Jerison-Lee Bubbles in CR Geometry

• Jerison-Lee bubbles are explicit extremal functions that solve the critical CR Yamabe equation, offering a complete classification of positive solutions in non-Euclidean settings.
• They are characterized by invariance under CR-conformal transformations, which include translations, dilations, rotations, and CR inversions on the Heisenberg group and sphere.
• Their stability, non-degeneracy, and quantitative interaction properties are established using techniques like the method of moving spheres and divergence identities.

Jerison–Lee bubbles are the unique, explicit family of extremal functions for the critical CR Yamabe equations in CR geometry, most notably on the Heisenberg group $\mathbb{H}^n$ and the sphere $S^{2m+1}$. They classify all positive solutions to certain nonlinear subelliptic equations, encapsulating the geometry and analysis of Sobolev extremals and constant scalar curvature contact forms in the CR setting. This construct generalizes sharp Euclidean results and connects to the full CR-conformal group, providing an orbit of extremal points under translations, dilations, rotations, and CR inversions.

1. Algebraic and Geometric Foundations

The Heisenberg group $\mathbb{H}^n \cong \mathbb{C}^n \times \mathbb{R}$ is equipped with the noncommutative group law

$$(z,t) \cdot (z',t') = (z+z',\, t+t'+2\,\mathrm{Im}\langle z, z'\rangle),$$

where $\langle z, z'\rangle = \sum_k z_k \overline{z'_k}$. The homogeneous Korányi norm, $|(z, t)|_h = (|z|^4 + t^2)^{1/4}$, and horizontal Laplacian

$$\Delta_h = \sum_{j=1}^n (X_j^2 + Y_j^2), \quad X_j = \partial_{x_j} + 2y_j \partial_t, \quad Y_j = \partial_{y_j} - 2x_j \partial_t,$$

define the structure for non-Euclidean analysis, with homogeneous dimension $Q = 2n + 2$ (Liu, 27 Dec 2025, Catino et al., 2023, Chen et al., 13 Jun 2025).

2. The Critical CR Yamabe Equation and Extremals

Jerison–Lee bubbles arise as the extremals to the critical CR Sobolev inequality and as the explicit positive solutions to the nonlinear equation

$$-\Delta_h u = u^{\frac{Q+2}{Q-2}} \quad \text{on } \mathbb{H}^n.$$

Every finite-energy positive solution is, up to CR-conformal transformations, a bubble of the explicit form

$$u(z, t) = K\, |t + i|z|^2 + \mu \cdot z + \kappa|^{-\frac{Q-2}{2}},$$

where $K > 0$, $\mu \in \mathbb{C}^n$, $\kappa \in \mathbb{C}$, and $\mathrm{Im} \kappa > |\mu|^2/4$ (Liu, 27 Dec 2025, Catino et al., 2023).

A normalized two-parameter family (center $\xi$, scale $\lambda>0$) is

$$U_{\lambda, \xi}(\zeta) = c_n \left( \frac{\lambda}{|\xi^{-1} \cdot \zeta|_h^2 + \lambda^2} \right)^{\frac{Q-2}{2}},$$

with the constant $c_n$ ensuring $U_{\lambda, \xi}$ solves the equation above (Liu, 27 Dec 2025, Catino et al., 2023).

On the CR sphere $S^{2m+1}$ with standard pseudohermitian structure $\theta_c$, Jerison–Lee bubbles are

$$\Phi_{t,a}(z) = \big|\cosh t + \sinh t\, \langle z, a \rangle\big|^{-2},$$

where $t \geq 0$, $a \in S^{2m+1} \subset \mathbb{C}^{m+1}$ (Wang, 2013).

3. Complete Classification Theorems

The main theorems, proven via the method of moving spheres and divergence identities, state:

• For each $n \geq 1$, any positive $C^2$ solution to $-\Delta_h u = u^{\frac{Q+2}{Q-2}}$ on $\mathbb{H}^n$ is necessarily a Jerison–Lee bubble. No finite-energy, decay, or symmetry hypotheses are required (Liu, 27 Dec 2025).
• On $\mathbb{H}^1$, all smooth positive solutions are bubbles without any decay assumption; for $n \geq 2$, mild decay $u(\xi) \leq C(1+|\xi|)^{-(Q-2)}$ at infinity suffices (Catino et al., 2023).
• On $S^{2m+1}$ or closed Einstein pseudohermitian manifolds of positive curvature, the only constant-scalar-curvature conformal deformations are trivial unless the manifold is CR isometric to the sphere, in which case bubbles exhaust the nontrivial solutions (Wang, 2013).

These results mirror the Caffarelli–Gidas–Spruck and Li–Zhu classification theorems for critical equations in Euclidean spaces, extending them to the Heisenberg group and CR geometry.

