Perelman's toolkit

Perelman entropy

Q: What is Perelman's entropy?

Perelman's entropy is the functional 𝓦(g,f,τ) = intMτ(R+|∇f|²)+f-n^(−n/2)e^(−f) dV, introduced in his 2002 paper on Ricci flow. It never decreases when g evolves by Ricci flow and (4πτ)^(−n/2)e^(−f) solves the conjugate heat equation. Its infimum over f, written μ(g,τ), is what Perelman used to prove that closed Ricci flows do not collapse locally in finite time.

Q: Why is it called entropy?

It is called entropy because it contains the term ∫ f u dV, which up to a constant is the Boltzmann–Shannon entropy -∫ ulog u dV of the density u. In Perelman's statistical analogy, 𝓦 is actually minus an entropy: his thermodynamic quantity is S = −𝓦. So 𝓦 increases while S decreases, and the analogy is a heuristic that the proofs do not use.

Q: Is Ricci flow a gradient flow?

Ricci flow is a gradient flow of Perelman's 𝓕-functional once it is modified by diffeomorphisms and a measure dm = e^(−f) dV is held fixed. For a fixed measure the L² gradient of 𝓕 is −(Ric+∇² f), so the flow partialt g = −2(Ric+∇² f) is its gradient flow. Different choices of measure give the same flow up to diffeomorphism, so Perelman compares the measure to a choice of gauge.

Q: What is the conjugate heat equation in Ricci flow?

The conjugate heat equation is partialt u = −Δu + Ru, which is solved backward in time along a Ricci flow. It is the adjoint of the heat equation partialt h = Δh for the evolving volume form, so intM hu dV stays constant in time. Perelman's monotonicity formulas hold when the weight e^(−f), or (4πτ)^(−n/2)e^(−f), solves this equation.

Q: What is Nash entropy in Ricci flow?

The pointed Nash entropy is 𝓝x,t(τ) = intM f dν− n/2, where dν= (4πτ)^(−n/2)e^(−f) dV is the conjugate heat kernel measure based at (x,t). It equals the average of Perelman's pointed 𝓦-entropy over scales from 0 to τ, and it is non-positive and non-increasing in τ. Hein and Naber defined it, and Bamler used it to measure local collapsedness at a single point, since e^(𝓝x,t(r²)) behaves like the volume ratio at scale r under suitable scalar curvature bounds.

Q: How does entropy prove no local collapsing?

A ball of radius r with |Rm|≤ r⁻² and a very small volume ratio gives a test function that makes μ(g(t),r²) very negative. Monotonicity bounds μ(g(t),r²) below by μ(g(0),t+r²), and on a closed manifold this is bounded below when t+r² stays bounded. So the volume ratio cannot go to zero in finite time, which is Perelman's no local collapsing theorem.

In briefPerelman found that Ricci flow is a gradient flow once a measure is fixed. His $F$ -functional and $W$ -entropy increase along the flow; a lower bound on $W$ rules out local collapsing on closed manifolds, and the equality cases are gradient solitons.

\mathcal W(g,f,\tau) = \int_M \big[\tau(R + |\nabla f|^2) + f - n\big]\,(4\pi\tau)^{-n/2}e^{-f}\,dV

34papers

4in the last 30 days

13key references

7guidebook chapters

At a glance

What they are. Integrals of a metric and a weight function that never decrease along Ricci flow when the weight solves the conjugate heat equation: Perelman's $\mathcal F$ and $\mathcal W$ (2002), and the pointed Nash entropy built from them.
In words. $\mathcal F$ is the weighted integral of scalar curvature plus the Dirichlet energy of the weight. $\mathcal W$ adds a scale $\tau$ and a relative-entropy term, which makes it scale invariant.
Monotonicity. The time derivative of each is the integral of a square that vanishes exactly on a gradient soliton: steady for $\mathcal F$ , shrinking for $\mathcal W$ .
Headline theorems. Perelman (2002): on closed manifolds there are no breathers other than gradient solitons, and a flow on a finite time interval is not locally collapsing.
Modern form. Hein and Naber (2012) proved sharp log-Sobolev inequalities for conjugate heat kernel measures. Bamler (2020) made the pointed Nash entropy the measure of local collapsedness in all dimensions.
Why it matters. Noncollapsing is what makes blow-up limits of singularities exist and stay $n$ -dimensional. Perelman describes it as removing the major stumbling block in Hamilton's approach.
What is open. Classifying the shrinkers that entropy singles out in dimension four and above, and deciding which singularities survive a generic perturbation.

The ideas that matter here

Key concepts

Each one in a few lines, with the equation that carries it and the stage of the Guidebook where it is learned.

Perelman's 𝓕-functional

\mathcal{F}(g,f) = \int_M \bigl(R + |\nabla f|^2\bigr)\,e^{-f}\,dV.

Couple the flow with the backward heat-type equation $\partial_t f = -\Delta f + |\nabla f|^2 - R$ . Then

\frac{d}{dt}\mathcal{F} = 2\int_M \bigl|\operatorname{Ric} + \nabla^2 f\bigr|^2 e^{-f}\,dV \;\ge\; 0.

Ricci flow is a gradient flow, modulo diffeomorphisms, and steady solitons are exactly the critical points. Hamilton's flow had been around for 20 years before anyone saw this.

Stage 12 · the target

Perelman's 𝓦-entropy

Add a scale parameter $\tau > 0$ with $\partial_t \tau = -1$ :

\mathcal{W}(g,f,\tau) = \int_M \Bigl[\tau\bigl(R + |\nabla f|^2\bigr) + f - n\Bigr](4\pi\tau)^{-n/2}e^{-f}\,dV,

subject to $\int_M (4\pi\tau)^{-n/2}e^{-f}\,dV = 1$ . It is nondecreasing, and constant exactly on shrinking solitons. Its infimum $\mu(g,\tau)$ drives the no-local-collapsing theorem.

(Non)collapsing

Stage 12 · the target

Nash entropy and 𝓕-convergenceadvanced

Bamler's reworking of Perelman's entropy into a compactness theory. Given a conjugate heat kernel measure $\nu_{\tau} = (4\pi\tau)^{-n/2}e^{-f}\,dg$ based at a spacetime point, the pointed Nash entropy is

\mathcal{N}(\tau) = \int_M f\,d\nu_\tau - \frac{n}{2},

which is monotone in $\tau$ and bounded by Perelman's μ-functional. Uniform Nash-entropy bounds give compactness in Bamler's $\mathbb{F}$ -topology, whose limits are metric flows: objects that are no longer smooth manifolds but still carry a heat flow and a conjugate heat flow.

Limit spaces & RCD Heat kernel

Stage 12 · the target

Where the concept cards came from: they were drafted by an AI at the site owner's request. The owner, who is still learning this material, chose which concepts to include but cannot vouch for every detail, so treat them as a well-drawn map and check the cited papers. Corrections are welcome: see how this site works.

Guide

In depth

What it is

Perelman's $\mathcal F$ -functional is $\mathcal F(g,f) = \int_M (R + |\nabla f|^2)\,e^{-f}\,dV$ . If $g$ flows by Ricci flow and $e^{-f}$ solves the conjugate (backward) heat equation, then

\frac{d}{dt}\mathcal F = 2\int_M |\operatorname{Ric} + \nabla^2 f|^2\, e^{-f}\,dV \;\ge\; 0.

So $\lambda(g) = \inf_f \mathcal F$ , taken over $f$ with $\int_M e^{-f}\,dV = 1$ , is non-decreasing. On a closed manifold it stays constant only on steady gradient solitons, which are Ricci-flat there. That rules out non-trivial steady breathers: solutions that return to themselves up to a diffeomorphism.

