Fractional Yamabe Flow

• Fractional Yamabe Flow is a nonlocal conformal geometric flow that replaces classical scalar curvature with fractional curvature using a conformally covariant fractional Laplacian.
• It employs degenerate elliptic extension methods and variational techniques to analyze existence, regularity, and convergence on Poincaré–Einstein manifolds.
• The flow framework also connects to fractional diffusion equations and bubbling phenomena, offering insights into energy quantization and positive mass challenges.

Fractional Yamabe flow is a nonlocal conformal geometric flow obtained by replacing the scalar curvature in the classical Yamabe flow by the fractional curvature associated with a conformally covariant fractional Laplacian. In the range $\gamma\in(0,1)$, it is studied on compact manifolds that arise as conformal infinities of asymptotically hyperbolic or Poincaré–Einstein manifolds, and it seeks to deform a metric within a fixed conformal class toward one of constant fractional curvature. Its stationary equation is the fractional Yamabe equation

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$

whose analytic, variational, and extension-theoretic foundations were developed by González–Qing (Gonzalez et al., 2010). The time-dependent flow was introduced on the conformal sphere by Jin–Xiong (Jin et al., 2011), and later extended to manifold settings with weak existence, smooth convergence, threshold convergence, and arbitrary-energy convergence results under additional hypotheses (Daskalopoulos et al., 2017, Chan et al., 2018, An et al., 29 Jul 2025).

1. Geometric and operator-theoretic framework

The geometric background is the conformal infinity of a conformally compact asymptotically hyperbolic manifold $(X^{n+1},g^+)$, or more specifically a Poincaré–Einstein manifold in much of the flow literature. If $\rho$ is a defining function of the boundary $M=\partial X$, then

$g=\rho^2 g^+$

extends smoothly to $\overline X$, and the induced boundary metric is defined only up to conformal factor, yielding a conformal class $(M,[h])$. In the Poincaré–Einstein case one has

$\operatorname{Ric}_{g^+}=-n\,g^+,$

and for each $h\in[h]$ there is a geodesic defining function $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$0 such that near $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$1,

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$2

These normal forms are the basic input for scattering theory and the Chang–González extension mechanism (Gonzalez et al., 2010, Mayer et al., 2017).

The fractional conformal Laplacian is defined from scattering theory. For

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$3

one solves

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$4

with asymptotic expansion

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$5

and defines the scattering operator $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$6. The normalized operator is

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$7

Its principal symbol agrees with that of $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$8, so it is the geometric fractional Laplacian attached to the filling metric (Gonzalez et al., 2010).

Conformal covariance is the structural identity behind the flow. If

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$9

then

$(X^{n+1},g^+)$0

The associated fractional curvature is

$(X^{n+1},g^+)$1

and the same quantity is denoted $(X^{n+1},g^+)$2 in several later flow papers. The conformal law gives

$(X^{n+1},g^+)$3

which is the exact identity used to rewrite the metric flow as a scalar PDE (Gonzalez et al., 2010).

A central analytic localization is the degenerate elliptic extension problem. With $(X^{n+1},g^+)$4, one obtains

$(X^{n+1},g^+)$5

in the compactified manifold, and after choosing a special defining function $(X^{n+1},g^+)$6 satisfying $(X^{n+1},g^+)$7,

$(X^{n+1},g^+)$8

in the bulk. The boundary operator is recovered through

$(X^{n+1},g^+)$9

with the usual $\rho$0 restriction when $\rho$1. This weighted local formulation is one of the basic analytic tools in the subject (Gonzalez et al., 2010).

2. Stationary fractional Yamabe problem and variational structure

The stationary problem underlying the flow is the fractional Yamabe problem: in a given conformal class $\rho$2, find a conformal metric $\rho$3 with constant fractional curvature. By conformal covariance, this is equivalent to

$\rho$4

It generalizes the classical Yamabe problem at $\rho$5, and the case $\rho$6 recovers Escobar’s boundary Yamabe problem (Gonzalez et al., 2010).

