Yamabe flow

What it is

Hamilton proposed the flow $\partial_t g = -(R - \bar r)\,g$, where $\bar r$ is the average scalar curvature, which keeps the volume fixed. Only the conformal factor changes, so for $n\ge 3$ writing $g = u^{4/(n-2)} g_0$ turns the flow into a single parabolic equation for the function $u$. Its fixed points are exactly the metrics of constant scalar curvature, and for $n\ge 3$ the total scalar curvature decreases along the flow. On surfaces it coincides with the normalised Ricci flow, since there $\operatorname{Ric} = \tfrac R2 g$.

The Yamabe problem asks whether every conformal class on a closed manifold of dimension $n\ge 3$ contains such a metric. Yamabe's 1960 argument had a gap. Trudinger, Aubin and Schoen completed the proof: Aubin treated $n\ge 6$ with the metric not locally conformally flat, and Schoen settled the remaining cases using the positive mass theorem. The flow asks for more: that a natural evolution finds the metric. The elliptic story is told step by step in the guidebook chapter on the Yamabe problem and critical exponents.

The equations

#The conformal factor

Set $c_n = \frac{4(n-1)}{n-2}$. Under $g = u^{4/(n-2)}g_0$ the scalar curvature transforms by

and the normalised flow becomes

This equation is parabolic as it stands. Ricci flow is only weakly parabolic and needs DeTurck's trick; here the conformal restriction removes the diffeomorphism invariance, so short-time existence is standard. On $\mathbb R^n$ the unnormalised flow, after rescaling time, reads $\partial_t(v^{(n+2)/(n-2)}) = \Delta v$. With $w = v^{(n+2)/(n-2)}$ this is the fast diffusion equation $\partial_t w = \Delta w^m$, $m = \frac{n-2}{n+2} < 1$, which ties the subject to a large PDE literature.

#Scalar curvature

Along the normalised flow,

The maximum principle gives $\partial_t R_{\min} \ge R_{\min}(R_{\min} - \bar r)$ in the barrier sense. Since $R_{\min}\le\bar r$, a negative minimum can only rise and a positive minimum stays positive, though it may fall. Note the factor $n-1$: for $n\ge 3$ the diffusion is faster than for the scalar curvature under Ricci flow, whose equation is $\partial_t R = \Delta R + 2|\operatorname{Ric}|^2$.

#The energy

The flow is the gradient flow of the Yamabe functional, which is the normalised total scalar curvature restricted to the conformal class:

Its infimum is the Yamabe constant $Y(M,[g_0])$. Aubin's inequality says $Y(M,[g_0])\le Y(S^n) = n(n-1)\operatorname{Vol}(S^n)^{2/n}$. Differentiating along the flow gives

so with volume fixed $\bar r$ decreases to a limit $\kappa$. At $n=2$ the right side vanishes, as Gauss–Bonnet requires.

Worked examples

Constant scalar curvature. If $R_{g_0}\equiv R_0$ is constant, the unnormalised flow $\partial_t g = -Rg$ is solved by $g(t) = (1 - R_0t)\,g_0$, and the normalised flow does not move at all. For $R_0>0$ the unnormalised solution disappears at $T = 1/R_0$; for $R_0<0$ it expands forever; for $R_0 = 0$ it is static.

The round sphere. A sphere of radius $\rho_0$ has $R_0 = n(n-1)/\rho_0^2$, so it vanishes at $T = \rho_0^2/(n(n-1))$. Under Ricci flow the same sphere vanishes at $\rho_0^2/(2(n-1))$. The two times agree only at $n=2$. From any metric conformal to a round one on $S^n$, the unnormalised flow also reaches extinction at a finite time, and the normalised flow converges to a metric of constant scalar curvature, which in this conformal class means a round one, by Obata's theorem.

A product. On $S^2(\rho)\times S^1$, with a circle of length $L$, Ricci flow shrinks the sphere factor and leaves the circle alone. The Yamabe flow scales the whole product, circle included, and its normalised version keeps the metric fixed. This product also shows that the flow need not find a minimiser. Its energy $\frac{2}{\rho^2}(4\pi\rho^2L)^{2/3}$ tends to infinity with $L$, so for large $L$ it exceeds $Y(S^3)$ and cannot be the minimum. The normalised flow from this metric nevertheless stays there forever. Brendle stresses the general point: the limit need not be a global minimiser, and a conformal class may contain infinitely many metrics of constant scalar curvature.