The $\mathcal W$ -entropy above adds a scale $\tau$ , and $\mu(g,\tau) = \inf_f \mathcal W$ is monotone in the same way. It is constant exactly on gradient shrinking solitons. $\mu(g,\tau)$ is the optimal constant in a logarithmic Sobolev inequality at scale $\sqrt\tau$ , and bounding it below gives κ-noncollapsing: balls with controlled curvature cannot be too thin.

The pointed Nash entropy, $\mathcal N = \int f\,d\nu - \tfrac n2$ computed with the conjugate heat kernel measure $\nu$ , is the version attached to a spacetime point, and it is at the centre of Bamler's modern theory.

Definitions and conventions

Throughout, $\partial_t g = -2\operatorname{Ric}$ , and $\tau = T - t$ runs backward from a chosen time $T$ . The conjugate heat operator is $\square^* = -\partial_t - \Delta + R$ . It is the adjoint of $\partial_t - \Delta$ for the evolving volume form, so a solution of $\square^* u = 0$ keeps $\int_M u\,dV$ constant. The heat kernel of this operator, started as a delta at a spacetime point, is the object behind all the pointed quantities below.

The energy $\mathcal F$ and the eigenvalue $\lambda$

With the normalisation $\int_M e^{-f}\,dV = 1$ and the substitution $w = e^{-f/2}$ ,

\mathcal F(g,f) = \int_M \big(4|\nabla w|^2 + Rw^2\big)\,dV,

so $\lambda(g)$ is the lowest eigenvalue of $-4\Delta + R$ . The scale-invariant version is $\bar\lambda = \lambda V^{2/n}$ , with $V$ the volume.

The entropy $\mathcal W$ and its infima

The test functions satisfy $u = (4\pi\tau)^{-n/2}e^{-f}$ and $\int_M u\,dV = 1$ . Then $\mathcal W(cg, f, c\tau) = \mathcal W(g, f, \tau)$ for $c > 0$ , and with $w = \sqrt u$ ,

\mathcal W = \int_M \big[\tau(4|\nabla w|^2 + Rw^2) - w^2\log w^2\big]\,dV - \frac n2\log(4\pi\tau) - n.

Define $\mu(g,\tau) = \inf_f \mathcal W(g,f,\tau)$ and $\nu(g) = \inf_{\tau>0}\mu(g,\tau)$ . On a closed manifold the infimum defining $\mu$ is attained by a smooth function (Rothaus, 1981). Perelman shows that $\mu(g,\tau)$ is negative for small $\tau$ and tends to $0$ as $\tau \to 0$ , and that $\mu(g,\tau) \to +\infty$ as $\tau \to \infty$ when $\lambda(g) > 0$ . Monotonicity gives, for $t_1 < t_2$ ,

\mu\big(g(t_1),\, \tau + t_2 - t_1\big) \le \mu\big(g(t_2),\, \tau\big).

The pointed quantities

Fix a spacetime point $(x_0,t_0)$ and let $d\nu = (4\pi\tau)^{-n/2}e^{-f}\,dV$ be the conjugate heat kernel measure based there, with $\tau = t_0 - t$ . Hein and Naber insert this $f$ into $\mathcal W$ to get the pointed entropy $\mathcal W_{x_0,t_0}(\tau)$ , and they define the pointed Nash entropy as its average over scales:

\mathcal N_{x_0,t_0}(\tau) = \frac1\tau\int_0^\tau \mathcal W_{x_0,t_0}(\sigma)\,d\sigma = \int_M f\,d\nu - \frac n2.

They note that it is closely related to the quantity Nash used to prove Hölder continuity of weak solutions. Equivalently $\frac{d}{d\tau}(\tau\mathcal N) = \mathcal W \le 0$ , and $\mathcal N$ is non-increasing in $\tau$ with $\mathcal N(0) = 0$ , so both are non-positive. Bamler records the comparison $\mathcal N_{x_0,t_0}(\tau) \ge \mu(g(t_0-\tau),\tau)$ , so on a finite time interval a lower bound on $\mu$ at the initial time bounds every Nash entropy below.

The monotonicity formulas

Why Ricci flow is a gradient flow

Let $\delta g_{ij} = v_{ij}$ , $\delta f = h$ , $v = g^{ij}v_{ij}$ . Perelman's first variation is

\delta\mathcal F = \int_M e^{-f}\Big[-v_{ij}\big(R_{ij} + \nabla_i\nabla_j f\big) + \big(\tfrac v2 - h\big)\big(2\Delta f - |\nabla f|^2 + R\big)\Big]\,dV.

The term $\tfrac v2 - h$ vanishes exactly when the measure $dm = e^{-f}\,dV$ is held fixed. So for a fixed measure the gradient of $\mathcal F$ is $-(\operatorname{Ric} + \nabla^2 f)$ , and the gradient flow is $\partial_t g = -2(\operatorname{Ric} + \nabla^2 f)$ , $\partial_t f = -R - \Delta f$ . Pulling back by the diffeomorphisms generated by $\nabla f$ gives Ricci flow coupled to

\partial_t f = -\Delta f + |\nabla f|^2 - R,

which says $\square^* e^{-f} = 0$ . Different measures lead to the same flow up to diffeomorphism. Perelman compares the choice of measure to a choice of gauge, as with DeTurck's trick.

Trace and Cauchy–Schwarz turn the monotonicity formula into $\frac{d}{dt}\lambda \ge \frac2n\lambda^2$ . So if $\lambda(g(0)) > 0$ , a closed flow must become singular by time $n/(2\lambda(g(0)))$ . On the round sphere of radius $r_0$ we have $\lambda = n(n-1)/r_0^2$ , and this bound equals the actual extinction time $r_0^2/(2(n-1))$ .

The $\mathcal W$ formula and its pointwise form

With $\partial_t f = -\Delta f + |\nabla f|^2 - R + \frac{n}{2\tau}$ and $\partial_t\tau = -1$ , Perelman's formula (3.4) reads

\frac{d}{dt}\mathcal W = \int_M 2\tau\,\Big|\operatorname{Ric} + \nabla^2 f - \frac{1}{2\tau}g\Big|^2 (4\pi\tau)^{-n/2}e^{-f}\,dV \ge 0.

It follows from a pointwise identity that works locally. For $v = [\tau(2\Delta f - |\nabla f|^2 + R) + f - n]u$ one has $\int_M v\,dV = \mathcal W$ and

\square^* v = -2\tau\,\Big|\operatorname{Ric} + \nabla^2 f - \frac{1}{2\tau}g\Big|^2 u.

If $u$ is a fundamental solution, then $v \le 0$ for all earlier times (Perelman, Corollary 9.3). This is a differential Harnack inequality of Li–Yau type. It also gives $f \le \ell$ , the reduced distance, which means $u \ge (4\pi\tau)^{-n/2}e^{-\ell}$ (Corollary 9.5). Perelman's pseudolocality theorem is proved by localising $v$ with a solution of the heat equation.

Relatives on a fixed background and on expanders

Lei Ni (2004) found the linear version. On a closed manifold with a fixed metric, for a positive solution of the heat equation $u = (4\pi t)^{-n/2}e^{-f}$ with $\int_M u\,dV = 1$ and $\mathcal W = \int_M(t|\nabla f|^2 + f - n)u\,dV$ ,

\frac{d}{dt}\mathcal W = -2t\int_M\Big(\Big|\nabla^2 f - \frac{1}{2t}g\Big|^2 + \operatorname{Ric}(\nabla f,\nabla f)\Big)u\,dV,

which is non-positive when $\operatorname{Ric} \ge 0$ . Feldman, Ilmanen and Ni (2005) gave the version for expanders: with $\sigma = t - t_0$ and $(4\pi\sigma)^{-n/2}e^{-f}$ solving the conjugate heat equation, the functional $\mathcal W_+ = \int_M[\sigma(R + |\nabla f|^2) - f + n](4\pi\sigma)^{-n/2}e^{-f}\,dV$ satisfies

\frac{d}{dt}\mathcal W_+ = 2\sigma\int_M\Big|\operatorname{Ric} + \nabla^2 f + \frac{1}{2\sigma}g\Big|^2(4\pi\sigma)^{-n/2}e^{-f}\,dV.

