CR Yamabe Flow: Convergence & Bubbling Analysis

• CR Yamabe flow is the pseudohermitian analogue of Hamilton’s Yamabe flow that deforms contact forms to attain constant Webster scalar curvature.
• It reformulates the CR Yamabe problem as a gradient flow aiming to reduce curvature variations while preserving normalized volume on compact CR manifolds.
• Recent studies highlight both smooth convergence under positivity assumptions and bubbling phenomena driven by negative pseudohermitian mass in dimension three.

Searching arXiv for core CR Yamabe flow papers and closely related variants. The CR Yamabe flow is the pseudohermitian analogue of Hamilton’s Yamabe flow in conformal Riemannian geometry. On a compact strictly pseudoconvex CR manifold $(M,\theta_0)$ of real dimension $2n+1$, it evolves a contact form within its CR conformal class so as to drive the Webster scalar curvature toward a constant. In its normalized form, the flow is

$$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$

where $R_{\theta(t)}$ is the Webster scalar curvature and

$$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$

The guiding problem is the CR Yamabe problem, namely the search for a conformal contact form with constant Webster scalar curvature. The long-time dynamics of the flow are now understood in several important regimes: convergence is known under hypotheses such as $n=1$, sphericality, or zero CR Yamabe invariant, while recent work has shown that non-convergence by bubbling can occur in dimension three on small Rossi deformations of $S^3$ (Ho et al., 2017, Sheng et al., 2018, Afeltra et al., 25 Jun 2026).

1. Geometric framework and defining equations

A compact strictly pseudoconvex CR manifold $(M^{2n+1},T_{1,0})$ carries a CR structure $T_{1,0}\subset \mathbb{C}TM$ with

$$T_{1,0}\cap T_{0,1}=\{0\}, \qquad [T_{1,0},T_{1,0}]\subset T_{1,0}.$$

A choice of global contact form $2n+1$0 determines a pseudohermitian structure. Associated objects include the Levi form, the Tanaka–Webster connection, the sub-Laplacian $2n+1$1, and the Webster scalar curvature $2n+1$2 (Ho et al., 2017, Sheng et al., 2018).

The normalized CR Yamabe flow is written as

$2n+1$3

with

$2n+1$4

If the evolving contact form is expressed conformally as

$2n+1$5

then the conformal factor satisfies

$2n+1$6

equivalently

$2n+1$7

These formulas make explicit that the flow is a weakly parabolic scalar evolution in the CR conformal class (Ho et al., 2017).

The conformal change law underlying the flow is

$2n+1$8

for $2n+1$9. This is the CR counterpart of the conformal scalar-curvature equation in the Riemannian Yamabe problem (Ho et al., 2017, Sheng et al., 2018).

A standard unnormalized variant is

$$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$0

which is used, for example, in eigenvalue monotonicity problems and in noncompact uniqueness theory (Ho, 2018, Ho et al., 2020).

2. Variational structure and the CR Yamabe problem

The CR Yamabe problem asks for a contact form conformal to $$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$1 whose Webster scalar curvature is constant. Its variational formulation uses the CR Yamabe constant

$$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$2

or equivalently, in the notation of some papers,

$$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$3

When $$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$4, the conformal class contains a contact form with zero pseudo-Hermitian scalar curvature; when $$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$5, the conformal class supports a favorable positive-curvature regime (Sheng et al., 2018, Ho et al., 2020).

The flow is presented as the CR analogue of Hamilton’s Yamabe flow. In the compact setting it preserves normalized volume and decreases the average curvature in the sense that

$$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$6

after normalization, and

$$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$7

Thus $$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$8 is nonincreasing and converges to a limit $$\frac{\partial}{\partial t}\theta(t) = -\big(R_{\theta(t)}-r_{\theta(t)}\big)\,\theta(t),$$9 (Ho et al., 2017). In the zero-invariant case one also has

$R_{\theta(t)}$0

so the average curvature is again monotone (Sheng et al., 2018).

This variational interpretation is central to both convergence and non-convergence results. A recent non-convergence construction states explicitly that the CR Yamabe flow is the gradient flow of the CR Yamabe functional, and exploits that structure near a bubble manifold to force concentration instead of compactness (Afeltra et al., 25 Jun 2026).

3. Fundamental evolution identities

Several evolution formulas recur throughout the theory. Under the normalized flow,

$R_{\theta(t)}$1

and the Webster scalar curvature evolves by

$R_{\theta(t)}$2

These formulas are used in maximum-principle arguments, curvature pinching estimates, and eigenvalue inequalities (Ho, 2018).

