CR Yamabe Invariant: Variational Analysis

• CR Yamabe invariant is a variational invariant defined on compact, strongly pseudoconvex CR manifolds by minimizing the normalized total Webster scalar curvature over a pseudohermitian conformal class.
• It leads to a subelliptic Euler–Lagrange equation whose minimizers yield constant Webster scalar curvature contact forms, with existence and uniqueness depending on curvature sign and normalization.
• Extensions of the invariant include applications in contact and Sasaki geometries, fully nonlinear CR k-Yamabe problems, and the global contact invariant σ₍c₎ that respects handle attachment and connected-sum operations.

Searching arXiv for recent and foundational papers on the CR Yamabe invariant. The CR Yamabe invariant is a variational invariant attached to a compact strongly pseudoconvex CR manifold, obtained by minimizing the normalized total Webster scalar curvature over a pseudohermitian conformal class. In the notation of the literature, this classwise invariant appears as $\lambda(X,H,J)$, $Y(M,T^{1,0})$, or $Y_{CR}(M,H,J)$, depending on the source. A closely related global contact invariant, introduced for contact manifolds that admit strictly pseudoconvex CR structures, is Dietrich’s $\sigma_c(M,H)$, defined as the supremum of the CR Yamabe constants over all compatible strictly pseudoconvex CR structures on the fixed contact distribution (Dietrich, 2018).

1. Variational definition and normalization conventions

Let $(X,H,J)$ be a compact, coorientable, strongly pseudoconvex CR manifold of real dimension $2n+1$, and let $\theta$ be a positive contact form with $\ker\theta=H$. Writing

$d\mu_\theta=\theta\wedge(d\theta)^n,$

the normalized total Webster scalar curvature is

$$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$

The CR Yamabe constant of the CR structure is the infimum

$Y(M,T^{1,0})$0

taken over all positive contact forms in the given pseudohermitian conformal class (Sung et al., 2022).

Fixing a reference form $Y(M,T^{1,0})$1, every other form in the class can be written as $Y(M,T^{1,0})$2 with $Y(M,T^{1,0})$3. In this parametrization,

$Y(M,T^{1,0})$4

Dietrich presents the same invariant as

$Y(M,T^{1,0})$5

where

$Y(M,T^{1,0})$6

and $Y(M,T^{1,0})$7 are defined via the Tanaka–Webster connection (Dietrich, 2018).

In dimension three, a standard normalization uses the CR Yamabe operator

$Y(M,T^{1,0})$8

and the constrained functional

$Y(M,T^{1,0})$9

The resulting invariant $Y_{CR}(M,H,J)$0 is independent of the initial choice of $Y_{CR}(M,H,J)$1 (Case et al., 2022).

The literature therefore uses closely related but not identical normalizations. Dietrich records

$Y_{CR}(M,H,J)$2

whereas Ivanov–Minchev–Vassilev identify the spherical value with the sharp CR-Sobolev constant in their normalization (Dietrich, 2018, Ivanov et al., 2015). This suggests that numerical formulas must be read together with the normalization convention adopted in each source.

2. Euler–Lagrange equation, solvability, and uniqueness

The variational problem leads to a subelliptic Euler–Lagrange equation. If $Y_{CR}(M,H,J)$3, Jerison–Lee and Gamara–Yacoub show that there exists a minimizer $Y_{CR}(M,H,J)$4 in $Y_{CR}(M,H,J)$5 which is smooth and satisfies

$Y_{CR}(M,H,J)$6

Equivalently, in dimension three,

$Y_{CR}(M,H,J)$7

If $Y_{CR}(M,H,J)$8, then $Y_{CR}(M,H,J)$9 has constant Webster scalar curvature and, under the volume normalization, $\sigma_c(M,H)$0 (Dietrich, 2018, Case et al., 2022).