Worked examples

The Gaussian. On flat $\mathbb R^n$ take $f = |x|^2/(4\tau)$ , so $u$ is the Euclidean heat kernel. Since $\int|x|^2 u\,dx = 2n\tau$ , we get $\mathcal W = 0$ . Every other admissible $f$ gives $\mathcal W \ge 0$ , and this statement is Gross's Gaussian logarithmic Sobolev inequality in another form. So $\mu(\mathbb R^n,\tau) = 0$ , and $\mu$ measures how far a manifold at scale $\sqrt\tau$ is from satisfying the Euclidean inequality.

Shrinking models. On a gradient shrinker $\operatorname{Ric} + \nabla^2 f = \frac{1}{2\tau}g$ , Perelman's Remark 3.2 shows that a minimiser of $\mu$ at the soliton's own scale, when one exists, satisfies the soliton equation, so the potential computes the entropy. For the round sphere the potential is constant, and

\mathcal W = -\frac n2 + \log\Big(\lvert S^n\rvert\Big(\frac{n-1}{2\pi}\Big)^{n/2}\Big).

$\mathcal W$ adds on products and a Gaussian factor contributes $0$ . In dimension three this gives:

Model	$\mathcal W$ at the soliton potential	$e^{\mathcal W}$
Gaussian $\mathbb R^3$	$0$	$1$
Round $S^3$	$-\tfrac32 + \log(2\sqrt\pi) \approx -0.235$	$\approx 0.791$
Cylinder $S^2\times\mathbb R$	$\log 2 - 1 \approx -0.307$	$\approx 0.736$

The derivation of these values is in the guidebook's chapter on the $\mathcal W$ -entropy.

Main theorems

No breathers (Perelman 2002, §§2–3). A breather is a solution with $g(t_2)$ isometric to $\alpha g(t_1)$ for some $t_1 < t_2$ , and it is steady, shrinking or expanding as $\alpha = 1$ , $\alpha < 1$ or $\alpha > 1$ . On a closed manifold every breather is a gradient soliton. The steady case follows from $\lambda$ . The expanding case uses the fact that $\bar\lambda$ is non-decreasing whenever it is non-positive. The shrinking case with $\lambda > 0$ needs $\mathcal W$ : then $\nu$ is negative and attained at some $\bar\tau$ , it is invariant under scaling and diffeomorphism, and equality in the monotonicity formula gives the soliton equation.

No local collapsing (Perelman 2002, §4). Let $g(t)$ be a smooth Ricci flow on a closed manifold for $t \in [0,T)$ with $T < \infty$ . Then there is $\kappa = \kappa(g(0),T) > 0$ such that $g(t)$ is κ-noncollapsed on the scale $\sqrt T$ : every ball $B$ of radius $r < \sqrt T$ with $|\operatorname{Rm}| \le r^{-2}$ on $B$ has volume at least $\kappa r^n$ .

The proof is short. Take a test function supported essentially on $B$ . Its normalising constant is roughly the logarithm of the volume ratio $r^{-n}\operatorname{Vol}(B)$ , and the curvature bound controls the other terms, so a tiny ratio makes $\mu(g(t), r^2)$ very negative. Monotonicity gives $\mu(g(t), r^2) \ge \mu(g(0), t + r^2)$ . Since $t + r^2$ stays bounded and $\mu(g(0),\cdot)$ is bounded below on bounded intervals, this is a contradiction. Kleiner and Lott write out the details. Combined with Hamilton's compactness theorem, it produces complete ancient limits at points of large curvature. Qi S. Zhang (2007) later proved a Sobolev inequality uniform along the flow, with long-time noncollapsing consequences that improve Perelman's local statement.

Sharp inequalities for heat kernel measures (Hein–Naber 2012). Let $\nu_s$ be the conjugate heat kernel measure at time $s < 0$ based at a point on the time- $0$ slice. With no curvature assumption beyond the standing bounded-geometry hypothesis of their paper,

\int_M \phi^2\log\phi^2\,d\nu_s \le 4\lvert s\rvert\int_M|\nabla\phi|^2\,d\nu_s \qquad\text{whenever } \int_M\phi^2\,d\nu_s = 1,

\int_M \phi^2\,d\nu_s \le 2\lvert s\rvert\int_M|\nabla\phi|^2\,d\nu_s \qquad\text{whenever } \int_M\phi\,d\nu_s = 0.

The constants are universal and match the Gaussian on $\mathbb R^n$ , and equality for non-constant $\phi$ forces the flow to split off a line. A consequence is the concentration inequality $\nu_s(A)\nu_s(B) \le \exp\big(-\operatorname{dist}_{g(s)}(A,B)^2/(8\lvert s\rvert)\big)$ .

ε-regularity. Hein and Naber show that the pointed entropy controls curvature: under a lower scalar curvature bound and a lower bound on $\mu$ , $\mathcal W_{x_0}(s) \ge -\varepsilon$ implies $|\operatorname{Rm}| \le \varepsilon^{-1}\lvert s\rvert^{-1}$ on a parabolic ball of radius $\sqrt{\varepsilon\lvert s\rvert}$ . Bamler (2020, Theorem 10.2) removes the extra hypotheses: there is $\varepsilon(n) > 0$ such that $\mathcal N_{x,t}(r^2) \ge -\varepsilon$ implies that the curvature scale at $(x,t)$ is at least $\varepsilon r$ .

Nash entropy as a local collapsing gauge (Bamler 2020). Bamler's bounds, for a Ricci flow on a compact manifold, include:

Gradient bound: $\lvert\nabla\mathcal N^*_s\rvert \le \big(\frac{n}{2(t-s)} - R_{\min}\big)^{1/2}$ , where $\mathcal N^*_s(x,t) = \mathcal N_{x,t}(t-s)$ and $R \ge R_{\min}$ at time $s$ . So nearby points have comparable entropy.
Lower volume bound: if $R \le r^{-2}$ on $B(x,t,r)$ then $\lvert B(x,t,r)\rvert_t \ge c\,e^{\mathcal N_{x,t}(r^2)}r^n$ . This extends Perelman's theorem and needs only an upper scalar curvature bound on the ball.
Upper volume bound: under a lower scalar curvature bound, $\lvert B(x,t,Ar)\rvert_t \le C\,e^{\mathcal N_{x,t}(r^2)}e^{C_0A^2}r^n$ for $A \ge 1$ .
Heat kernel bound: $K(x,t;y,s) \le C\,e^{-\mathcal N_{x,t}(t-s)}(t-s)^{-n/2}$ , again under a lower scalar curvature bound, with a Gaussian refinement around $H_n$ -centres.

