman's W-entropy monotonicity, the concavity of Shannon entropy power, and the Li-Yau-Hamilton-Perelman Harnack inequality from smooth super Ricci flows to time-dependent metric measure spaces, in particular to closed (K,n,N)-super Ricci flows defined via a synthetic Bakry-Emery Ricci measure. It also claims applications to the equivalence between volume non-collapsing and lower boundedness of the W-entropy on RCD(0,N) spaces, and to logarithmic Sobolev inequalities. The main new statements are the W-entropy dissipation formulas in Theorems 4.4 and 4.5, with the explicit estimate d/dt W_{N,K}(u) ≤ −(2t/N) ∫ u (L log u + N/(2t) − NK/2)^2 dμ, and the resulting monotonicity under (K,n,N)-super Ricci flow.

Significance. If the main results are correct, the paper would give a genuinely synthetic extension of Perelman's entropy monotonicity, going beyond the static RCD(0,N) results of Kuwada-Li and complementing the recent RCD(K,n,N) work of Li-Zhang and Brena. The explicit dissipation estimate and the Harnack inequalities would be valuable tools for the analysis of time-dependent metric measure spaces. The paper also contains a potentially significant equivalence result for non-collapsing on RCD(0,N) spaces. However, the current manuscript does not establish the central theorems as written because a load-bearing Bochner identity is invoked but never stated or proved, and several supporting estimates and proofs are omitted. These gaps block acceptance and require substantive revision.

major comments (4)

[Section 5.3, proof of Theorems 4.4 and 4.5] The proof begins with 'Under the condition that the Riemannian Bochner formula (??) holds', but the formula is never stated, located, or proved. The identity required is not the paper's Eq. (3.17), which defines Ric∞,n as a measure-valued remainder; rather, the proof needs a pointwise/measure identity expressing Γ2(L)(log u, log u) + (1/t−K) Tr∇² log u + (n/4)(1/t−K)² as the sum of a Hilbert–Schmidt square, a Ric_{N,n}(L) term, and |(L−Tr∇²) log u|²/(N−n). The stated hypotheses on u (u ∈ W^{1,2} ∩ D(L) ∩ L∞, Lu ∈ L∞) do not place −log u in the RCD test-function class TestF, so even the existence of the terms in such a decomposition is not justified. This gap propagates directly to the dissipation estimate (4.8), the Shannon entropy power concavity in Section 6, and the Harnack inequality in Section 7.4, all of which rely on the same decomposition. The central monotonicity claim is therefore unsupported as written.
[Section 8, proof of Theorem 8.1] The proof asserts, without proof, that 'Based on the Li-Yau upper bound estimate (8.1), we can prove that ∫ d²(x,y) u(x,y,τ) dμ(y) ≤ C4(N)'. This estimate is then substituted 'into (31)', but no equation (31) exists in the manuscript. The estimate is essential for bounding −∫ v² log v² dμ and hence for deriving the non-collapsing volume lower bound (8.11) from the lower boundedness of W_N. Without a proof or a precise reference for this estimate, the claimed equivalence between volume non-collapsing and lower boundedness of the W-entropy is not established.
[Theorems 6.2, 6.3, and 9.4] Several results that are presented as new theorems of this paper have omitted proofs. Theorem 6.2 says 'the proof has been essentially given by S. Li-Li [37]' with no detail; Theorem 6.3 explicitly omits the second of its two announced proofs; Theorem 9.4 says 'the proof is similar' and leaves the Euler–Lagrange derivation and the monotonicity of μ_K(t) unstated. Since these theorems are part of the paper's claims on metric measure spaces, citing a smooth-manifold proof is not sufficient unless the cited argument is shown to extend verbatim to the stated RCD/super-Ricci-flow setting. This is a completeness gap.
[Theorem 4.5, Eq. (4.11)] The formula in Theorem 4.5 contains a symbol error: the second integral has 'm − n' in the denominator, although m is not defined in the theorem and the theorem is about the (K,n,N) case; this should be 'N − n'. Additionally, the definition line (4.10) writes 'W_{m,K}(u)' where the theorem is defining W_{N,K}. These typos obscure the already delicate algebraic structure of the entropy formula.

minor comments (5)