The natural quotient is the fractional Yamabe functional. In one notation,

$\rho$7

and the corresponding Yamabe constant is

$\rho$8

In later flow papers the same object is written as

$\rho$9

Critical points of these quotients are positive smooth solutions of the constant-curvature equation, so they are precisely the equilibria of the fractional Yamabe flow (Gonzalez et al., 2010, Chan et al., 2018).

The extension formulation yields a local weighted energy. With $M=\partial X$0,

$M=\partial X$1

and the Euler–Lagrange equation of the constrained minimization reproduces the fractional Yamabe equation. Sharp trace inequalities and model bubbles play the same role as in the local Yamabe theory. Equality in the Euclidean sharp trace inequality is attained by the standard bubbles

$M=\partial X$2

and the spherical model constant is

$M=\partial X$3

The universal upper bound

$M=\partial X$4

is the analogue of Aubin’s inequality (Gonzalez et al., 2010).

The stationary literature also isolates the compactness defects that later reappear in flow theory. For locally flat conformal infinities of Poincaré–Einstein manifolds, the variational analysis exhibits a local regime and a global regime, and the latter involves quantization phenomena and bubbling at multiples of the spherical energy. Bahri–Coron’s algebraic topological argument was used to bypass the absence of a fractional positive mass theorem in that setting and prove existence of a constant fractional scalar curvature metric on any conformal infinity of a Poincaré–Einstein manifold of dimension either $M=\partial X$5 or $M=\partial X$6 and locally flat (Mayer et al., 2017).

3. Formulation of the flow

On the conformal sphere, Jin–Xiong introduced the normalized fractional Yamabe flow

$M=\partial X$7

where $M=\partial X$8 is the $M=\partial X$9-curvature and $g=\rho^2 g^+$0 is its average. Writing

$g=\rho^2 g^+$1

the flow is equivalent to

$g=\rho^2 g^+$2

Under stereographic projection it becomes

$g=\rho^2 g^+$3

on $g=\rho^2 g^+$4. The unnormalized version is

$g=\rho^2 g^+$5

so the geometric flow is directly linked to a fractional porous medium or fast diffusion equation (Jin et al., 2011).

On general compact manifolds, one common normalized form is

$g=\rho^2 g^+$6

with

$g=\rho^2 g^+$7

and $g=\rho^2 g^+$8. The conformal factor then satisfies

$g=\rho^2 g^+$9

For local weak existence, the corresponding unrescaled equation is

$\overline X$0

a fractional fast diffusion equation on the manifold (Daskalopoulos et al., 2017).

A closely related normalization, used in the later convergence literature, is

$\overline X$1

where $\overline X$2 and

$\overline X$3

Equivalently,

$\overline X$4

The flow preserves volume: $\overline X$5 and, under the normalization $\overline X$6, the energy equals the averaged fractional curvature,

$\overline X$7

Its basic dissipation identity is

$\overline X$8

so $\overline X$9 is monotone decreasing (Chan et al., 2018).

The extension formulation remains integral to the parabolic theory. For $(M,[h])$0, if $(M,[h])$1 solves

$(M,[h])$2

in the bulk, then

$(M,[h])$3

and manifold-level flow papers repeatedly use the corresponding weighted integration-by-parts identities, positivity arguments, and Harnack estimates (Daskalopoulos et al., 2017, Chan et al., 2018).

4. Existence, regularity, and convergence results

The first broad manifold-level theory established global weak existence under positivity assumptions. For a compact smooth boundaryless manifold $(M,[h])$4 that is the conformal infinity of a Poincaré–Einstein manifold $(M,[h])$5, assuming

$(M,[h])$6

and, when $(M,[h])$7, also $(M,[h])$8, the fractional Yamabe flow exists for all times in the sense of mild and weak solutions. The construction uses Crandall–Liggett semigroup theory and an implicit elliptic resolvent problem in the extension manifold. A key ingredient is the $(M,[h])$9-contractivity estimate