Key results

Year	Result	Who
2002	$\mathcal F$ and $\mathcal W$ monotone; no breathers; no local collapsing; differential Harnack; pseudolocality	Perelman
2004	Entropy formula for the linear heat equation on a fixed metric	Ni
2004	Second variation of $\lambda$ and $\nu$ ; instability of certain Einstein manifolds	Cao, Hamilton, Ilmanen
2005	Entropy $\mathcal W_+$ for expanders	Feldman, Ilmanen, Ni
2007	Uniform Sobolev inequality along the flow	Zhang
2008	Sharp log-Sobolev inequalities on gradient solitons	Carrillo, Ni
2008	Scalar curvature and diameter bounds for Kähler–Ricci flow, after Perelman	Šešum, Tian
2009	Entropy of mean curvature flow; the stable self-shrinkers	Colding, Minicozzi
2012	Sharp log-Sobolev and Poincaré inequalities for conjugate heat kernels; ε-regularity	Hein, Naber
2020	Nash entropy bounds; compactness and structure theory	Bamler

Years are those of the version cited in the reading lists below.

Why it matters

Monotonicity turns the flow into something you can control globally. Perelman used $\mathcal W$ to prove noncollapsing, which rules out the cigar soliton as a blow-up limit and makes limits at singularities exist in the first place. The equality case singles out shrinkers. In dimension three Perelman identifies the asymptotic shrinkers of $\kappa$ -solutions with the reduced volume instead, so the two monotone quantities do complementary work on the way to the Poincaré conjecture.

Stability of singularity models is also read through entropy. Cao, Hamilton and Ilmanen computed the second variation of $\lambda$ and $\nu$ and showed that certain Einstein metrics are unstable: generic perturbations acquire higher entropy and cannot return near the original metric.

Perelman also noted that $\mathcal F$ and its first variation appear in string theory, where $\mathcal F$ describes the low energy effective action and $f$ is called the dilaton. He presented his link with the renormalization group as speculation. Current work on this is collected under physics.

Analogues in mean curvature flow

Mean curvature flow has direct analogues. Huisken's monotonicity formula says that the Gaussian-weighted area of a moving hypersurface is non-increasing, with equality on self-shrinkers. Colding and Minicozzi take the supremum over all Gaussians, of every centre and scale, to define an entropy of a hypersurface. They prove that spheres, cylinders and planes are the only stable self-shrinkers, so other singularities can be perturbed away.

The directions are opposite. Their entropy is non-increasing and takes its smallest value on the plane. The pointed $\mathcal W$ is non-decreasing in $t$ and reaches its largest value, $0$ , only on flat space. Also $\nu$ is an infimum where their entropy is a supremum, so loosely $-\nu$ plays the part of the logarithm of their entropy. Ricci flow has no ambient space supplying a family of Gaussians, and the conjugate heat kernel takes that role.

Extensions

Kähler. Perelman announced applications to Kähler–Ricci flow on Fano manifolds, and Šešum and Tian wrote out his scalar curvature and diameter bounds. Tian, Zhang and Zhu prove a Łojasiewicz-type inequality for Perelman's entropy along that flow.
Generalized and $G_2$ flows. Kopfer and Streets prove sharp Poincaré and log-Sobolev inequalities along generalized Ricci flow, and Li carries over analogues of Bamler's bounds. He and Yan estimate the time derivative of the pointed Nash entropy, which gives a proof of backward pseudolocality that does not use Bamler's compactness and structure theory and also applies to the $G_2$ -Laplacian flow.
Synthetic and singular spaces. Brena justifies the $\mathcal W$ formula on finite-dimensional RCD spaces, and Dai and Wang define the functionals on manifolds with non-isolated conical singularities.
The source of the formula. Perelman explains his reduced volume through an auxiliary high-dimensional space but gives no such origin for the entropy. Bustamante and Reiris derive it as a limit of Colding's monotonic volume on that space.

Where it stands

Entropy is now a precision tool. Bamler's compactness and structure theory rests on Nash entropy and heat kernel bounds: a uniform lower bound on Nash entropy gives compactness in the $\mathbb F$ -topology, and non-collapsed limits are smooth away from a singular set of parabolic codimension at least 4, with tangent flows that are gradient shrinking solitons, possibly singular. Hein and Naber's concentration estimate has been sharpened: Koirala proves a Gaussian isoperimetric inequality for conjugate heat kernel measures that recovers them. Carrillo and Ni proved sharp log-Sobolev inequalities on gradient shrinkers, with the constant set by a geometric invariant of the soliton.

Chan, Ma and Zhang show that a local $\nu$ -functional depends only on the Nash entropy at the centre of the disk. Bamler, Chan, Ma and Zhang prove that steady gradient solitons with bounded Nash entropy have volume growth at least $r^{(n+1)/2}$ , which is the Bryant soliton's rate. For singularities, Li, Yan and Zhang use a Łojasiewicz–Simon inequality for Perelman's entropy to prove strong uniqueness of tangent flows when one tangent flow comes from an asymptotically conical shrinker.

Open problems

Which shrinkers occur. Entropy says that tangent flows are shrinkers but not which ones. Shrinkers in low dimensions are understood far better than those in dimension four and above, where no classification is known to us.
Generic singularities. A Ricci flow counterpart of the Colding–Minicozzi list of stable models is, as far as we know, not available. Cao, Hamilton and Ilmanen give the linear stability framework, but the full list of stable shrinkers and a perturbation theory for the flow remain unsettled.
Uniqueness of tangent flows. Strong uniqueness is known in cases such as the asymptotically conical one above. We know of no general theorem.
Collapsed flows. Bamler's theory needs a Nash entropy lower bound. Collapsed behaviour falls outside it and is much less understood.

Quick answers

Common questions

What is Perelman's entropy?

Perelman's entropy is the functional $\mathcal W(g,f,\tau) = \int_M[\tau(R+|\nabla f|^2)+f-n](4\pi\tau)^{-n/2}e^{-f}\,dV$ , introduced in his 2002 paper on Ricci flow. It never decreases when $g$ evolves by Ricci flow and $(4\pi\tau)^{-n/2}e^{-f}$ solves the conjugate heat equation. Its infimum over $f$ , written $\mu(g,\tau)$ , is what Perelman used to prove that closed Ricci flows do not collapse locally in finite time.

Why is it called entropy?

It is called entropy because it contains the term $\int f\,u\,dV$ , which up to a constant is the Boltzmann–Shannon entropy $-\int u\log u\,dV$ of the density $u$ . In Perelman's statistical analogy, $\mathcal W$ is actually minus an entropy: his thermodynamic quantity is $S = -\mathcal W$ . So $\mathcal W$ increases while $S$ decreases, and the analogy is a heuristic that the proofs do not use.

Is Ricci flow a gradient flow?

Ricci flow is a gradient flow of Perelman's $\mathcal F$ -functional once it is modified by diffeomorphisms and a measure $dm = e^{-f}\,dV$ is held fixed. For a fixed measure the $L^2$ gradient of $\mathcal F$ is $-(\operatorname{Ric}+\nabla^2 f)$ , so the flow $\partial_t g = -2(\operatorname{Ric}+\nabla^2 f)$ is its gradient flow. Different choices of measure give the same flow up to diffeomorphism, so Perelman compares the measure to a choice of gauge.

What is the conjugate heat equation in Ricci flow?

The conjugate heat equation is $\partial_t u = -\Delta u + Ru$ , which is solved backward in time along a Ricci flow. It is the adjoint of the heat equation $\partial_t h = \Delta h$ for the evolving volume form, so $\int_M hu\,dV$ stays constant in time. Perelman's monotonicity formulas hold when the weight $e^{-f}$ , or $(4\pi\tau)^{-n/2}e^{-f}$ , solves this equation.

What is Nash entropy in Ricci flow?