[Section 3.3, Eq. (3.17)] Equation (3.17) is called a 'distributional Bochner formula', but as written it is essentially a definition of Ric∞,n as the remainder Γ2(L)(f,f) − ‖∇²f‖²_{HS}. The paper should clearly distinguish this definition from a genuine Bochner identity, which would require proving that Ric∞,n is a measure or a tensor with the expected properties.
[Section 3.3, Definition 3.7] The paper assumes the global geometric dimension n of the RCD spaces is constant in t, but this is not automatic for a time-dependent family and is not discussed. The assumption should be stated as an explicit hypothesis of the theorems that use it, not as a general standing assumption.
[Theorem 7.12 and Lemma 7.11] The sign and the role of the commutator term [∂_τ, L] log H are not consistent between Lemma 7.11 and Theorem 7.12: the lemma includes a term 2τ[∂_τ,L] log H in the expression for □*w_m, but the theorem's final inequality ≤ 2τP*([∂_τ,L] log H H) seems to drop the negative definite part of WH without stating the necessary inequality WH ≤ 0. This needs clarification.
[Throughout] There are numerous typos and infelicities that should be corrected: 'mnaifolds' for 'manifolds', 'rôle' for 'role', 'c Lap lacian' for 'Laplacian', 'Riccatti' for 'Riccati', 'Contempo-rary' for 'Contemporary', 'dimensional' for 'dimensional', and 'dimensioal' for 'dimensional'. Reference [46] is cited as 'arxiv2504.01864' with a missing colon. Theorems 4.6–4.8 should either include the precise 'reasonable growth condition' or state that they are quoted verbatim from [27] and [46].
[Section 7.2, Remark 7.8] The remark candidly notes that the results of Section 7.2 were not submitted earlier because the Gaussian lower bound for L ≠ Δ is not true in general. This is a useful caution, but it also means the reader should be told explicitly which of the Harnack statements in Section 7.2 are actually proved under the stated hypotheses and which are conditional on unverified assumptions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the W-entropy monotonicity is a conditional theorem, not an input–output loop; the §5.3 '(??)' Bochner formula is a missing justification, not a circular step.

full rationale

No circularity of the kind measured by the score scale is established. The W-entropy monotonicity in Theorems 4.3–4.5 is obtained by differentiating the explicitly defined W_{N,K} and completing squares; once the 'Riemannian Bochner formula (??)' invoked in Section 5.3 is granted, the sign of d/dt W is controlled by the defining inequality of a (K,n,N)-super Ricci flow (Definition 3.7), so this is the standard assumption–conclusion pattern rather than an input–output loop. The unnumbered '(??)' in Section 5.3 is a genuine missing proof or reference: the identity is neither stated nor cited for the regularity class of log u, so the theorems as stated on all closed RCD spaces are unsupported. But an omitted proof is a correctness risk, not a circular reduction to the theorem's conclusion. The self-citations to Li–Li [30] and Li–Zhang [46] supply smooth and static RCD antecedents while the present proof is reproduced in the text, and the Kuwada–Li theorem is an external published benchmark. Hence there is no load-bearing self-citation chain and no definitional identity that forces the claimed result by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The central proofs rest on a distributional Bochner formula that is not stated as a theorem or given a reference, on compactness of the spaces, on the constancy of local dimension, and on existing heat kernel estimates in RCD theory. There are no data-fitted parameters; the axioms are structural assumptions about the synthetic spaces.

assumptions (4)

domain assumption Distributional Bochner formula for the time-dependent Witten Laplacian on closed RCD spaces: Gamma_2(L_t)(f,f) = ||Hess_t f||^2_HS + Ric_{N,n}(L_t)(grad f, grad f).
Assumed implicitly in the proof of Theorems 4.4 and 4.5 (Section 5.3, 'the Riemannian Bochner formula (??)'). The theorem statements only say 'closed RCD spaces', so this is an extra, unstated hypothesis.
domain assumption The local geometric dimension n of each RCD space is a constant independent of t (Section 3.3, paragraph before Definition 3.6).
Needed to define Tr Hess f and the (K,n,N)-Ricci tensor; the constancy in time is assumed without proof.
domain assumption Closed (compact) metric measure spaces so that integration by parts and the heat semigroup have no boundary terms (used throughout).
The theorems restrict to closed RCD spaces; compactness is not proved but is part of the setting.
domain assumption On RCD(-K,N) spaces, the two-sided Gaussian heat kernel bounds (8.1) of Jiang-Li-Zhang and the Li-Yau Harnack inequality of Jiang and Zhang-Zhu hold (used in Section 8).
These are deep results cited from the literature; the paper relies on them for the non-collapsing equivalence.