Why it matters

The Yamabe flow is a test of Hamilton's idea of reaching a canonical metric through an evolution equation, in a setting simple enough that the convergence question has been answered in wide generality. Its difficulties are instructive for Ricci flow. The scalar curvature equation is critical for the Sobolev embedding, so solutions can concentrate ("bubble"), and controlling bubbles requires the positive mass theorem. That links a geometric flow to general relativity, a connection that returns with scalar curvature and physics.

The contrast with Perelman is exact. His $\mathcal W$-entropy is built on the logarithmic Sobolev inequality, which is subcritical, so its minimisers always exist on a closed manifold. The Yamabe functional sits at the critical exponent, where a concentrating sequence can escape at fixed energy cost. Daskalopoulos and Šešum noted in 2011 that an analogue of Perelman's monotonicity formula was still lacking for the Yamabe flow, and we do not know of a later replacement.

Key results

Year	Result	Who
1989	The flow is introduced; long-time existence on closed manifolds (unpublished); exponential convergence when the initial scalar curvature is negative	Hamilton
1992	Convergence for locally conformally flat metrics with positive Ricci curvature, with a Harnack inequality	Chow
1994	Convergence for every locally conformally flat metric, by moving planes	Ye
2003	Convergence for $3\le n\le 5$ below an energy threshold	Schwetlick, Struwe
2005	Convergence for $3\le n\le 5$ or locally conformally flat, any initial energy	Brendle
2007	Convergence for $n\ge 6$ if $M$ is spin or the Weyl tensor condition holds	Brendle
2011	Classification of complete locally conformally flat gradient solitons with positive sectional curvature	Daskalopoulos, Šešum
2012	Type II ancient solutions: towers of spheres	Daskalopoulos, del Pino, Šešum
2014	Convergence rates: exponential if the limit is integrable, polynomial in general, and examples of slow convergence	Carlotto, Chodosh, Rubinstein

The convergence theorems

#Long-time existence and the non-positive case

Hamilton showed in unpublished work of the late 1980s that the normalised flow on a closed manifold exists for all $t\ge 0$. Long-time existence is therefore not the issue; the question is what happens as $t\to\infty$. Hamilton conjectured that the flow always converges to a metric of constant scalar curvature. When $Y(M,[g_0])\le 0$ convergence follows from the maximum principle, so the case that matters is $Y(M,[g_0])>0$.

#Locally conformally flat metrics

Bennett Chow (1992) treated locally conformally flat metrics of positive Ricci curvature. In that setting the unnormalised flow satisfies

and Hamilton's tensor maximum principle preserves $\operatorname{Ric}\ge 0$. Chow also proved a Harnack inequality in the style of Hamilton's: for $\operatorname{Ric}\ge 0$ and any vector field $X$,

Setting $n=2$ in this expression and minimising over $X$ gives the quantity in Hamilton's Harnack estimate on surfaces. Rugang Ye (1994) removed the Ricci condition. When the Yamabe constant is positive, the case that remains, a locally conformally flat manifold has a universal cover that maps conformally onto a dense domain of $S^n$ through the developing map (Schoen–Yau). On that domain the method of moving planes gives a gradient estimate for the conformal factor, hence uniform upper and lower bounds.

#Low dimensions

Schwetlick and Struwe (2003) dropped conformal flatness for $3\le n\le 5$, assuming

Simon Brendle (2005) removed the energy assumption. His theorem: if $3\le n\le 5$ or $(M,g_0)$ is locally conformally flat, the metrics $g(t)$ converge to a metric of constant scalar curvature as $t\to\infty$. For flat conformal classes this reproves Ye's result by a different route.

#Dimension six and higher

For $n\ge 6$, let $Z$ be the set of points $p$ where the Weyl tensor vanishes to an order greater than $[\frac{n-6}{2}]$:

This set depends only on the conformal class. Brendle (2007) proved convergence when either $M$ is spin or $Z=\emptyset$. The two hypotheses correspond to two ways of building test functions. Away from $Z$ the Weyl tensor supplies them, extending a construction of Aubin. At a point $p\in Z$ one needs the ADM mass of $M\setminus\{p\}$ with the metric $G^{4/(n-2)}g$ to be positive, where $G$ is the Green's function of the conformal Laplacian with pole at $p$; the spin condition guarantees this. Carron, Lye and Vertman (2021) note that, assuming the 2017 Schoen–Yau argument for the positive mass theorem is correct, Brendle's two papers cover all closed manifolds not conformally equivalent to spheres. The round conformal class falls under Ye's theorem.