The pointed Nash entropy is $\mathcal N_{x,t}(\tau) = \int_M f\,d\nu - \frac n2$ , where $d\nu = (4\pi\tau)^{-n/2}e^{-f}\,dV$ is the conjugate heat kernel measure based at $(x,t)$ . It equals the average of Perelman's pointed $\mathcal W$ -entropy over scales from $0$ to $\tau$ , and it is non-positive and non-increasing in $\tau$ . Hein and Naber defined it, and Bamler used it to measure local collapsedness at a single point, since $e^{\mathcal N_{x,t}(r^2)}$ behaves like the volume ratio at scale $r$ under suitable scalar curvature bounds.

How does entropy prove no local collapsing?

A ball of radius $r$ with $|\operatorname{Rm}|\le r^{-2}$ and a very small volume ratio gives a test function that makes $\mu(g(t),r^2)$ very negative. Monotonicity bounds $\mu(g(t),r^2)$ below by $\mu(g(0),t+r^2)$ , and on a closed manifold this is bounded below when $t+r^2$ stays bounded. So the volume ratio cannot go to zero in finite time, which is Perelman's no local collapsing theorem.

What is the difference between Perelman's entropy and reduced volume?

Both are monotone along Ricci flow and both prove noncollapsing, but they are built differently. The $\mathcal W$ -entropy uses a solution of the conjugate heat equation and is a log-Sobolev quantity, while reduced volume uses the $\mathcal L$ -length from a base point and comes from a volume comparison. They are linked by Perelman's Corollary 9.5: the conjugate heat kernel satisfies $f \le \ell$ , where $\ell$ is the reduced distance.

If you are new to this

Foundational papers

The papers the rest of the topic builds on, in reading order. Titles open the paper.

2002
The entropy formula for the Ricci flow and its geometric applications Grisha Perelman · arXiv:math/0211159 Perelman: the entropy formula for the Ricci flow and its geometric applications. View paper Connections
2004
The entropy formula for linear heat equation Lei Ni · The Journal of Geometric Analysis, vol. 14 Ni: the entropy formula on a fixed metric, the simplest place to see the computation. DOI 10.1007/bf02921867
2004
Gaussian densities and stability for some Ricci solitons Huai-Dong Cao, Richard S. Hamilton, Tom Ilmanen · arXiv:math/0404165 Cao–Hamilton–Ilmanen: second variation of $λ$ and $ν$ and linear stability of solitons. PDF arXiv:math/0404165
2005
Entropy and reduced distance for Ricci expanders Michael Feldman, Tom Ilmanen, Lei Ni · The Journal of Geometric Analysis, vol. 15 Feldman–Ilmanen–Ni: the entropy formula for expanders. DOI 10.1007/bf02921858
2007
A uniform Sobolev inequality under Ricci flow Qi S. Zhang · arXiv:0706.1594 Zhang: a Sobolev inequality uniform along the flow, with long-time noncollapsing. PDF arXiv:0706.1594
2008
Sharp logarithmic Sobolev inequalities on gradient solitons and applications Jose Carrillo, Lei Ni · arXiv:0806.2417 Carrillo–Ni: sharp log-Sobolev inequalities on gradient solitons. PDF arXiv:0806.2417
2008
BOUNDING SCALAR CURVATURE AND DIAMETER ALONG THE KÄHLER RICCI FLOW (AFTER PERELMAN) Natasa Sesum, Gang Tian · Journal of the Institute of Mathematics of Jussieu, vol. 7 Šešum–Tian: Perelman's scalar curvature and diameter bounds along Kähler–Ricci flow. DOI 10.1017/s1474748008000133
2012
New logarithmic Sobolev inequalities and an ε-regularity theorem for the Ricci flow Hans-Joachim Hein, Aaron Naber · arXiv:1205.0380 Hein–Naber: new log-Sobolev inequalities and ε-regularity for Ricci flow. PDF arXiv:1205.0380
2020
Entropy and heat kernel bounds on a Ricci flow background Richard H Bamler · arXiv:2008.07093 Bamler: entropy and heat kernel bounds, the foundation of the higher-dimensional theory. View paper Connections
2020
Structure theory of non-collapsed limits of Ricci flows Richard H Bamler · arXiv:2009.03243 Bamler: structure of non-collapsed limits, where Nash entropy bounds are put to use. View paper Connections
2009
Generic mean curvature flow I; generic singularities Tobias H. Colding, William P. Minicozzi · arXiv:0908.3788 Colding–Minicozzi: the entropy of mean curvature flow and generic singularities. PDF arXiv:0908.3788

Where to start reading

Surveys & books

Longer, gentler routes in, and the references specialists keep on their desks.

2006
Notes on Perelman's papers Bruce Kleiner, John Lott · Geom. Topol. 12 (2008) 2587-2855 Kleiner–Lott: detailed proofs of Perelman's monotonicity formulas. View paper Connections Journal version
2006
Lectures on the Ricci Flow Peter Topping · Cambridge University Press Topping's lectures, whose later chapters build the $F$ and $W$ functionals from scratch. DOI 10.1017/CBO9780511721465

Keep going

Across this site

Where Perelman entropy connects to the rest of Neckpinch.

Learn it step by step

Ricci flow as a gradient flow: the first variation, $\lambda$ , and steady and expanding breathers.
The $\mathcal W$ -entropy: the log-Sobolev link, no shrinking breathers, and the statistical analogy.
Eigenvalue problems and Perelman's $\lambda$ : the spectrum of $-4\Delta + R$ .
Entropy, information and diffusion and self-similar variables and monotonicity: the same ideas for the plain heat equation.
Optimal transport and Ricci curvature and Fokker–Planck and diffusion models: the entropy picture seen from transport and from probability.
Stage 12 of the Guidebook: where all of this sits in the route to the proof.

Neighbouring topics

Noncollapsing: the main consequence, and what happens when it fails.
Reduced volume: the other monotone quantity.
Heat kernel and Harnack inequalities: the analysis under the pointed quantities.
Ricci solitons and ancient solutions: the equality cases and the blow-up limits.
Limit spaces and singularities: where Nash entropy is put to work.

Concept cards

Perelman's $\mathcal F$ -functional, the $\mathcal W$ -entropy and Nash entropy and $\mathbb F$ -convergence: the short versions.
Ancient κ-solutions and synthetic and super Ricci flows: where noncollapsing and the entropy ideas lead.

Papers and people

Perelman's first preprint and Kleiner and Lott's notes: read these side by side.
Bamler's survey: the map of the modern field.
All papers on this topic and who has written most on it here.
A rigidity result for ancient flows by Zhang, using a log entropy condition near infinity, and the Fisher metric of the heat kernel by Chow and Koirala, which relates its trace to the pointed Nash entropy.

Reading trails and applications

Perelman's proof and the expositions that checked it and Ricci flow in every dimension: ordered reading through this material.
Heat kernel methods and optimal transport in machine learning: the nearest applications. We list none that use Perelman's entropy itself.

Learn it step by step

In the guidebook

Chapters of the Guidebook that build perelman entropy from the ground up, with proofs, figures and exercises.

6A.12
Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives
Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.
6A.13
Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
6B.16
Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity
Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.
6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.
6D.4
Fokker–Planck, Time Reversal and Diffusion ModelsBook 6D · Diffusions, Finance and Computation
How densities of diffusions evolve, relative entropy and convergence to equilibrium, reversing a diffusion in time with the score, and the generative models built on it, with Perelman's conjugate heat equation as the geometric twin.
12A.2
Ricci Flow as a Gradient FlowBook 12A · Entropy and Noncollapsing
The 𝓕 functional, its monotonicity, and no steady breathers.
12A.3
The 𝓦-EntropyBook 12A · Entropy and Noncollapsing
A scale-invariant entropy, log-Sobolev, and no shrinking breathers.