invented entities (2)

(K,n,N)-super Ricci flow on metric measure spaces (Definition 3.7)
purpose: Defines the class of time-dependent spaces on which the W-entropy monotonicity and entropy power concavity are claimed; combines Sturm's super Ricci flow with a constant local dimension and a Witten potential.
New definition in this paper; no external falsifiable handle. Its usefulness depends on the Bochner formula assumption, which is not established here.
N-dimensional Bakry-Emery Ricci curvature measure Ric_{N,n}(L_t) on mm spaces (Definition 3.6)
purpose: Formalizes the curvature term in the dissipation formulas for time-dependent Witten Laplacians on synthetic spaces.
Defined formally through the Bochner formula; the definition is only meaningful if the Hessian and Gamma_2 exist on the space, which is part of the unstated assumption.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On Perelman's $W$-entropy and Shannon entropy power for super Ricci flows on metric measure spaces." pith.science (2026). https://pith.science/paper/WZETJ5JV

@misc{pith2026250503202,
  author       = {Pith},
  title        = {Pith review of: On Perelman's $W$-entropy and Shannon entropy power for super Ricci flows on metric measure spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZETJ5JV}},
  note         = {Machine review of arXiv:2505.03202}
}

abstract

In this paper, we extend Perelman's $W$-entropy formula and the concavity of the Shannon entropy power from smooth Ricci flow to super Ricci flows on metric measure spaces. Moreover, we prove the Li-Yau-Hamilton-Perelman Harnack inequality on super Ricci flows. As a significant application, we prove the equivalence between the volume non-local collapsing property and the lower boundedness of the $W$-entropy on RCD$(0, N)$ spaces. Finally, we use the $W$-entropy to study the logarithmic Sobolev inequality with optimal constant on super Ricci flows on metric measure spaces.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 57 canonical work pages

[37]

Tohoku Math

Li, S., Li, X.-D.: On the Shannon entropy power on Riemannian manif olds and Ricci ﬂow. Tohoku Math. J. (2) 76, 577-608 (2024)

work page 2024
[1]

Ambrosio, L., Gigli, N., Savar´ e, G.: Calculus and heat ﬂow in metric me asure spaces and appli- cations to spaces with Ricci bounds from below. Invent. Math. 195, 289-391 (2014)

work page 2014
[2]

Duke Math

Ambrosio, L., Gigli, N., Savar´ e, G.: Metric measure spaces with Riem annian Ricci curvature bounded from below. Duke Math. J. 163 , 1405-1490 (2014) 39

work page 2014
[3]

S´ eminaire de probabilit´ es, XIX, 1983/1984

Bakry, D., ´Emery, M.: Diﬀusions Hypercontractives. S´ eminaire de probabilit´ es, XIX, 1983/1984. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985)

work page 1985
[4]

Bakry, D., Ledoux, M.: A logarithmic Sobolev form of the Li-Yau par abolic inequality, Rev. Mat. Iberoamericana 22 (2006), no. 2, 683-702

work page 2006
[5]

Bacher, K., Sturm, K.-T.: Localization and tensorization propert ies of the curvature-dimension condition for metric measure spaces. J. Funct. Anal. 259, 28-56 (2010)

work page 2010
[6]

Brena, C.: Perelman’s entropy and heat kernel bounds on RCD sp aces, arXiv:2503.03017v1, 4 Mar 2025

arXiv 2025
[7]

Brena, C., Gigli, N., Honda, S., Zhu, X.: Weakly non-collapsed RCDspa ces are strongly non- collapsed. J. ReineAngew. Math. 794, 215-252 (2023)

work page 2023

Show all 61 references

[8]

Brena, C., Gigli, N.: Fine Representation of Hessian of Convex Func tions and Ricci Tensor on RCD Spaces, Potential Analysis https://doi.org/10.1007/s11118- 024-10153-5

doi
[9]