A central existence criterion is the strict inequality below the spherical constant. Dietrich states that

$\sigma_c(M,H)$1

whenever $\sigma_c(M,H)$2 is non-spherical or $\sigma_c(M,H)$3, and in that case a minimizer always exists (Dietrich, 2018). More generally, the survey on the positive mass theorem emphasizes that the positive mass theorem plays a key role in finding a solution to the Yamabe equation with minimum energy for the positive curvature case, especially in the delicate spherical and low-dimensional settings (Cheng, 18 Jun 2026).

Uniqueness is markedly asymmetric with respect to the sign of the invariant. Sung–Takeuchi state that once $\sigma_c(M,H)$4, the CR-Yamabe minimizer is unique up to scaling (Sung et al., 2022). Dietrich similarly records that if $\sigma_c(M,H)$5 is pseudo-Einstein, then any constant-$\sigma_c(M,H)$6 contact form in $\sigma_c(M,H)$7 must itself be Einstein, and in the non-spherical case it is unique up to scale (Dietrich, 2018).

The three-sign trichotomy parallels the Riemannian Yamabe problem but with specifically CR features. The recent survey formulates it as follows: if $\sigma_c(M,H)$8, the conformal class contains a unique contact form of constant negative curvature; if $\sigma_c(M,H)$9, one gets a contact form with $(X,H,J)$0; if $(X,H,J)$1 and $(X,H,J)$2, minimizers exist by compactness of subcritical approximations (Cheng, 18 Jun 2026).

3. Contact invariant $(X,H,J)$3, handle attachment, and connected sum

For a compact contact manifold $(X,H,J)$4 admitting at least one strictly pseudoconvex CR structure, Dietrich defines the global contact invariant

$(X,H,J)$5

where $(X,H,J)$6 is the set of all strictly pseudoconvex CR structures on $(X,H,J)$7 (Dietrich, 2018).

The central structural result is monotonicity under strictly pseudoconvex CR handle attachment. If $(X,H,J)$8 is obtained from an SPC manifold $(X,H,J)$9 by one or more strictly pseudoconvex CR handle attachments, then

$2n+1$0

Dietrich’s proof proceeds by locally making the structure spherical near attaching points, introducing cylindrical end coordinates in Heisenberg normal form, gluing long cylinders by inversion–dilation, extracting a slice with small boundary energy by a layer-cake argument, and cutting off a near-minimizer to recover a test function on the original punctured manifold (Dietrich, 2018).

For disjoint unions, the CR Yamabe constant satisfies the explicit formula

$2n+1$1

where $2n+1$2. Combining this with handle-attachment gives the connected-sum inequality

$2n+1$3

hence

$2n+1$4

This is the CR analogue of the Kobayashi connected-sum inequality in conformal geometry (Dietrich, 2018).

In dimension three, Dietrich also gives a lower-bound computation for circle bundles over Riemann surfaces using the Burns–Epstein invariant. If $2n+1$5 is a circle bundle over a Riemann surface $2n+1$6 of genus $2n+1$7 carrying its canonical Einstein contact form $2n+1$8, then

$2n+1$9

Using Cauchy–Schwarz and the fact that the Einstein form is a Yamabe form in the negative case, one obtains

$\theta$0

with equality, and therefore

$\theta$1

Among the basic examples recorded by Dietrich are

$\theta$2

and for the Boothby–Wang Einstein circle bundle over $\theta$3,

$\theta$4

All of these formulas are stated explicitly in Dietrich’s paper (Dietrich, 2018).

4. Sign, rigidity, and three-dimensional classification

In dimension three, the sign of the CR Yamabe invariant has direct geometric implications. Case–Chanillo–Yang summarize the trichotomy as follows: if $\theta$5, there exists a contact form of strictly positive Webster curvature; if $\theta$6, one obtains a torsion-free, zero-curvature flat contact form; if $\theta$7, the CR Yamabe problem still has a unique, up to scaling, contact form of constant negative Webster curvature, although no global topological classification is given there (Case et al., 2022).