If you follow the research

Latest papers · 34 papers

Filter on Papers

The newest 20, updated every weekday from arXiv. Follow this topic with its own RSS feed, or see who writes most on it.

math.DGNov 4, 2026arXiv:2611.06678

Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres

Reto Buzano, Shengwen Wang

We study the Gromov-Hausdorff stability of round spheres under perturbations of Perelman's nu-entropy. We show that in dimension two almost maximal entropy, together with an area normalisation, implies Gromov-Hausdorff closeness to the round two-sphere. In contrast, in dimensions at least three, we construct smooth metrics on the sphere whose entropies converge to that of the round sphere, but which remain a definite Gromov-Hausdorff distance away from it. These metrics can be chosen conformal and uniformly bi-Lipschitz equivalent to the round metric, with the same volume as the round sphere. Finally, in dimension three, we contrast this Gromov-Hausdorff instability with a dynamical stability result: metrics with nu-entropy above that of the round shrinking cylinder converge exponentially fast to round metrics under the volume-normalised Ricci flow.

Ricci flow Perelman entropy 3-manifolds Limit spaces & RCD Stability & uniqueness Long-time existence & convergence

View paper Connections

math.DGOct 30, 2026arXiv:2610.41345

On the geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness

Yu Li, Longhuan Yan, Wenjia Zhang

We prove strong uniqueness of tangent flows at the first singular time of a closed Ricci flow when one tangent flow is generated by a complete smooth asymptotically conical Ricci shrinker. The modified flow converges to this shrinker in a fixed gauge, with quantitative estimates on expanding domains. The main ingredients are a Łojasiewicz-Simon inequality for Perelman's entropy, based on the global weighted divergence-free gauge constructed in the preceding paper, and estimates controlling the geometric radius under Ricci flow.

Ricci flow Ricci solitons Singularities Perelman entropy Stability & uniqueness Boundaries & non-compact

View paper Connections

math.DGOct 28, 2026arXiv:2610.38418

Backward Pseudolocality of $G_{2}$ -Laplacian Flow

Weiyong He, Longhuan Yan

In this paper, we establish a new estimate on the time-derivative of pointed Nash entropy and use it to give a new proof of backward pseudolocality for Ricci flow. The proof does not rely on Bamler's compactness theorem and structure theory concerning limits of Ricci flows. Our methods also work for other reasonable geometric flows, like the $G_{2}$ -Laplacian, at a scale depending on its deviation from Ricci flow and its scalar curvature.

Ricci flow Generalized & G₂ flows Perelman entropy Scalar curvature Limit spaces & RCD

View paper Connections

math.DGOct 28, 2026arXiv:2610.38408

Pseudolocality of $G_{2}$ -Laplacian Flow

Weiyong He, Longhuan Yan

In this paper, we give a new proof of Perelman's pseudolocality theorem for Ricci flow, which does not rely on Perelman's Harnack-type inequality. The key step is an estimate relating initial local $μ$ -entropy to pointed Nash entropy at later spacetime points. The argument also works for other reasonable geometric flows, like the $G_{2}$ -Laplacian flow, at a scale depending on its deviation from Ricci flow and its scalar curvature lower bound. As a consequence, we can also prove the quantitative rigidity results concerning pointed Nash entropy due to Hein-Naber and Bamler for such geometric flows. In the appendix, we also show that the other results in Bamler's work (arXiv:2008.07093), when suitably modified, remain valid for Laplacian flow with bounded torsion. These include log-Sobolev/Poincaré inequality for conjugate heat kernel measure, heat kernel estimates, and dependence of the pointed Nash entropy on its basepoint.

Ricci flow Generalized & G₂ flows Perelman entropy Harnack inequalities Heat kernel Scalar curvature Stability & uniqueness

View paper Connections

math.DGSep 23, 2026arXiv:2609.27644

On the singularity formation of gauge fields coupled to Ricci flow

Andoni Royo Abrego

We study a family of coupled geometric evolution equations describing the deformation of a Riemannian metric and a non-abelian gauge field on closed manifolds, which generalizes the Ricci–Yang–Mills flow. We derive interior curvature estimates, find a preserved integral curvature condition, and discover a scaling invariant monotone functional analogue to Perelman's entropy. In particular, we prove strong dominance of the Riemannian curvature over the gauge curvature at finite-time singularities in all dimensions. We also provide a non-trivial explicit example of a shrinking self-similar solution on a $S U (2)$ bundle over $S^{4}$ .

Ricci flow Singularities Perelman entropy

View paper Connections

math.DGSep 16, 2026arXiv:2609.18594

Heat kernel on Ricci shrinker metric measure spaces

Bing Wang, Jie Wang

As a metric measure space possessing positive Bakry-Émery curvature, a Ricci shrinker with potential function $f$ admits a heat kernel $H_{f}$ under the weighted volume measure $e^{- f} d v$ . In this paper, we study $H_{f}$ systematically and establish a series of fundamental estimates. Moreover, we clarify the equivalence between $H_{f}$ and the spacetime heat kernel $H (x, t; y, s)$ under Ricci flow induced by a Ricci shrinker. Inspired by this relation, we extend corresponding analysis tools of Ricci flows to Ricci shrinker metric measure spaces, such as Nash entropy and reduced distance. As a direct application, we prove that $∣ \nabla R ∣ = o (f^{3/2})$ implies $R = o (f)$ .

Ricci flow Ricci solitons Perelman entropy Reduced volume Heat kernel Limit spaces & RCD

View paper Connections

math.DGAug 14, 2026arXiv:2608.14478

The Fisher Metric of the Ricci Flow Heat Kernel

Bennett Chow, Robert Koirala

We introduce and study a Fisher information metric $g_{τ}^{F}$ associated to the conjugate heat kernel of a Ricci flow $(M^{n}, g_{t})$ . This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that $g_{τ}^{F}$ is monotone in scale and satisfies $g_{τ}^{F} \leq g_{t}$ . We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect $g_{t} - g_{τ}^{F} .$ This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities $0 < g_{τ}^{F} < g_{t}$ at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for $φ$ -divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to $0$ to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's $ε$ -regularity theorem.

Ricci flow Perelman entropy Heat kernel Stability & uniqueness Eigenvalues & spectrum

View paper Connections

cond-mat.stat-mechJun 12, 2026arXiv:2606.14186

Stochastic Thermodynamics on Time-Evolving Curved Spaces

Rihito Nagase, Shoki Sugimoto, Asuka Takatsu, Takahiro Sagawa

We construct stochastic thermodynamics of overdamped Langevin systems on nonrelaticvistic curved spaces with time-dependent metrics. The time dependence of the metric contributes to the energy balance by performing work on the kinetic energy, which is instantaneously dissipated as heat in the overdamped regime. This contribution makes our framework thermodynamically consistent so that entropy production satisfies the second law of thermodynamics. As a special case, when the metric evolves according to backward Ricci flow, the entropy balance exhibits a structure similar to Perelman's entropy functional. Our framework provides a way to quantify thermodynamic costs in dynamics on time-evolving spaces such as diffusion on membranes.