Bru` e, E., Semola, D.: Constancy of the dimension for RCD( K,N ) spaces via regularity of La- grangian ﬂows, arXiv:1804.07128

arXiv
[10]

Cao, H.-D., Zhu, X.-P.: A complete proof of the Poincar´ e and geo metrization conjectures: ap- plication of the Hamilton-Perelman theory of the Ricci ﬂow, Asian J. M ath. 10 (2006), no. 2, 165-492, and Asian J. Math. 10 (2006), no. 4, 663

2006
[11]

S.-T.: A lower bound for the heat kernel

Cheeger, J., Yau. S.-T.: A lower bound for the heat kernel. Comm un. Pure Appl. Math. 34, 465–480 (1981)

1981
[12]

B., Lu, P., Ni, L.: Hamilton’s Ricci Flow, Lectures in Comtempo rary Maths., Sciences Press, Beijing, Amer

Chow. B., Lu, P., Ni, L.: Hamilton’s Ricci Flow, Lectures in Comtempo rary Maths., Sciences Press, Beijing, Amer. Math. Soc. 2006

2006
[13]

Cavalletti, F., Milman, E.: The globalization theorem for the curvat ure dimension condition, arXiv:1612.07623

arXiv
[14]

IEEE Trans

Costa, M.: A new entropy power inequality. IEEE Trans. Inform . Theory. 31, 751-760 (1985)

1985
[15]

Cover, T., Thomas, J.: Elements of Information Theory, Secon d Edition, Wiley InterScience, A John Wiley Sons, INC., Publication, (2006)

2006
[16]

Cambridge Unive rsity Press, Cambridge (1990)

Davies, E.B.: Heat Kerel and Spectral Theory. Cambridge Unive rsity Press, Cambridge (1990)

1990
[17]

Erbar, M., Kuwada, K., Sturm, K.-T.: On the equivalence of the en tropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Inven t. Math. 201, 993-1071 (2015)

2015
[18]

De Gruyter Studies in Mathematics

Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmet ric Markov Processes. De Gruyter Studies in Mathematics. 19. Walter de Gruyter & Co., Berlin. x+392 pp (1994)

1994
[19]

Gigli, N., Mondino, A., Savar´ e, G.: Convergence of pointed non-c ompact metric measure spaces and stability of Ricci curvature bounds and heat ﬂows. Proc. Lond . Math. Soc. (3) 111(5), 1071–1129 (2015)

2015
[20]

Gigli, N.: On the diﬀerential structure of metric measure spaces and applications. Mem. Am. Math. Soc. 236, vi+91 (2015)

2015
[21]

Hamilton, The formation of singularities in the Ricci ﬂow, Surv eys in Diﬀerential Geometry, 2, 7-136, International Press, 1995

R.S. Hamilton, The formation of singularities in the Ricci ﬂow, Surv eys in Diﬀerential Geometry, 2, 7-136, International Press, 1995

1995
[22]

Discrete Contin

Han, B.-X.: New characterizations of Ricci curvature on RCD me tric measure spaces. Discrete Contin. Dyn. Syst. 38(10), 4915–4927 (2018)

2018
[23]

Han, B.-X.: Ricci tensor on RCD ∗(K,N ) spaces. J. Geom. Anal. 28(2), 1295–1314 (2018) 40

2018
[24]

Jiang, R.: The Li-Yau inequality and heat kernels on metric measur e spaces. J. Math. Pures Appl.104 29-57 (2015)

2015
[25]

Jiang, R., Li, H., Zhang, H.: Heat kernel bounds on metric measur e spaces and some applications, Potential Analysis, 44, 601-627 (2016)

2016
[26]

Kleiner, J

B. Kleiner, J. Lott, Notes on Perelman’s papers, Geom. Topol. 1 2 (2008), no. 5, 2587-2855

2008
[27]

Manuscripta Math

Kuwada, K., Li X.-D.: Monotonicity and rigidity of the W-entropy o n RCD(0,N) spaces. Manuscripta Math. 164, 119-149 (2021)

2021
[28]

Li, H.: Sharp heat kernel bounds and entropy in metric measure spaces. Sci. China Math. 61, 487-510 (2018)

2018
[29]