Under the unnormalized flow,

$R_{\theta(t)}$3

These identities are the CR analogues of the standard Riemannian Yamabe-flow formulas (Ho, 2018).

In the zero CR Yamabe invariant setting, the flow analysis also relies on a CR Poincaré inequality

$R_{\theta(t)}$4

together with the stronger estimate

$R_{\theta(t)}$5

and a CR Gagliardo–Nirenberg interpolation inequality of the form

$R_{\theta(t)}$6

These inequalities are used to convert curvature-decay information into higher-order decay and smooth convergence (Sheng et al., 2018).

A different but related line of work studies the flow on complete noncompact manifolds. There the unnormalized flow is written in conformal form $R_{\theta(t)}$7, and the scalar PDE is analyzed by maximum principles under geometric exhaustion hypotheses called Condition A and Condition B. Under these assumptions, uniqueness of the CR Yamabe flow follows on complete noncompact CR manifolds (Ho et al., 2020).

4. Convergence theory

The strongest convergence theorem for the compact normalized flow states that if $R_{\theta(t)}$8 is a compact strongly pseudoconvex CR manifold with

$R_{\theta(t)}$9

and either $$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$0 with nonnegative CR Paneitz operator, or $$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$1 is spherical but not CR equivalent to the standard sphere $$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$2 together with the stated mass hypotheses, then the CR Yamabe flow exists for all time and converges to a contact form with constant Webster scalar curvature (Ho et al., 2017).

The proof uses a CR concentration-compactness theorem. For a sequence $$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$3, the corresponding conformal factors satisfy a compactness-or-bubbling dichotomy: $$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$4 where $$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$5 are the standard CR bubbles modeled on the Heisenberg group. The decisive step is to rule out bubbling by combining an improved energy-control estimate, finite total curvature dissipation,

$$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$6

uniform local smallness of conformal mass, and a CR positive mass input. Once bubbling is excluded, one obtains uniform bounds

$$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$7

Hölder control in space-time, and then smooth convergence to a constant-Webster-curvature contact form (Ho et al., 2017).

A more specialized but sharper result holds when the CR Yamabe invariant vanishes. If $$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$8 is compact, strictly pseudoconvex, and

$$r_{\theta(t)} = \frac{\int_M R_{\theta(t)}\,dV_{\theta(t)}}{\int_M dV_{\theta(t)}}.$$9

then the CR Yamabe flow exists for all time and converges exponentially to a contact form with flat pseudo-Hermitian scalar curvature: $n=1$0 The argument uses the CR Poincaré inequality and a CR Gagliardo–Nirenberg interpolation inequality to prove exponential decay of the average curvature and of the curvature defect, ultimately yielding exponential convergence of the conformal factor to a constant (Sheng et al., 2018).

These two convergence theories occupy different parts of the parameter space. The first addresses positive CR Yamabe constant under structural assumptions such as $n=1$1 or sphericality (Ho et al., 2017). The second handles the zero-invariant case and proves exponential convergence to a scalar-flat limit (Sheng et al., 2018).

5. Non-convergence and bubbling phenomena

A major recent development is the construction of a genuine non-convergence example in dimension three. On small Rossi deformations of the standard CR three-sphere, there exists a smooth contact form whose CR Yamabe flow does not converge. More precisely, the flow develops a one-bubble concentration regime (Afeltra et al., 25 Jun 2026).

The setting is the three-dimensional case $n=1$2, where the conformal factor equation becomes

$n=1$3

The paper uses three ingredients: the gradient-flow nature of the CR Yamabe functional, the negativity of the pseudohermitian mass on small Rossi spheres, and a Lyapunov–Schmidt decomposition near a finite-dimensional bubble manifold (Afeltra et al., 25 Jun 2026).

The bubble manifold consists of CR bubbles transported from the Heisenberg group. The one-bubble regime is described by

$n=1$4

Instead of converging to a smooth limiting contact form, the solution stays near the bubble manifold while the scale diverges (Afeltra et al., 25 Jun 2026).

The key analytic quantity is the pseudohermitian mass. If $n=1$5 is the Green function of the conformal sublaplacian, then in CR normal coordinates

$n=1$6

For Rossi spheres,

$n=1$7

so $n=1$8 for small nonzero $n=1$9. In the expansion of the CR Yamabe functional near a bubble,

$S^3$0

equivalently

$S^3$1

Because the mass is negative, the flow is energetically driven toward larger $S^3$2, hence toward stronger concentration rather than a smooth limit (Afeltra et al., 25 Jun 2026).

