Cherrier-Escobar Invariant

• Cherrier-Escobar Invariant is a conformally covariant boundary operator combining normal derivatives and mean curvature to interpolate between Dirichlet and Neumann conditions.
• It underpins well-posed elliptic boundary problems and influences the spectral theory of Robin Laplacians by controlling eigenvalue behavior.
• Utilizing tools like tractor calculus, it extends to a family of higher-order conformal boundary operators with applications in conformal field theory and eigenvalue prescription.

The conformal Robin operator is the canonical first-order, conformally covariant boundary differential operator acting on conformal densities or weighted functions on Riemannian manifolds with boundary. It plays a central role in conformal geometry, spectral theory, and boundary conformal field theory, providing the paradigm for a family of conformally invariant boundary operators and interpolating between Dirichlet and Neumann conditions. The operator is also essential for formulating well-posed elliptic boundary problems for conformally covariant interior differential operators and appears in the analysis of Robin Laplacians, eigenvalue prescription on manifolds and domains, and in the study of conformal symmetry breaking in representation theory (Gover et al., 2018, He et al., 3 Mar 2025, Freitas et al., 2018, Bourgine et al., 2016).

1. Definition and Construction

Let $(M^n, g)$ be a Riemannian manifold with smooth boundary $\Sigma = \partial M$, and let $\mathcal{E}[w]$ denote the bundle of conformal densities of weight $w$. The conformal Robin operator

$$\delta_{g, w} : \mathcal{E}[w] \longrightarrow \mathcal{E}[w-1]|_{\Sigma}$$

is defined for any real $w$ and section $\sigma \in \Gamma(\mathcal{E}[w])$ by

$$\delta_{g, w} \sigma = n^a \nabla_a \sigma - w H \sigma,$$

where $n^a$ is the outward unit normal (with $n^a n_a = 1$), and $H = \frac{1}{n-1} \nabla_a n^a$ is the mean curvature of the boundary $\Sigma$. For ordinary functions $f \in C^\infty(M) = \mathcal{E}[0]$, this becomes the familiar

$\delta_{g, 0} f = n^a \nabla_a f.$

The operator combines normal derivative (Neumann-type) and boundary value (Dirichlet-type) data, making it a mixed (Robin-type) operator (Gover et al., 2018).

2. Conformal Covariance and Transformation Laws

Under conformal rescaling $\hat{g}_{ab} = e^{2\Upsilon}g_{ab}$ with

$\hat{\sigma} = e^{w\Upsilon}\sigma,$

the transformation laws are

$$\hat{n}^a = e^{-\Upsilon} n^a, \qquad \hat{H} = e^{-\Upsilon}(H + n^a\nabla_a\Upsilon),$$

and

$$\hat{\nabla}_a \sigma = e^{w\Upsilon}(\nabla_a \sigma + w\sigma \nabla_a \Upsilon).$$

A direct computation shows

$$\hat{\delta}_{\hat{g}, w}(e^{w\Upsilon}\sigma) = e^{(w-1)\Upsilon} \delta_{g, w} \sigma,$$

establishing conformal covariance of bidegree $(-w, 1-w)$. This property is critical for constructing conformally invariant elliptic boundary problems and defining differential boundary conditions consistent under metric rescaling (Gover et al., 2018, He et al., 3 Mar 2025, Freitas et al., 2018).

3. Boundary Value Problems and Self-Adjointness

The conformal Robin operator provides the natural boundary term for conformally covariant interior elliptic operators. For the conformal Laplacian (the second-order GJMS operator), the pair $(P_2, \delta_{g, 1-n/2})$ forms a formally self-adjoint boundary problem. The associated Dirichlet-to-Neumann map $\mathcal{P}_1$ arises: $$\mathcal{P}_1(f) = \delta_{g, 1-n/2}(\sigma_f), \quad \sigma_f|_\Sigma = f,\,\, P_2\sigma_f = 0,$$ with $\mathcal{P}_1$ having leading symbol $(-\Delta)^{1/2}$, i.e., realizing the half-order conformal fractional Laplacian (Gover et al., 2018). The boundary problem

$$\left\{\begin{array}{ll} P_2 \sigma = 0 & \text{in } M, \ \sigma|_{\Sigma} = f, & \ \delta_{g, 1-n/2}(\sigma) = 0 & \text{on } \Sigma \end{array}\right.$$

is elliptic, conformally invariant, and formally self-adjoint when the kernel is trivial.