Two sphere-type classification theorems sharpen the nonnegative case. If $\theta$8 is closed and embeddable, and

$\theta$9

then $\ker\theta=H$0 is CR-equivalent to a compact quotient of the Heisenberg group. If $\ker\theta=H$1 is closed and universally embeddable, and

$\ker\theta=H$2

then $\ker\theta=H$3 is contact-diffeomorphic to a finite quotient of the standard contact three-sphere $\ker\theta=H$4 (Case et al., 2022).

These theorems rest on curvature identities involving the CR Paneitz operator, the $\ker\theta=H$5-operator, and the $\ker\theta=H$6-curvature. In particular,

$\ker\theta=H$7

and, for a $\ker\theta=H$8-flat form, the total integral $\ker\theta=H$9 is a CR invariant (Case et al., 2022).

The positive-sign regime also interacts strongly with embeddability. For a closed pseudo-Hermitian $d\mu_\theta=\theta\wedge(d\theta)^n,$0-manifold, the condition $d\mu_\theta=\theta\wedge(d\theta)^n,$1 is conformally invariant, and the paper on embeddability proves that if $d\mu_\theta=\theta\wedge(d\theta)^n,$2 and the CR Yamabe constant $d\mu_\theta=\theta\wedge(d\theta)^n,$3, then $d\mu_\theta=\theta\wedge(d\theta)^n,$4 admits a CR embedding into some $d\mu_\theta=\theta\wedge(d\theta)^n,$5 (Chanillo et al., 2010). The same work shows that $d\mu_\theta=\theta\wedge(d\theta)^n,$6 alone does not guarantee embeddability: Rossi’s non-embeddable spheres have positive Webster curvature but a Paneitz operator with negative eigenvalues (Chanillo et al., 2010).

The rigidity of the spherical model is a recurring theme across the literature. The recent survey on the positive mass theorem states that in dimension three, under positivity of $d\mu_\theta=\theta\wedge(d\theta)^n,$7 and nonnegativity of the CR Paneitz operator, the blow-up mass is nonnegative and vanishes only for the standard sphere; in dimension five, the analogous p-mass rigidity singles out the Heisenberg model (Cheng, 18 Jun 2026).

5. Deformation theory and inequivalent CR structures

Sung–Takeuchi develop an integral-analytic description of the CR Yamabe constant that is well adapted to deformation questions. If $d\mu_\theta=\theta\wedge(d\theta)^n,$8, then for every $d\mu_\theta=\theta\wedge(d\theta)^n,$9,

$$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$0

and if the CR Yamabe problem is solvable, equality holds. If $$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$1, then for every $$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$2,

$$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$3

and each infimum is attained exactly at a CR-Yamabe minimizer (Sung et al., 2022).

The same paper constructs an infinite-dimensional family of strongly pseudoconvex CR structures with varying CR Yamabe constants on one fixed manifold. Starting from a compact Kähler–Hodge manifold $$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$4 of constant scalar curvature, one considers the principal $$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$5-bundle $$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$6 with Euler class $$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$7 and the family of connection forms $$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$8 determined by Kähler potentials $$\mathcal F(\theta) = \frac{\int_X R_\theta\,d\mu_\theta}{\left(\int_X d\mu_\theta\right)^{n/(n+1)}}.$$9. For each $Y(M,T^{1,0})$00, the lifted pseudohermitian structure $Y(M,T^{1,0})$01 is strongly pseudoconvex, and the map

$Y(M,T^{1,0})$02

is continuous in the $Y(M,T^{1,0})$03-topology on $Y(M,T^{1,0})$04. Moreover,

$Y(M,T^{1,0})$05

is independent of $Y(M,T^{1,0})$06, and if the base-point form is already a minimizer, then

$Y(M,T^{1,0})$07

for every $Y(M,T^{1,0})$08, where $Y(M,T^{1,0})$09 denotes the constant-scalar-curvature locus. This yields infinitely many pairwise inequivalent CR structures with strictly smaller CR Yamabe constants than the base-point (Sung et al., 2022).