#Rates

Once a flow converges, how fast? Carlotto, Chodosh and Rubinstein showed the rate depends on the limit. If $g_\infty$ is an integrable critical point of the Yamabe functional (for instance non-degenerate), convergence is exponential. In general it is at worst polynomial, by a Łojasiewicz–Simon inequality. Under an Adams–Simon condition there are flows converging at exactly a polynomial rate, and they construct explicit examples, which they describe as apparently the first slowly converging solutions to a geometric flow. The round sphere is integrable by Obata's theorem, so convergence to it is exponential. These questions belong to the wider topic of stability.

How convergence is proved

The outline below follows the surveys of Brendle and of Brendle and Marques.

Step 1: from flow to Palais–Smale. With volume one, $\bar r\downarrow\kappa$ and $\lVert R - \kappa\rVert_{L^{2n/(n+2)}}\to 0$. So at any sequence of times the conformal factors $u_\nu$ almost solve the Yamabe equation in the dual norm.

Step 2: bubble decomposition. By Struwe's theorem, after passing to a subsequence $u_\nu$ converges weakly in $H^1$ to some $u_\infty$, and for some integer $m\ge 0$, points $y_{i,\nu}$ and scales $\varepsilon_{i,\nu}>0$,

Each term is a spherical bubble and each costs a fixed amount of energy. If $u_\infty\ne 0$ then $\kappa^{n/2} = E(u_\infty)^{n/2} + m\,Y(S^n)^{n/2}$, and if $u_\infty = 0$ then $\kappa^{n/2} = m\,Y(S^n)^{n/2}$. This explains the Schwetlick–Struwe threshold: below it the arithmetic allows at most one bubble, which they exclude with the positive mass theorem.

Step 3: many bubbles. Above the threshold several bubbles might form and interact, and that is the harder case Brendle handles. The test functions glue an Aubin–Talenti bubble to the Green's function, following Schoen, and the positive mass theorem again rules out concentration. Brendle's key estimate has Łojasiewicz form: there is $\gamma>0$ with

for large $\nu$. Where $u_\infty\ne 0$ this rests on a Łojasiewicz inequality for real analytic functions.

Step 4: finite length. Combine this with $\bar r(t)-\kappa = \frac{n-2}{2}\int_t^\infty\lVert R-\bar r\rVert_{L^2}^2\,d\tau$. An ODE lemma gives

So volume cannot concentrate in small balls, which forces uniform upper and lower bounds on $u$, and convergence follows. The same tools now serve other conformal flows, as the section on variants shows.

Solitons and ancient solutions

#Yamabe solitons

A gradient Yamabe soliton satisfies $(R-\rho)\,g = \nabla^2 f$ for a constant $\rho$: shrinking if $\rho>0$, steady if $\rho=0$, expanding if $\rho<0$. These are the self-similar solutions of $\partial_t g = -Rg$, the counterpart of Ricci solitons, and one of the variations on the soliton equation covered elsewhere on this site.

Daskalopoulos and Šešum proved three things. A compact gradient Yamabe soliton has constant scalar curvature, with no assumption on curvature sign or conformal flatness. A complete locally conformally flat gradient soliton with positive sectional curvature is rotationally symmetric, and they classify these: the shrinkers include the spheres and a family of cigar solutions containing the Barenblatt solutions, the steady ones are logarithmic cigars, and the expanders form a further family. And an eternal solution with positive Ricci curvature whose scalar curvature attains an interior space-time maximum is a steady soliton. Solitons are also studied for the fully non-linear $k$-Yamabe flow.

#Ancient solutions

Ancient solutions exist for all $t\in(-\infty,T)$. One calls such a solution type I if $\limsup_{t\to-\infty}|t|\max|\operatorname{Rm}|<\infty$, and type II otherwise. On $S^n$ the contracting spheres are type I shrinking solitons. The King solutions, which look like two self-similar pieces glued together as $t\to-\infty$, are type I but not solitons.

Ricci flow on $S^2$ has only the spheres and the King–Rosenau solutions (Daskalopoulos, Hamilton, Šešum). The Yamabe flow in dimension $n\ge 3$ has far more. Daskalopoulos, del Pino and Šešum constructed rotationally symmetric type II solutions converging to a tower of two spheres joined by a short neck, whose curvature operator changes sign. Chen, Kim and Musso (2026) construct uncountably many families on $S^n$ for every $n\ge 3$, none conformally equivalent to a rotationally symmetric solution, whose backward limit is a wedge sum of finitely many copies of $S^n$.