Pure and Appl

Kopfer, E., Sturm, K.-T.: Heat Flow on Time-Dependent Metric Me asure Spaces and Super-Ricci Flows, Comm. Pure and Appl. Math., Vol. LXXI, 2500-2608 (2018)

2018
[30]

Paciﬁc J

Li, S., Li, X.-D.: The W -entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials. Paciﬁc J. Math. 278, 173-199 (2015)

2015
[31]

Li, S., Li, X.-D.: Hamilton diﬀerential Harnack inequality and W -entropy for Witten Laplacian on Riemannian manifolds. J. Funct. Anal. 274, 3263-3290 (2018)

2018
[32]

Li, S., Li, X.-D.: On Harnack inequalities for Witten Laplacian on Riema nnian manifolds with super Ricci ﬂows. Asian J. Math. 22, 577-597 (2018)

2018
[33]

arXiv:2001.11184, (2020)

Li, S., Li, X.-D.: On the R´ enyi entropy power and the Gagliardo-N irenberg-Sobolev inequality on Riemannian manifolds. arXiv:2001.11184, (2020)

2020 arXiv
[34]

61 (2018), 1 385-1406

Li, S., Li, X.-D.: W -entropy formulas on super Ricci ﬂows and Langevin deformation o n Wasser- stein space over Riemannian manifolds, Sci China Math. 61 (2018), 1 385-1406

2018
[35]

49 (2019), no

Li, S., Li, X.-D.: On the Li-Yau-Hamilton Harnack inequalities on Ricci ﬂow and super Ricci ﬂows (in Chinese), Sci Sin Math. 49 (2019), no. 11, 1613-1632

2019
[36]

Li, S., Li, X.-D.: W -entropy, super Perelman Ricci ﬂows and ( K,m )-Ricci solitons, J. Geom. Anal. 30 (2020), no. 3, 3149-3180

2020
[38]

Li, S., Li, X.-D.: W-entropy formulas and Langevin deformation of ﬂows on Wasserstein space over Riemannian manifolds. Probab. Theory Related Fields. 188, 911-955 (2024)

2024
[39]

Li, Perelman’s W-entropy for the Fokker-Planck equation over complete Riemannian man- ifolds, Bull

X.-D. Li, Perelman’s W-entropy for the Fokker-Planck equation over complete Riemannian man- ifolds, Bull. Sci. math. 135 (2011) 871-882

2011
[40]

X.-D. Li, Diﬀerential Harnack inequality and Perelman’s entropy f ormula on complete Rieman- nian manifolds, invited talk in 2008 Workshop on Markov Processes an d Related Fields, organized by Prof. Mufa Chen, Beijing Normal University and Anhui Normal Un iversity, Wuhu, July 24,

2008
[41]

Li.: Sobolev inequalities on forms and Lp,q - cohomology on complete Riemannian manifolds, J Geom Anal (2010) 20: 354-387 DOI 10.1007/s12220-009-9114- 7

X.-D. Li.: Sobolev inequalities on forms and Lp,q - cohomology on complete Riemannian manifolds, J Geom Anal (2010) 20: 354-387 DOI 10.1007/s12220-009-9114- 7

2010 doi
[42]

Li, X.-D.: Perelman’s entropy formula for the Witten Laplacian on R iemannian manifolds via Bakry-Emery Ricci curvature. Math. Ann. 353, 403-437 (2012)

2012
[43]

Li, X.-D.: Hamilton’s Harnack inequality and the W -entropy formula on complete Riemannian manifolds, Stochastic Process. Appl. 126, no. 4, 1264-1283 (2016)

2016
[44]

Li, Z.: The globalization theorem for CD( K,N ) on locally ﬁnite spaces, arXiv:2212.07962 41

arXiv
[45]

Li, J., Xu, X.: Diﬀerential Harnack inequalities on Riemannian manifo lds I: linear heat equation, Adv. Math. 226:5, 4456-4491 (2011)

2011
[46]

Li, X.-D., Zhang, E.: On the W -entropy and Shannon entropy power on RCD( K,N ) and RCD(K,n,N ) spaces, arxiv2504.01864

arXiv
[47]

Lott, J., Villani, C.: Ricci curvature for metric measure spaces v ia optimal transport. Ann. of Math. (2) 169, 903-991 (2009)