This resolves a possible misconception in the older convergence theory. The positivity assumptions used in known convergence results are not merely technical: the Rossi-sphere example shows that mass positivity is genuinely tied to large-time dynamics (Afeltra et al., 25 Jun 2026).

6. Spectral, noncompact, and prescribed-curvature extensions

The CR Yamabe flow has also been used as a tool in spectral geometry. On a compact strictly pseudoconvex CR manifold $S^3$3 of real dimension $S^3$4, if $S^3$5 denotes the first positive eigenvalue of $S^3$6 along the normalized flow, then one has Dini-derivative bounds

$S^3$7

and

$S^3$8

Under the negativity assumption

$S^3$9

the flow preserves negativity and yields a comparison theorem between the first eigenvalue of the initial contact form and that of the limiting constant-curvature contact form $(M^{2n+1},T_{1,0})$0: $(M^{2n+1},T_{1,0})$1 The same paper studies the unnormalized flow and proves monotonicity of the first eigenvalue of $(M^{2n+1},T_{1,0})$2 under suitable sign conditions (Ho, 2018).

For complete noncompact CR manifolds, the unnormalized flow

$(M^{2n+1},T_{1,0})$3

admits a uniqueness theory based on maximum principles. If the evolving pseudohermitian structures satisfy Condition B and the solution is positive and bounded, then the scalar PDE for the conformal factor has a unique solution with given initial data (Ho et al., 2020). The same work studies CR Yamabe solitons, defined by

$(M^{2n+1},T_{1,0})$4

or equivalently

$(M^{2n+1},T_{1,0})$5

On the Heisenberg group, the corresponding potential functions are classified explicitly (Ho et al., 2020).

A further extension is the prescribed Webster scalar curvature flow

$(M^{2n+1},T_{1,0})$6

with normalization

$(M^{2n+1},T_{1,0})$7

When $(M^{2n+1},T_{1,0})$8, this reduces to the CR Yamabe flow. In dimension three, this framework yields existence results for the prescribed Webster scalar curvature problem under nonnegative CR Yamabe constant and additional conditions such as nonnegativity of the CR Paneitz operator or sign conditions on $(M^{2n+1},T_{1,0})$9 (Ho et al., 2021).

7. Related Yamabe-type flows and conceptual boundaries

The CR Yamabe flow is often discussed alongside several structurally related but distinct flows. The most important distinction is with the fractional Yamabe flow, which evolves a metric in a conformal class using a nonlocal conformally covariant operator $T_{1,0}\subset \mathbb{C}TM$0 and fractional curvature $T_{1,0}\subset \mathbb{C}TM$1 for $T_{1,0}\subset \mathbb{C}TM$2. On the sphere, the normalized fractional flow takes the form

$T_{1,0}\subset \mathbb{C}TM$3

or, on a general compact manifold,

$T_{1,0}\subset \mathbb{C}TM$4

These flows are not CR flows, but they are explicitly described as fractional analogues of Hamilton’s Yamabe flow. The case $T_{1,0}\subset \mathbb{C}TM$5 is connected to Escobar’s boundary Yamabe problem and therefore provides the closest bridge to boundary or CR-type geometric settings (Jin et al., 2011, Daskalopoulos et al., 2017).

This suggests a useful conceptual boundary. The classical CR Yamabe flow is driven by the local Webster scalar curvature and the sub-Laplacian in pseudohermitian geometry (Ho et al., 2017, Ho, 2018). The fractional Yamabe flow instead uses nonlocal conformally covariant fractional operators on the conformal infinity of asymptotically hyperbolic manifolds (Jin et al., 2011), or on compact manifolds via scattering theory and the Chang–González extension (Daskalopoulos et al., 2017). The similarity is philosophical rather than literal.

Another relevant boundary is between the compact and noncompact theories. The compact normalized flow targets constant Webster scalar curvature in a fixed volume class (Ho et al., 2017, Sheng et al., 2018). The noncompact theory in (Ho et al., 2020) focuses on the unnormalized flow, maximum principles, uniqueness, and solitons. These are complementary rather than competing formulations.

Finally, the recent non-convergence result makes clear that one should not assume unconditional convergence of the CR Yamabe flow in dimension three. A plausible implication is that the correct global picture is stratified: some regimes admit smooth convergence, some admit exponential convergence, and others admit bubbling governed by pseudohermitian mass and the fine structure of the CR manifold (Ho et al., 2017, Sheng et al., 2018, Afeltra et al., 25 Jun 2026).