Yamabe flow and Ricci flow

The two flows agree on surfaces and part ways from dimension three. The Yamabe flow preserves the conformal class, so it cannot change the conformal structure. Ricci flow changes the conformal class and detects topology through its singularities, which on closed 3-manifolds calls for surgery. The normalised Yamabe flow on a closed manifold has no finite-time singularity, and the only possible failure is concentration as $t\to\infty$. A fixed point of the normalised Yamabe flow has constant scalar curvature, one scalar condition, whereas a fixed point of normalised Ricci flow is an Einstein metric. Ricci flow's analysis is a tensor reaction–diffusion system with curvature pinching; the Yamabe flow's main analytic issue is a critical exponent.

Brendle's survey treats the two side by side and goes on to the sphere theorems, which are proved with Ricci flow.

Variants

CR Yamabe flow. Afeltra, Pinamonti and Ho (2026) construct a contact form on small Rossi deformations of the CR 3-sphere whose flow fails to converge, developing one concentrating bubble. The construction rests on the negative pseudohermitian mass of those spheres. So mass positivity is tied to the large-time behaviour of the flow and is not only a technical assumption.

Fractional and $Q$-curvature flows. An-Lacroix, Chan and Ho prove convergence of the fractional flow for arbitrary initial energy, whenever a fractional positive mass conjecture holds. Gong, Lee and Wei prove global convergence of the Gursky–Malchiodi flow in dimensions $n\ge 5$, for initial metrics of positive scalar curvature and $Q$-curvature, with a non-local Łojasiewicz–Simon inequality. Ho and Lee study the rate of $Q$-curvature flow, following Carlotto, Chodosh and Rubinstein.

Other conformal flows. The weighted Yamabe flow of Yan lives on smooth metric measure spaces. Other relatives include the generalized Yamabe flows of Lye, Vertman and Vitti, the dual flow of Xiong, and the Chern–Yamabe flow on compact Hermitian manifolds, a neighbour of Kähler geometry. The $(\alpha,\beta)$-Ricci–Yamabe flow generalises both Ricci and Yamabe flow.

Boundary, non-compact and singular spaces. On manifolds with boundary, Almaraz and Sun prove convergence with minimal boundary in dimensions up to seven, and in any dimension if the manifold is spin. Carron, Lye and Vertman prove convergence on a class of singular spaces with positive Yamabe constant under a low-energy condition, and study a non-convergent example on Eguchi–Hanson space. On complete manifolds much less is known. Takahashi and Yamamoto find flows that are complete at each time but converge to an incomplete metric. On asymptotically flat manifolds with $Y\le 0$, the flow diverges but converges after rescaling (Carron, Chen, Wang). The flow has also been used as a tool: Cheng applies it to prove a pinching result for locally conformally flat manifolds, and Hsiao and Lee obtain a gap theorem.

Combinatorial flows. There is a discrete theory on triangulations, with results on hyperbolic bordered surfaces by Li, Xu and Zhou, infinite triangulations of surfaces and in three dimensions. See discrete and graph Ricci.

Where it stands

Chow (1992) and Ye (1994) proved convergence for locally conformally flat manifolds, the first with positive Ricci curvature. Schwetlick and Struwe (2003) handled low dimensions when the initial energy is below a threshold. Brendle then proved convergence from arbitrary initial data, first in dimensions 3 to 5 and then, under additional hypotheses, in higher dimensions. On closed manifolds, what the case $n\ge 6$ with $M$ not spin and $Z\ne\emptyset$ depends on is the positive mass theorem itself.

Current work concerns the flow on manifolds with boundary, on non-compact and singular spaces, and variants such as the CR and fractional Yamabe flows. Open directions we can point to from the papers above:

• The complete non-compact flow: Daskalopoulos and Šešum called it unsettled in 2011, and Carron, Lye and Vertman still described understanding as limited in 2021.
• Whether the compact ancient solutions can be classified, given how many are now known.
• Singularity models for finite-time singularities of the complete flow, where solitons are expected to play the role they play for Ricci flow.
• Which variants converge. The CR example shows that mass can decide the outcome, and the fractional result is conditional on a positive mass conjecture.

New papers appear under the Yamabe flow tag.