Conformally Invariant Elliptic Equations

• Conformally invariant elliptic equations are PDEs that preserve their form under metric rescaling, exemplified by the Yamabe equation.
• Advanced analytical techniques like moving sphere methods and variational approaches are used to establish rigidity, compactness, and Liouville-type theorems.
• These equations have broad applications in geometric analysis, quantum field theory, and curvature problems on manifolds with boundary.

A conformally invariant elliptic equation is an elliptic partial differential equation (PDE), system, or nonlocal equation whose form is preserved under conformal transformations of the underlying geometric space. More precisely, these equations admit invariance under conformal rescalings of the metric, or, in the flat case, under the action of the Möbius group. The paper of such equations originates in geometric analysis and representation theory, encompasses a significant class of critical nonlinear PDEs (including the Yamabe problem, critical power nonlinearities, GJMS operators, and fully nonlinear extensions), and underpins profound classification, compactness, and rigidity phenomena in mathematical physics and differential geometry.

1. Defining Features and Examples

The canonical prototype is the Yamabe equation on ℝⁿ,

$$- \Delta u = u^{\frac{n+2}{n-2}}, \quad u > 0 \quad \text{in } \mathbb{R}^n,$$

which is invariant under the full conformal group. More generally, consider

$f(\lambda(A^u)) = 1 \quad \text{in } \mathbb{R}^n$

where $A^u$ denotes the conformal Hessian (often related to the Schouten tensor under conformal metric deformation), $\lambda(A^u)$ is the vector of its eigenvalues, and $f$ is a symmetric function homogeneous of degree one, positive in a symmetric convex cone $\Gamma$ and vanishing on $\partial\Gamma$.

Other central examples include:

• Conformally invariant integral equations involving Poisson-type kernels or Riesz potentials, e.g.,

$$u(x) = \int_{B} K(x, y) [u(y)]^{\frac{n+2\sigma}{n-2\sigma}} dy + h(x)$$

• High-order conformally invariant equations such as those involving the GJMS operator $P_g$,

$\alpha P_g v + 2Q_g = 2Q_g e^{nv}$

with $Q_g$ the Q-curvature.

Systems coupling conformally invariant operators of different order, or equations with exponentially increasing nonlinearity, also arise; for instance, the classification of solutions to

$$\begin{cases} (-\Delta)^{1/2} u(x) = e^{pv(x)},\ -\Delta v(x) = u^4(x),\ \end{cases}$$

in $\mathbb{R}^2$ (Dai et al., 2021).

Nonlocal critical nonlinearity problems, such as the Choquard equation

$$- u = I_{\mu}\left(|u|^{2_{\mu}^*}\right)|u|^{2_{\mu}^* - 2}u, \;\; 2_{\mu}^* = \frac{2n-\mu}{n-2}$$

are also conformally invariant (up to scaling) when the convolution kernel and nonlinearity are balanced (Almutairi et al., 14 Sep 2025).

2. Structural Properties and Conformal Invariance

A unifying principle is invariance under conformal transformations, including Möbius transformations in Euclidean space or conformal rescalings on Riemannian manifolds. The conformal Hessian (or Schouten tensor in the geometric setting) transforms covariantly, and symmetric functions of its eigenvalues produce formally invariant equations (see (Chu et al., 2023)). Under metric scaling $g \to e^{2\rho}g_0$, the Schouten tensor changes as

$$A_{g} = A_{g_0} - \nabla^2 \rho + d\rho \otimes d\rho - \frac{1}{2} |\nabla \rho|^2 g_0.$$

The function $f$ is designed so that, for admissible tuples $\lambda = \lambda(A^u)$, $f(\lambda)$ is invariant under these transformations.

For nonlocal equations, the invariance is reflected in the invariance of associated functionals under the Möbius group, and, in high order cases, in the conformal covariance of operators such as the GJMS sequence.

Systems and integral equations with critical exponents (determined by scaling) inherit conformal invariance appropriately under simultaneous rescalings of all dependent and independent variables.

3. Rigidity, Classification, and Liouville Theorems

Liouville-type theorems are fundamental in the analysis of conformally invariant elliptic equations. For many such equations, the only positive (or admissible) entire solutions are explicit "bubble" solutions,

$$u(x) = \left( \frac{a}{1 + b|x - x_0|^2} \right)^{(n-2)/2},$$

where the constants $a, b > 0$ and center $x_0$ are free parameters corresponding to conformal transformations (Chu et al., 2023, Li et al., 2016, Jin et al., 2019). For equations of the form $f(\lambda(A^u))=1$ in $\mathbb{R}^n$ (with suitable cones and structural conditions), these are the only $C^{1,1}_{\text{loc}}$ solutions.

For viscosity or weak solutions, classification proceeds by first establishing strong comparison principles and maximum principles, even in settings lacking uniform ellipticity (Li et al., 2019, Li et al., 2012). For higher-order conformally invariant equations—including GJMS operators or fractional Laplacians—bubble-type and periodic (Fowler-type) solutions again play the central role (Jin et al., 2019).

In dimensions and parameter ranges where only explicit solutions exist, rigidity results can be interpreted as sharp uniqueness assertions: for example, the only solutions to $$\alpha P_{\mathbb{S}^n} v + 2Q_{g_{\mathbb{S}^n}} = 2Q_{g_{\mathbb{S}^n}} e^{nv}$$ in the regime $\alpha > 1$ are $v\equiv 0$ (Zhang, 2022).