Sung–Takeuchi also give a simply-connected example carrying two inequivalent strongly pseudoconvex CR structures with opposite-sign CR Yamabe constants. For each $Y(M,T^{1,0})$10, the manifold $Y(M,T^{1,0})$11 is taken as the circle bundle $Y(M,T^{1,0})$12 over

$Y(M,T^{1,0})$13

One lifted CR structure has $Y(M,T^{1,0})$14 because the base admits a strictly positive Ricci form, while a second lifted CR structure obtained by a topological–almost-complex argument has $Y(M,T^{1,0})$15. Section 6 of the paper verifies that the two cooriented contact structures are not isomorphic because their first Chern classes in $Y(M,T^{1,0})$16 differ, mod $Y(M,T^{1,0})$17 if $Y(M,T^{1,0})$18, or literally if $Y(M,T^{1,0})$19 (Sung et al., 2022).

The same paper isolates open problems. In particular, it remains unknown whether a single cooriented contact manifold can admit two CR structures with opposite-sign CR Yamabe constants; the constructed example realizes opposite signs only on inequivalent contact distributions (Sung et al., 2022).

6. Extensions, analogues, and current directions

One important extension replaces integrable CR structures by contact Riemannian manifolds, where the horizontal almost complex structure $Y(M,T^{1,0})$20 need not be integrable. Wu–Wang define the corresponding Yamabe functional using the Tanaka–Webster–Tanno connection and show that if the complex structure is not integrable, the Yamabe invariant on a contact Riemannian manifold is always less than the Yamabe invariant of the Heisenberg group. Consequently, the Yamabe problem on a contact Riemannian manifold is always solvable (Wu et al., 2015). This places the CR Yamabe problem inside a broader sub-Riemannian variational framework.

A second extension is the fully nonlinear CR $Y(M,T^{1,0})$21-Yamabe problem. Barbosa–Carneiro–Montenegro define the pseudohermitian Schouten tensor

$Y(M,T^{1,0})$22

set $Y(M,T^{1,0})$23 to be the $Y(M,T^{1,0})$24th elementary symmetric polynomial of the eigenvalues of $Y(M,T^{1,0})$25, and define

$Y(M,T^{1,0})$26

When the pseudohermitian Cotton tensor vanishes, the problem is variational and its Euler–Lagrange equation is a fully nonlinear complex $Y(M,T^{1,0})$27-Hessian equation. The paper proves the upper bound

$Y(M,T^{1,0})$28

for compact CR manifolds with positive CR $Y(M,T^{1,0})$29-Yamabe invariant, and equality implies that $Y(M,T^{1,0})$30 is locally CR-equivalent to the sphere (Barbosa et al., 2012).

A third direction is Sasaki geometry. The paper on constant scalar curvature Sasaki metrics defines a $Y(M,T^{1,0})$31-equivariant CR Yamabe energy

$Y(M,T^{1,0})$32

and shows that the Einstein–Hilbert functional on the Sasaki–Reeb cone has a minimum

$Y(M,T^{1,0})$33

which is a topological constant and gives the natural upper bound

$Y(M,T^{1,0})$34

If the CR Yamabe invariant attains this topological value and the Sasaki–Reeb cone contains a regular Reeb vector field, then the corresponding Sasaki manifold is K-semistable. In the non-positive average scalar curvature case, the equality $Y(M,T^{1,0})$35 is equivalent to the existence of approximately constant scalar curvature Sasaki structures, and the paper derives from this a numerical criterion for K-semistability of polarized compact complex manifolds (Lahdili et al., 31 Aug 2025).

Finally, the positive-mass viewpoint has become structurally important in the positive CR Yamabe class. The 2026 survey emphasizes that the positive mass theorem is now part of the standard analytic toolkit for the positive-curvature case, while the Rossi spheres show that nonembeddable structures may have

$Y(M,T^{1,0})$36

without attaining the infimum, because the p-mass is negative for small deformation parameter $Y(M,T^{1,0})$37 (Cheng, 18 Jun 2026). This indicates that, unlike the classical Riemannian situation, the CR Yamabe problem is sensitive not only to curvature sign and compactness, but also to embeddability and mass-type obstructions.