2009
[48]

J. W. Morgan, G. Tian, Ricci ﬂow and the Poincar´ e conjecture , Clay Mathematics Monographs,
[49]

Ni, L.: The entropy formula for linear equation. J. Geom. Anal. 14, 87-100 (2004)

2004
[50]

American Mathematical Society, Providence, RI; Clay Mathemat ics Institute, Cambridge, MA, 2007

2007
[51]

, http://arXiv.org/abs/maths0211159

Perelman, G.: The entropy formula for the Ricci ﬂow and its geom etric applications. , http://arXiv.org/abs/maths0211159
[52]

The entropy formula for linear equation

L. Ni, Addenda to “The entropy formula for linear equation”, J. Geom. Anal. 14 (2), 329-334, (2004)

2004
[53]

Bell Sys tem Tech

Shannon, C.: A mathematical theory of communication. Bell Sys tem Tech. J. 27, 379-423, 623- 656 (1948)

1948
[54]

Rothaus, O.S.: Logarithmic Sobolev inequalities and the spectrum of Schrodinger operators. J. Funct. Anal. 42(1), 110–120 (1981)

1981
[55]

Sturm, K.-T.: Super-Ricci ﬂows for metric measure spaces Jou rnal of Functional Analysis 275(2018) 3504-3569

2018
[56]

Acta Math

Sturm, K.-T.: On the geometry of metric measure spaces. Acta Math. 196, 65-131 (2006)

2006
[57]

Nonlinear Anal

Wu, J.-Y.: The logarithmic entropy formula for the linear heat equ ation on Riemannian manifolds. Nonlinear Anal. 75, 4862-4872 (2012)

2012
[58]

Varopoulos, N.Th.: Hardy-Littlewood theory for semigroups. J . Funct. Anal. 63, 240-260 (1985)

1985
[59]

Zhang, H.-C., Zhu, X.-P.: Local Li-Yau’s estimates on RCD ∗(K,N ) metric measure spaces. Calc. Var. PDE 55, 93 (2016). Xiang-Dong Li, State Key Laboratory of Mathematical Sciences, A cademy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, Zhonggua ncun ...

2016
[60]

Ye, R.-G,: The Log entropy functional along the Ricci ﬂow, arXiv :0708.2008v3 (2007)

2007 arXiv
[2008]

See http://math0.bnu.edu.cn/probab/Workshop2008/Talk s/XiangdongLi.pdf

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [37]

Tohoku Math

Li, S., Li, X.-D.: On the Shannon entropy power on Riemannian manif olds and Ricci ﬂow. Tohoku Math. J. (2) 76, 577-608 (2024)

work page 2024

[2] [1]

Ambrosio, L., Gigli, N., Savar´ e, G.: Calculus and heat ﬂow in metric me asure spaces and appli- cations to spaces with Ricci bounds from below. Invent. Math. 195, 289-391 (2014)

work page 2014

[3] [2]

Duke Math

Ambrosio, L., Gigli, N., Savar´ e, G.: Metric measure spaces with Riem annian Ricci curvature bounded from below. Duke Math. J. 163 , 1405-1490 (2014) 39

work page 2014

[4] [3]

S´ eminaire de probabilit´ es, XIX, 1983/1984

Bakry, D., ´Emery, M.: Diﬀusions Hypercontractives. S´ eminaire de probabilit´ es, XIX, 1983/1984. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985)

work page 1985

[5] [4]

Bakry, D., Ledoux, M.: A logarithmic Sobolev form of the Li-Yau par abolic inequality, Rev. Mat. Iberoamericana 22 (2006), no. 2, 683-702

work page 2006

[6] [5]

Bacher, K., Sturm, K.-T.: Localization and tensorization propert ies of the curvature-dimension condition for metric measure spaces. J. Funct. Anal. 259, 28-56 (2010)

work page 2010

[7] [6]

Brena, C.: Perelman’s entropy and heat kernel bounds on RCD sp aces, arXiv:2503.03017v1, 4 Mar 2025

arXiv 2025

[8] [7]

Brena, C., Gigli, N., Honda, S., Zhu, X.: Weakly non-collapsed RCDspa ces are strongly non- collapsed. J. ReineAngew. Math. 794, 215-252 (2023)

work page 2023