4. Boundary Phenomena and Rigidity Results

Conformally invariant elliptic equations exhibit subtle interactions with boundary geometry. In the context of conformal metrics $g = e^{2\rho}g_0$ on subdomains of $\mathbb{S}^n$, rigidity results can be obtained under boundary conditions involving umbilicity and lower mean curvature bounds. The main rigidity theorem asserts that if $f(\lambda(g)) \geq 1$ and $H_g \geq H_r$ on the boundary (where $H_r$ is the mean curvature of a geodesic sphere in $\mathbb{S}^n$), then $g$ is isometric to the standard metric on a spherical domain of radius $r$ (Barbosa et al., 2017).

These results extend classical rigidity theorems (such as the Min-Oo conjecture and Toponogov's theorem) to fully nonlinear, non-concave equations involving arbitrary symmetric functions of the Schouten tensor eigenvalues, even in the absence of concavity.

Boundary geometry enters through sharp mean curvature estimates and through representation formulas which realize the conformal metric as a hypersurface in hyperbolic space by a conformal support function.

5. Isolated Singularities, Blow-up, and Local Analysis

A detailed analysis of solutions near singularities and blow-up points provides deep understanding of global compactness and the moduli space of solutions. In the neighborhood of an isolated singularity, a dichotomy is observed: either the singularity is removable (the solution extends regularly) or the solution displays precise quantitative blow-up, taking the form

$u(x) \sim |x|^{-(n-2)}$

for the second-order Yamabe case, or $|x|^{-(n-2\sigma)}$ for order-$2\sigma$ problems (Jin et al., 2019). For higher-order and nonlocal equations, the same dichotomy persists, but the rates and structures must be computed via integral representations and refined moving-sphere arguments.

Harnack inequalities of the sup–inf type hold for these equations—if $u$ is a positive solution in a ball, then

$\sup_{B} u \cdot \inf_{B} u^{-1} \leq C$

for $C$ depending only on geometric and equation data; see (Li et al., 2012, Chu et al., 2023, Li et al., 2016). These inequalities, together with explicit asymptotics, enable fine control of concentration phenomena and underlie compactness theorems for families of solutions.

Singularity removability and extension across isolated points is characterized by lower- or upper-conical behavior, with necessary and sufficient conditions expressed in terms of the conformal Hessian (Chu et al., 2023).

6. Methods of Construction and Analytical Techniques

The production and analysis of conformally invariant elliptic equations employs a variety of geometric and analytic frameworks:

• Weyl-to-Riemann method: One writes homogeneous equations in Weyl geometry using the Weyl covariant derivative, ensuring conformal invariance and then selecting combinations (and conformal weights) so that the equations reduce to Riemannian conformally invariant equations, including for higher-spin fields (Faci, 2012).
• Variational methods on symmetric spaces: By lifting nonlocal or critical equations to the sphere (via stereographic projection) and employing group symmetries, the lack of compactness in variational embeddings on $\mathbb{R}^n$ can be circumvented, leading to the construction of infinitely many (nodal or sign-changing) solutions in the presence of appropriate group actions (Almutairi et al., 14 Sep 2025).
• Integral and nonlocal formulations: Some critical exponent problems are most naturally formulated as nonlocal integral equations involving Poisson-type kernels, for which compactness and existence can be understood via symmetry constraints (e.g., antipodal symmetry) and blow-up analysis (Xiong, 2017, Du et al., 2023).
• Moving sphere or moving plane methods: These symmetry techniques provide powerful tools for classification and uniqueness, particularly for equations and systems with degenerate or fully nonlinear structure (Chu et al., 2023, Zhang, 2022, Li et al., 2016).
• Prolongation and algebraic constraints: For overdetermined systems, e.g., scalar-flat Möbius Einstein–Weyl equations, local polynomial obstructions are derived by prolongation methods, and the existence of solutions is tied to vanishing of resultants built from conformal invariants (Randall, 2012).

Partial uniform ellipticity theory quantifies the directions in which maximum principles and regularity estimates hold, allowing for a non-uniform (but robust) regularity theory that underpins a priori estimates for Yamabe–type and Loewner–Nirenberg problems in the fully nonlinear context (Yuan, 2022).

7. Impact and Applications

Conformally invariant elliptic equations and systems have significant applications in geometric analysis and mathematical physics:

• Yamabe-type problems: Prescribing scalar curvature or more general curvature invariants (e.g., $\sigma_k$-curvature, Q-curvature) under conformal deformation of metrics on compact or complete manifolds (Chu et al., 2023, Yuan, 2022).
• Rigidity and sharp inequalities: Uniqueness and classification theorems furnish rigidity results, which underwrite sharp Sobolev, Beckner, and Onofri inequalities. For example, the moving plane classification of GJMS operator equations leads directly to uniqueness of extremals for the Beckner inequality (Zhang, 2022).
• Quantum field and gauge theory: New conformally invariant Yang–Mills–type actions and equations on higher-dimensional manifolds relate to the paper of Fefferman–Graham obstruction tensors, Q-curvature, and higher-order gauge theories (Gover et al., 2021).
• Manifolds with boundary and noncompact problems: The generalization of the Loewner–Nirenberg problem and the paper of complete conformal metrics with prescribed singular behavior on domains with boundary arise in physical and geometric problems involving singularities.
• Nonlocal, integral, and system equations: Prescribing nonlocal curvatures, such as fractional Laplacians or integral curvature quantities, and the analysis of critical systems involving mixed-order or Hartree-type nonlinearities extend the theory into nonlocal domains, with applications in both geometry and nonlinear analysis (Dai et al., 2021, Almutairi et al., 14 Sep 2025).

The field continually advances through deeper classification, novel variational methods, and the exploitation of geometric and analytic invariances, providing a unified approach to rigidity, compactness, and the structure of critical PDEs in conformal geometry.