Ricci flow Perelman entropy

View paper Connections

math.DGJun 4, 2026v2arXiv:2606.06619

On the structure of complete $G_{2}$ -solitons

Haozhao Li, Yuanqing Ma, Kai Zheng

In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed $G_{2}$ -structures. Under a lower scalar-curvature bound and a distance-dependent bound on the gradient of the soliton potential, we prove pointed measured Gromov-Hausdorff compactness. With a uniform lower bound for the localised Perelman entropy, the Gromov-Hausdorff convergence improves to pointed $C^{1, α}$ convergence. Our principal result shows that this $C^{1, α}$ convergence upgrades to smooth convergence on the regular set. More precisely, after passing to a subsequence, the metrics, the defining positive $3$ -forms, and the soliton potentials converge smoothly on every compact subset of the regular set, and the limiting data define a gradient Laplacian soliton. The proof develops a local entropy method adapted to $G_{2}$ -solitons. Since the available $C^{1, α}$ control does not directly close an elliptic bootstrap for the $G_{2}$ -soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy $ε$ -regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed $C^{1, α}$ convergence gives an almost-Euclidean local isoperimetric inequality, which in turn verifies the required small-entropy condition automatically. The resulting curvature bounds are then combined with $G_{2}$ -specific differential identities and quantitative interior estimates to control the soliton data. Finally, at the critical exponent in dimension seven, we show that a uniform weighted $L^{\frac{7}{2}}$ -curvature bound then yields pointed $C^{\infty}$ compactness.

Generalized & G₂ flows Perelman entropy Scalar curvature Limit spaces & RCD Long-time existence & convergence Geometric inequalities

View paper Connections

math.DGMay 20, 2026arXiv:2605.21193

Sharp Gaussian Isoperimetry along a Ricci Flow

Robert Koirala

We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set concentration estimate. In particular, this recovers the exponential concentration estimate of Hein–Naber from a sharper isoperimetric profile. We also derive Gaussian rearrangement inequalities, recover the sharp Hein–Naber log-Sobolev inequality, and identify the universal Gaussian-model constants in Bamler's $L^{p}$ -Poincaré inequalities. Further applications include Gaussian-profile localization near Bamler's $H_{n}$ -centers, convex-order and moment estimates for logarithmic derivatives of the conjugate heat kernel, reverse hypercontractivity, entropy-regular profile stability, and a path-space Bobkov inequality.

Ricci flow Perelman entropy Heat kernel Stability & uniqueness Geometric inequalities

View paper Connections

math.DGJan 5, 2026arXiv:2601.01863

A Spinorial Perelman's Functional: Critical Points and Gradient Flow

Tsz-Kiu Aaron Chow, Frederick Tsz-Ho Fong

In this article, we introduce an energy functional on closed Riemannian spin manifolds which unifies Perelman's W- and F-functionals, Baldauf-Ouzch's E-functional, and Dirchlet energy for spinors. We compute its first variation formula, and show that its critical points under natural constraints are twisted Ricci solitons and eigen-spinsors of the weighted Dirac operator. We introduce a negative L^2-gradient flow of this functional, and establish its short-time existence and uniqueness via contraction mapping methods.

Ricci solitons Perelman entropy Stability & uniqueness Harmonic maps & DeTurck trick

View paper Connections

gr-qcAug 13, 2025arXiv:2508.10939

The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory

Sergiu I. Vacaru, Elşen V. Veliev

The gravitational field equations in general relativity (GR) consist of a sophisticated system of nonlinear partial differential equations. Solving such equations in some generic off-diagonal forms is usually a hard analytic or numeric task. Physically important solutions in GR were constructed using a diagonal ansatz for metrics with a maximum of 4 independent coefficients. The Einstein equations can be solved in exact or parametric forms determined by some integration constants for corresponding assumptions on spherical or cylindrical spacetime symmetries. The anholonomic frame and connection deformation method allows us to construct generic off-diagonal solutions described by 6 independent coefficients of metrics depending, in general, on all spacetime coordinates. New types of exact and parametric solutions are determined by generating and integration functions and (effective) generating sources. They may describe vacuum gravitational and matter fields solitonic hierarchies; locally anisotropic polarizations of physical constants for black holes, wormholes, black toruses, or cosmological solutions; various types of off-diagonal deformations of horizons, etc. The additional degrees of freedom (related to off-diagonal coefficients) can be used to describe dark energy and dark matter configurations and elaborate locally anisotropic cosmological scenarios. In general, the generic off-diagonal solutions do not involve certain hypersurface or holographic configurations and can't be described in the framework of the Bekenstein-Hawking thermodynamic paradigm. We argue that generalizing the concept of G. Perelman's entropy for relativistic Ricci flows allows us to define and compute geometric thermodynamic variables for all possible classes of solutions in GR.

Ricci flow Perelman entropy Physics

View paper Connections

math.DGJul 7, 2025v2arXiv:2507.05032

On a parabolic curvature lower bound generalizing Ricci flows

Marco Flaim, Erik Hupp

Optimal transport plays a major role in the study of manifolds with Ricci curvature bounded below. Some results in this setting have been extended to super Ricci flows, revealing a unified approach to analysis on Ricci nonnegative manifolds and Ricci flows. However we observe that the monotonicity of Perelman's functionals ( $F$ , $W$ , reduced volume), which hold true for Ricci flows and Ricci nonnegative manifolds, cannot be strictly generalized to super Ricci flows. In 2010 Buzano introduced a condition which still generalizes Ricci flows and Ricci nonnegative manifolds, and on which Perelman's monotonicities do hold. We provide characterizations of this condition using optimal transport and understand it heuristically as Ricci nonnegativity of the space-time. This interpretation is consistent with its equivalence to Ricci nonnegativity on Perelman's infinite dimensional manifold. More precisely, we prove that for smooth evolutions of Riemannian manifolds, this condition is equivalent to a Bochner inequality (resembling Perelman's Harnack inequality but for the forward heat flow), a gradient estimate for the heat flow, a Wasserstein contraction along the adjoint heat flow, the convexity of a modified entropy along Wasserstein geodesics, and an Evolutionary Variational Inequality (EVI). The optimal transport statements use Perelman's $L$ distance as cost, as first studied on Ricci flows by Topping and by Lott. We also consider dimensionally improved and weighted versions of these conditions. The dimensional Bochner inequality and all gradient estimates for the forward heat equation, along with the EVIs, appear to be new even for general Ricci flows, and are related to the Hamiltonian perspective on the $L$ distance. Most of our proofs do not use tensor calculus or Jacobi fields, suggesting the possibility of future extensions to more singular settings.

Ricci flow Perelman entropy Reduced volume Harnack inequalities Weak & singular flows Limit spaces & RCD

View paper Connections

cs.LGMar 25, 2025arXiv:2503.19867

Geometric Meta-Learning via Coupled Ricci Flow: Unifying Knowledge Representation and Quantum Entanglement

Ming Lei, Christophe Baehr

This paper establishes a unified framework integrating geometric flows with deep learning through three fundamental innovations. First, we propose a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology, formally proved to preserve isometric knowledge embedding (Theorem ). Second, we derive explicit phase transition thresholds and critical learning rates (Theorem ) through curvature blowup analysis, enabling automated singularity resolution via geometric surgery (Lemma ). Third, we establish an AdS/CFT-type holographic duality (Theorem ) between neural networks and conformal field theories, providing entanglement entropy bounds for regularization design. Experiments demonstrate 2.1 $\times$ convergence acceleration and 63% topological simplification while maintaining $O (N lo g N)$ complexity, outperforming Riemannian baselines by 15.2% in few-shot accuracy. Theoretically, we prove exponential stability (Theorem ) through a new Lyapunov function combining Perelman entropy with Wasserstein gradient flows, fundamentally advancing geometric deep learning.

Ricci flow Singularities Surgery Perelman entropy Stability & uniqueness Discrete & graph Ricci

View paper Connections

math.DGMar 4, 2025arXiv:2503.03017

Perelman's entropy and heat kernel bounds on RCD spaces

Camillo Brena

We study Perelman's W-entropy functional on finite-dimensional RCD spaces, a synthetic generalization of spaces with Bakry-Émery Ricci curvature bounded from below. We rigorously justify the formula for the time derivative of the W-entropy and derive its monotonicity and rigidity properties. Additionally, we establish bounds for solutions of the heat equation, which are of independent interest.