$\operatorname{Ric}_{g^+}=-n\,g^+,$0

which yields uniqueness and comparison. Under stronger geometric hypotheses—$\operatorname{Ric}_{g^+}=-n\,g^+,$1 compact, locally conformally flat, positive Yamabe constant, and initial metric with nonnegative fractional curvature—the same work proves that the flow exists smoothly for all $\operatorname{Ric}_{g^+}=-n\,g^+,$2 and converges in $\operatorname{Ric}_{g^+}=-n\,g^+,$3 for every integer $\operatorname{Ric}_{g^+}=-n\,g^+,$4 to a smooth metric $\operatorname{Ric}_{g^+}=-n\,g^+,$5 with constant fractional curvature (Daskalopoulos et al., 2017).

The later convergence theory adapts the classical Struwe–Brendle strategy to the nonlocal setting. For $\operatorname{Ric}_{g^+}=-n\,g^+,$6, under

$\operatorname{Ric}_{g^+}=-n\,g^+,$7

and, for $\operatorname{Ric}_{g^+}=-n\,g^+,$8, $\operatorname{Ric}_{g^+}=-n\,g^+,$9, a convergence theorem was proved for a restricted class of initial data satisfying the one-bubble threshold

$h\in[h]$0

Under the additional positive mass assumption, the flow converges to a stationary solution

$h\in[h]$1

hence to a metric of constant fractional curvature. The argument proves asymptotic $h\in[h]$2-flatness of curvature for every

$h\in[h]$3

uniform-in-time upper and lower bounds on the conformal factor, a profile decomposition for time slices, and a Łojasiewicz–Simon type estimate leading to polynomial decay

$h\in[h]$4

for some $h\in[h]$5 (Chan et al., 2018).

A full convergence theorem for arbitrary initial energy was obtained later, again under a fractional positive mass assumption. In that result, for $h\in[h]$6, if

$h\in[h]$7

and, when $h\in[h]$8, $h\in[h]$9, then the normalized fractional Yamabe flow

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$00

converges without any small-energy restriction, provided the Positive Mass Conjecture holds with $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$01. The decisive estimate is a Brendle-type superlinear energy-defect inequality: after passing to a subsequence $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$02,

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$03

Combined with the dissipation identity, this yields convergence for arbitrary initial energy (An et al., 29 Jul 2025).

Several limitations remain explicit. The manifold theory is developed for $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$04; for $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$05, the maximum principle fails and the parabolic theory is largely open. The general arbitrary-energy convergence theorem is conditional on a fractional positive mass conjecture that is not known in full generality. In the weak theory, the case $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$06 is excluded because contractivity breaks down, and nonuniqueness can occur even for spatially homogeneous solutions (Daskalopoulos et al., 2017, Chan et al., 2018, An et al., 29 Jul 2025).

5. Bubbling, compactness defects, and the positive mass problem

Critical fractional conformal problems exhibit the same local-versus-global dichotomy as the classical Yamabe problem, but with additional nonlocal complications. In the locally flat Poincaré–Einstein setting, the variational landscape contains quantized bubbling phenomena, energies quantizing at multiples of the spherical fractional Yamabe constant, and a missing positive mass theorem that blocks the direct Aubin–Schoen minimization strategy. The stationary existence theory in that regime constructs Euclidean standard bubbles

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$07

their projective extensions, Green kernels, and curved Schoen-type bubbles, and defines a fractional mass $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$08 from the Green-function expansion. A positive mass analogue would force strict inequality below the sphere level, but such a theorem is not known in general (Mayer et al., 2017).

The convergence theory for the flow inherits exactly these compactness defects. Time slices $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$09 form approximate solutions of the stationary equation, and concentration-compactness yields decompositions into a background solution plus finitely many bubbles. In the threshold theorem, the possible asymptotic alternatives reduce to a compact case $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$10 and a one-bubble case $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$11, with the initial energy bound ruling out multiple bubbles. This is why the condition

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$12

is the natural one-bubble threshold (Chan et al., 2018).