4. Family of Higher-Order Conformal Boundary Operators

The conformal Robin operator is the $K=1$ member of a broader family $d_K$ of conformally covariant natural differential operators

$$d_K : \mathcal{E}[w] \to \mathcal{E}[w-K]|_{\Sigma},$$

constructed inductively using tractor calculus. The construction utilizes the Thomas $D$-operator $D_A$ and the normal tractor $N^A$: $d_{K+1} = N^A d_K D_A,$ for suitable normalization in $w$ and $n$. For $K=1$, $d_1 = \delta_{g, w}$ (Gover et al., 2018). This provides a systematic route to higher-order conformal boundary operators (e.g., the Chang–Qing operator at $K=3$), all sharing conformal covariance and naturality.

5. Conformal Robin Boundary Value Spectrum and Eigenvalue Prescription

The Robin Laplacian (for functions) is defined by

$$\Delta_g u + \lambda u = 0 \text{ in } M, \qquad \delta_{g,0} u + \rho u = 0 \text{ on }\partial M,$$

for a boundary parameter $\rho \in C_+(\partial M)$. The spectrum is discrete and monotone: $$0 < \lambda_1(M, g, \rho) < \lambda_2(M, g, \rho) \leq \lambda_3(M, g, \rho) \leq \dots \to \infty.$$ Under a conformal change $g \to \tilde{g} = e^{2w}g$, the Robin parameter transforms $\tilde{\rho} = e^{-w}\rho$, crucial for spectral problems in a given conformal class (He et al., 3 Mar 2025).

Prescription results include the ability (in $n \geq 3$) to set a finite collection of Robin eigenvalues, control their multiplicities, or fix both spectrum and volume within a given conformal class. The proofs employ reduction to large-parameter Dirichlet limits and quantum graph gluing, establishing that for any finite nondecreasing list $(a_1, \dots, a_m)$, and $V>0$, there exists a metric $g$ for which $\operatorname{Vol}(M, g)=V$ and $\lambda_k(M,g,\rho)=a_k$ for $1 \leq k \leq m$ (He et al., 3 Mar 2025).

6. Conformal Robin Operator in Classical and Representation-Theoretic Contexts

On the standard conformal hemisphere $(S^n_+, g_0)$, the boundary $\Sigma = S^{n-1}$ is totally geodesic $(H \equiv 0)$, so $\delta_{g_0, w}(f) = n^a\nabla_a f$ is simply the normal derivative. In this model, all calculations reduce to classical harmonic analysis, and the construction yields symmetry-breaking intertwiners between the principal series representations of $SO(n,1)$ and $SO(n-1,1)$. The conformal Robin operator thus realizes the symmetry-breaking mechanism central to the analytic theory of boundary representations (Gover et al., 2018).

7. Conformal Robin Operator in Integrable Models and Logarithmic CFT

In the context of logarithmic minimal models $\mathcal{LM}(p,p')$ on strips, the “Robin” boundary condition is realized as a linear combination of Dirichlet and Neumann tiles at the lattice boundary and is labeled by Kac parameters $(r, s-\tfrac{1}{2})$. The continuum scaling limit identifies the Robin boundary state as the Virasoro highest-weight representation with conformal dimension

$$\Delta^{p, p'}_{r, s - \frac{1}{2}} = \frac{(r p' - (s - \frac{1}{2})p)^2 - (p' - p)^2}{4 p p'},$$

with $r \in \mathbb{Z}$, $s \in \mathbb{N}$. The Robin operator here geometrizes the interpolation between boundary conditions and encodes fusion rules and scaling fields in logarithmic CFTs (Bourgine et al., 2016). The lattice realization through the Robin boundary ensures integrability and provides analytic access to boundary free energies and scaling spectra.

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Summary Table: Conformal Robin Operator—Key Properties

Setting	Definition / Role	Conformal Transformation
Riem. Manifold, $(M,g)$	$$\delta_{g, w} \sigma = n^a \nabla_a \sigma - w H \sigma$$	$$\hat{\delta}_{\hat{g}, w}(e^{w\Upsilon}\sigma) = e^{(w-1)\Upsilon} \delta_{g, w} \sigma$$
Laplacian Spectrum	Boundary term in Robin BVP: $\partial_\nu u + \rho u = 0$	$\tilde{\rho} = e^{-w}\rho$ under $g \to e^{2w}g$
Hemisphere, $S^n_+$	Reduces to normal derivative: $n^a \nabla_a \sigma$ if $H=0$	Invariant splitting under conformal group
Conformal Field Theory	Lattice operator interpolating Dirichlet/Neumann (Kac $s-\frac12$)	Robin boundary field realized as half-integer scaling operator

The conformal Robin operator underlies the analysis of conformally invariant boundary value problems in geometry and mathematical physics, provides a prototype for higher-order boundary operators, and manifests in both analytic and algebraic settings as the fundamental mixed boundary operator respecting conformal symmetry (Gover et al., 2018, He et al., 3 Mar 2025, Freitas et al., 2018, Bourgine et al., 2016).