Perelman entropy Heat kernel Weak & singular flows Limit spaces & RCD Stability & uniqueness

View paper Connections

math.DGJan 22, 2025v3arXiv:2501.12949

Deriving Perelman's entropy from Colding's monotonic volume

Ignacio Bustamante, Martin Reiris

In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop-Gromov volume comparison applied to a suitably constructed $N$ -space, which becomes Ricci-flat as $N \to \infty$ , Perelman did not provide a corresponding explanation for the origin of the entropy. In this article, we demonstrate that Perelman's entropy emerges as the limit of Colding's monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman's $N$ -space.

Ricci-flat & Calabi–Yau Perelman entropy Reduced volume

View paper Connections

hep-thDec 13, 2024arXiv:2412.10364

Navigating string theory field space with geometric flows

Saskia Demulder, Dieter Lust, Thomas Raml

The Swampland Distance Conjecture postulates the emergence of an infinite tower of massless states when approaching infinite-distance points in moduli space. However, most string backgrounds are supported by fluxes, and therefore depart from the purely geometric paradigm. This fact requires an extension of the Swampland conjectures to scalar field spaces with non-trivial potentials, rather than just moduli spaces. To address this task, we utilise geometric flows, in particular generalised Ricci flow, to probe the associated scalar field spaces. Considering internal spaces supported by three-form fluxes, we first show that the distance defined in terms of the Perelman entropy functional needs to be refined in order to encompass fluxes. Doing so, we extend the Ricci Flow Conjecture to include Kalb-Ramond flux besides the metric and the dilaton field. This allows us to probe infinite-distance points within these scalar field spaces in a purely geometric way. We subsequently construct a geometric flow for internal manifolds supported by Ramond-Ramond fluxes and discuss its role in the Ricci Flow Conjecture. Our analysis suggests that in the presence of fluxes the Distance Conjecture might be better characterised in terms of a cost function on the space of metrics, rather than a genuine distance.

Ricci flow Generalized & G₂ flows Perelman entropy Smale conjecture & diffeomorphisms Physics

View paper Connections

gr-qcSep 24, 2024arXiv:2410.03700

Nonmetric geometric flows and quasicrystalline topological phases for dark energy and dark matter in $f (Q)$ cosmology

L. Bubuianu, E. Nurlan, J. O. Seti + 2 more

We elaborate on nonmetric geometric flow theory and metric-affine gravity with applications in modern cosmology. Two main motivations for our research follow from the facts that 1) cosmological models for $f (Q)$ modified gravity theories, MGTs, are efficient for describing recent observational data provided by the James Webb Space Telescope; and 2) the statistical thermodynamic properties of such nonmetric locally anisotropic cosmological models can be studied using generalizations of the concept of G. Perelman entropy. We derive nonmetric distorted R. Hamilton and Ricci soliton equations in such canonical nonholonomic variables when corresponding systems of nonlinear PDEs can be decoupled and integrated in general off-diagonal forms. This is possible if we develop and apply the anholonomic frame and connection deformation method involving corresponding types of generating functions and generating sources encoding nonmetric distortions. Using such generic off-diagonal solutions (when the coefficients of metrics and connections may depend generically on all spacetime coordinates), we model accelerating cosmological scenarios with quasi-periodic gravitational and (effective) matter fields; and study topological and nonlinear geometric properties of respective dark energy and dark matter, DE and DM, models. As explicit examples, we analyze some classes of nonlinear symmetries defining topological quasicrystal, QC, phases which can modified to generate other types of quasi-periodic and locally anisotropic structures. The conditions when such nonlinear systems possess a behaviour which is similar to that of the Lambda cold dark matter ( $Λ$ CDM) scenario are stated. We conclude that nonmetric geometric and cosmological flows can be considered as an alternative to the $Λ$ CDM concordance models and speculate on how such theories can be elaborated.

Anisotropic & nonlocal flows Ricci solitons Perelman entropy Physics

View paper Connections

hep-thSep 24, 2024arXiv:2410.03698

Nonassicative cosmological solitonic R-flux deformations in gauge gravity and G. Perelman geometric flow thermodynamics

L. Bubuianu, J. O. Seti, S. Vacaru, E. V. Veliev

We elaborate on a model of nonassociative and noncommutative gauge gravity for the de Sitter gauge group $S O (4, 1)$ embedding extensions of the affine structure group $A f (4, 1)$ and the Poincaré group $I S O (3, 1)$ . In string theory, such nonassociative gauge gravity theories are determined by star product R-flux deformations. They are new avenues to quantum gravity and geometric and quantum information theories. We analyze physically important and geometric thermodynamic properties of new classes of generic off-diagonal cosmological solitonic solutions encoding nonassociative effective sources. Particularly, we focus on modelling by such solutions of locally anisotropic and inhomogeneous dark matter and dark energy structures generated as nonassociative solitonic hierarchies. Such accelerating cosmological evolution scenarios can't be described in the framework of the Bekenstein-Hawking thermodynamic formalism. This motivates a change in the gravitational thermodynamic paradigm by considering nonassociative and relativistic generalizations of the concept of W-entropy in the theory of Ricci flows. Finally, we compute the corresponding modified G. Perelman's thermodynamic variables and analyze the temperature-like evolution of cosmological constants determined by nonassociative cosmological flows.

Ricci flow Perelman entropy Homogeneous spaces Physics

View paper Connections

math.DSSep 15, 2024arXiv:2410.02457

Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds

Zeraoulia Rafik, Sobhan Sobhan Allah

This paper explores a novel approach to modeling the positional dynamics of stars using discrete dynamical systems. We define star evolution through discrete-time update rules based on right ascension, declination, and distance, incorporating chaotic behavior via nonlinear functions and external perturbations. By applying Ricci flow and Riemannian metrics, we provide new insights into the positional dynamics of stars. Theoretical computations of Perelman entropy are used to assess system complexity, with high-precision Runge-Kutta methods ensuring accurate solutions for our chaotic model. We quantify chaos using Lyapunov exponents and perform bifurcation analysis to study how parameter variations affect the dynamics. Comparing our model to the Lorenz attractor reveals both similarities and unique characteristics in stellar dynamics. Our results show that entropy increases exponentially, indicating that predicting star positions with precision becomes increasingly challenging over time. This study advances the understanding of chaos in celestial systems and contributes to dynamical systems theory by integrating chaos theory with astronomical modeling.

Ricci flow Perelman entropy

View paper Connections

All 34 papers on Perelman entropy, by year

2026 2025 2024 2023 2022 2021

Perelman entropy

Nash entropy and 𝓕-convergenceadvanced

What it is

Definitions and conventions

#The energy F\mathcal FF and the eigenvalue λ\lambdaλ

#The entropy W\mathcal WW and its infima

#The pointed quantities

The monotonicity formulas

#Why Ricci flow is a gradient flow

#The W\mathcal WW formula and its pointwise form

#Relatives on a fixed background and on expanders

Worked examples

Main theorems

Key results

Why it matters

Analogues in mean curvature flow

Extensions

Where it stands

Open problems

What is Perelman's entropy?

Why is it called entropy?

Is Ricci flow a gradient flow?

What is the conjugate heat equation in Ricci flow?

What is Nash entropy in Ricci flow?

How does entropy prove no local collapsing?

What is the difference between Perelman's entropy and reduced volume?

The energy $\mathcal F$ and the eigenvalue $\lambda$

The entropy $\mathcal W$ and its infima

The pointed quantities

Why Ricci flow is a gradient flow

The $\mathcal W$ formula and its pointwise form

Relatives on a fixed background and on expanders