4. Methodologies: Moving Spheres and Divergence Identities

Classification is achieved via two methodologies:

Method of Moving Spheres (Heisenberg Group)

• Generalized CR Inversion: For center $\xi$, radius $\lambda$, and $\beta \in \mathbb{R}$, a CR automorphism $\Phi_{\xi, \lambda}^\beta$ is defined with the property $$d_h(\Phi_{\xi, \lambda}^\beta(\zeta), \xi) \cdot d_h(\zeta, \xi) = \lambda^2$$.
• Kelvin Transform: For $u$ a solution, $$u_{\xi, \lambda}^\beta(\zeta) = (\lambda/d_h(\zeta, \xi))^{Q-2} u(\Phi_{\xi, \lambda}^\beta(\zeta))$$ is also a solution.
• Sphere Expansion: By maximum principle and comparison, one shows $u$ must coincide with its Kelvin transform across some sphere, implying constancy and yielding the bubble form (Liu, 27 Dec 2025).

Jerison–Lee Differential Identity (CR Sphere and Heisenberg Group)

• A divergence formula translates geometric and analytic properties into an identity whose nonnegative right-hand side (sum of squares of "error tensors") vanishes for extremals. Integration by cut-off arguments forces each tensor to vanish, revealing the bubble structure of solutions.
• Vanishing of the error tensors in the sphere case implies the CR structure is torsion-free and Einstein, so the conformal factor is necessarily a bubble (Wang, 2013, Catino et al., 2023).

5. Stability, Non-degeneracy, and Interaction

Jerison–Lee bubbles have quantitative stability and non-degeneracy properties under the CR–Yamabe equation:

• Stability Inequalties: For any $u$ close in $\mathcal{D}^1$ (the Folland–Stein space) to the sum of $m$ weakly interacting bubbles,

$$\|u - \sum_{i=1}^m g_{\lambda_i, \xi_i} U\|_{\mathcal{D}^1} \lesssim \| \Delta_{H^n} u + |u|^{2/n} u \|_{\mathcal{D}^{-1}}$$

with the rate depending on $n$ (linear for $n=1$, logarithmic for $n=2$, power-law for $n \geq 3$) (Chen et al., 13 Jun 2025).

• Interaction Parameter: For two bubbles with parameters $(\lambda_i, \xi_i)$ and $(\lambda_j, \xi_j)$,

$$\epsilon_{ij} = \min\left\{ \frac{\lambda_i}{\lambda_j}, \frac{\lambda_j}{\lambda_i}, \frac{1}{\lambda_i \lambda_j |\xi_i^{-1} \circ \xi_j|^2} \right\},$$

quantifies "weak interaction" via separation or scale.

• Non-degeneracy: The linearized operator $L = -\Delta_{H^n} - p U^{p-1}$ has a $(2n+2)$-dimensional kernel, matching the infinitesimal CR-symmetries (Chen et al., 13 Jun 2025).
• Orthogonality and coercivity: For weakly interacting bubble configurations, orthogonality in $\mathcal{D}^1$ is preserved and the linearized operator is invertible off the span of parameter derivatives.

6. Geometric, Sobolev, and Pseudohermitian Consequences

• CR-conformal group orbit: Bubbles form a full orbit under left-translations, non-isotropic dilations $(z, t) \mapsto (\lambda z, \lambda^2 t)$, unitary rotations, and CR inversions.
• Constant pseudohermitian scalar curvature contact forms: If $\Theta$ is the standard contact form, then $\Theta_u = u^{2/n} \Theta$ has constant pseudohermitian scalar curvature precisely when $u$ is a bubble.
• Folland–Stein Sobolev inequality extremals: Bubbles attain equality in the sharp CR Sobolev inequality, fully characterizing extremal functions (Chen et al., 13 Jun 2025).

7. Analogies and Generalizations in CR and Euclidean Settings

• The Jerison–Lee bubble classification in $\mathbb{H}^n$ is the CR-analogue of the Caffarelli–Gidas–Spruck and Gidas–Ni–Nirenberg theorems for critical semilinear equations in $\mathbb{R}^n$.
• On closed Einstein pseudohermitian manifolds, the only conformal constant-scalar-curvature pseudohermitian structures are global scalings or pulled-back Jerison–Lee bubbles, up to CR isometry; nontrivial cases are CR equivalent to the sphere (Wang, 2013).
• The underlying methodology—divergence identities, moving spheres, and invariance principles—extends to higher-dimensional CR geometry and remains valid for general settings where the ambient structure is Einstein and torsion-free.

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Summary Table: Jerison–Lee Bubble Features

Property	Heisenberg Group $\mathbb{H}^n$	Sphere $S^{2m+1}$ / Einstein CR Manifolds
Explicit Form	$$K\,|t + i|z|^2 + \mu \cdot z + \kappa|^{-\frac{Q-2}{2}}$$	$$\Phi_{t,a}(z) = |\cosh t + \sinh t\,\langle z, a\rangle|^{-2}$$
Classification Theorem	Unique up to CR conformal transform, no symmetry/decay required (Liu, 27 Dec 2025)	Unique up to CR automorphism (Wang, 2013)
Extremality	Sharp Folland–Stein Sobolev inequality	Sharp sphere Sobolev inequality
Invariance Group	Left-translations, dilations, unitary rotations, CR inversion	CR automorphisms of the sphere

All claims and technical details above are explicitly documented in (Liu, 27 Dec 2025, Chen et al., 13 Jun 2025, Catino et al., 2023), and (Wang, 2013) according to the classification, stability, and uniqueness theorems, divergence identities, explicit formulas, and underlying geometric interpretations.