The arbitrary-energy theorem replaces that threshold by a full Brendle-type bubble analysis. For a Palais–Smale sequence extracted from the flow, one has

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$13

together with the energy quantization identity

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$14

The analysis then requires interaction estimates between fractional Schoen bubbles, coercivity of the linearized operator on orthogonal complements, a finite-dimensional reduction around nonzero limits, and delicate control of bubble errors in the critical dual norm. A notable technical innovation is a new pointwise gradient estimate in the extension manifold, used to compare “glue-and-extend” and “extend-and-glue” constructions and prove the decisive estimate

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$15

(An et al., 29 Jul 2025).

The recurrent obstruction is the positive mass problem. The locally flat stationary theory states that the Aubin–Schoen minimizing strategy would require a fractional analogue of the Schoen–Yau positive mass theorem, but such a theorem is not known. The convergence theory for arbitrary initial energy therefore remains conditional on the Positive Mass Conjecture with $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$16, formulated through the Green-function expansion

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$17

This is one of the central unresolved structural issues in the subject (Mayer et al., 2017, An et al., 29 Jul 2025).

6. Singular stationary profiles, diffusion models, and related directions

Not all work relevant to fractional Yamabe flow is itself parabolic. The stationary theory of singular solutions in $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$18 is directly relevant because blow-up limits and candidate singularity models for the flow satisfy the same critical equation

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$19

For such isolated singular solutions, an Emden–Fowler reduction converts the radial problem into a one-dimensional nonlocal equation

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$20

and every singular solution has infinite Morse index. The same work also proves that if a singular solution satisfies either $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$21 or $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$22, where

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$23

then $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$24. These results do not formulate a time-dependent fractional Yamabe flow, but they identify stationary singular profiles as extremely unstable objects in the variational sense, and they are therefore pertinent to blow-up analysis and the exclusion of certain singular limits (Cruz-Blázquez et al., 2022).

The sphere model also reveals a second major direction: the geometric flow is equivalent, after stereographic projection and removal of the volume normalization, to a fractional fast diffusion equation

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$25

Jin–Xiong used the convergence of the normalized flow on $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$26 to obtain extinction profiles for that Euclidean equation. For positive $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$27 data satisfying the Kelvin-transform regularity condition, if $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$28 is the extinction time, then

$$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$29

with the precise remainder control stated in the theorem. The same dynamical framework also yields a monotone quantity along the fast diffusion flow and a quantitative improvement of the sharp fractional Sobolev inequality when $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$30 (Jin et al., 2011).

Two misconceptions are repeatedly corrected by the literature. First, the fractional Yamabe flow is not merely the Euclidean equation with $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$31; in the geometric setting the operator is the scattering-theoretic fractional conformal Laplacian associated with a filling metric. Second, several foundational papers most relevant to the subject are stationary rather than parabolic: González–Qing provides the conformal covariance, extension law, Hopf-type maximum principle, weighted trace inequalities, and existence theory that underlie the flow, while locally flat stationary analyses and singular-profile results describe the equilibria, bubble models, and compactness defects that the flow must confront (Gonzalez et al., 2010, Mayer et al., 2017, Cruz-Blázquez et al., 2022).

Taken together, the subject presents a now well-defined nonlocal analogue of classical Yamabe flow. Its core ingredients are the scattering-defined operator $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$32, the fractional curvature $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$33 or $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$34, the critical quotient $$P_\gamma^h(w)=c\,w^{\frac{n+2\gamma}{n-2\gamma}},\qquad w>0,$$35, the weighted extension formulation, and the concentration-compactness theory of fractional bubbles. Its strongest current dynamical results establish global convergence on the sphere, smooth global convergence on locally conformally flat compact manifolds under positivity assumptions, threshold convergence under a one-bubble energy condition, and arbitrary-energy convergence conditional on fractional positive mass. The principal open structural issue is still the general positive mass theory required to make the full arbitrary-energy convergence theorem unconditional (Jin et al., 2011, Daskalopoulos et al., 2017, Chan et al., 2018, An et al., 29 Jul 2025).