Conformally Covariant Tridifferential Operators

• Conformally covariant tridifferential operators are trilinear differential operators that transform predictably under metric rescaling, providing a framework for analyzing nonlinear geometric PDEs and variational invariants.
• They are constructed via ambient metric techniques and symmetrization methods, yielding a (k+1)-dimensional family of operators with formal self-adjointness at specific conformal weights.
• Their applications include nonlinear analogues of the Paneitz operator, the formulation of sharp boundary inequalities, and realizing representation-theoretic constructs on the sphere.

A conformally covariant tridifferential operator is a trilinear differential operator acting on triple products of functions or densities on a conformal manifold and transforming in a prescribed way under conformal changes of the metric. These operators generalize the well-studied linear (GJMS) and bilinear (e.g., Ovsienko–Redou) conformally covariant constructions and arise naturally both in the paper of geometric variational invariants (such as the σ₂-curvature) and in conformal representation theory on the sphere via residues of invariant trilinear forms. Such operators are central in the explicit analysis of nonlinear geometric PDEs, higher-order boundary problems, and the representation theory of the conformal group, and their construction reveals refined algebraic and analytic properties distinguishing the tridifferential case from lower rank analogues.

1. Definitions and Foundational Properties

A natural ℓ-differential operator on a smooth Riemannian manifold $(M^n,g)$ is a multilinear differential map $D^{(g)}: (C^\infty(M))^\ell \to C^\infty(M)$ expressible as universal linear combinations of complete contractions involving covariant derivatives of the arguments, the metric $g_{ij}$ and $g^{ij}$, and the Riemann tensor $R_{ijkl}$ and its derivatives.

A tridifferential operator is an operator with $\ell=3$; it is called conformally covariant of bidegree $(a,b)$ if it satisfies

$$D^{\hat g}(u_1,u_2,u_3) = e^{-b\omega} D^g(e^{a\omega}u_1, e^{a\omega}u_2, e^{a\omega}u_3)$$

under metric rescaling $\hat g = e^{2\omega}g$. Formal self-adjointness holds if, for any compactly supported test-tuple $(u_0,u_1,u_2,u_3)$, the integral

$\int_M u_0 D(u_1,u_2,u_3) \,\mathrm{dvol}_g$

is symmetric under permutation of all four arguments (Case et al., 2020, Case et al., 13 Nov 2025).

2. Ambient Metric Construction and Tangentiality

Following Fefferman–Graham, the ambient metric encodes the conformal class $(M^n, c)$ via an $(n+2)$-dimensional ambient pseudo-Riemannian manifold $(\widetilde{M},g)$, with the conformal bundle $G$ sitting as the light cone. Homogeneous functions on $G$, corresponding to densities of given conformal weight, extend to homogeneous functions on $\widetilde{M}$. A tangential operator on $[\vec{w}]$—homogeneous tensors of weights $w_j$—is an ambient differential operator descending to the base manifold and respecting conformal invariance.

For tridifferential operators of total order $2k$,

$$D(U_1 \otimes U_2 \otimes U_3) = \sum_{\alpha\in I_k^5} \binom{k}{\alpha}\, A_\alpha\, \Delta^{\alpha_1} \Bigl( \Delta^{\alpha_2} \bigl(\Delta^{\alpha_3} U_1 \cdot \Delta^{\alpha_4} U_2 \bigr) \cdot \Delta^{\alpha_5} U_3 \Bigr)$$

with multi-indices $\alpha = (\alpha_1,\dots,\alpha_5)$, $I_k^5 = \{\sum\alpha_i = k\}$, and $A_\alpha$ determined by tangentiality relations—explicitly, the requirement that $[D,Q]_j$ vanishes for each slot, with $Q$ the ambient quadratic form and $$[ \Delta^m, Q ] = -2m \Delta^{m-1}(2X + n + 4 - 2m)$$ (Case et al., 13 Nov 2025, Case et al., 2020).

For generic input weights $(w_1,w_2,w_3)$ and $n\geq 2k$ (if $n$ is even), there is a $(k+1)$-dimensional solution space of coefficient functions $A\in F_k^5$, yielding a $(k+1)$-parameter family of order-$2k$ conformally covariant tridifferential operators.

3. Symmetrization and Formal Self-Adjointness

Formal self-adjointness in the trilinear setting is achieved by imposing equal input weights $w_1 = w_2 = w_3 = -\frac{n-2k}{4}$ so that the output weight shift $w_1 + w_2 + w_3 - 2k = -n$. Symmetrization over cyclic permutations of the slots,

$$D'(U_1, U_2, U_3) := D(U_1, U_2, U_3) + D(U_2, U_3, U_1) + D(U_3, U_1, U_2),$$

ensures formal self-adjointness, justified by index-swapping identities among ambient coefficients and integration by parts in the flat ambient volume (Case et al., 13 Nov 2025).

In explicit examples for $k=1$ (order $2$) and $k=2$ (order $4$), the symmetrized operator is unique (up to scale) for $k=1$ and has a $2$-dimensional family for $k=2$, reflecting the increased complexity versus the linear and bilinear cases.

4. Explicit Constructions: The Nonlinear Analogue of the Paneitz Operator

A distinguished example is the unique, formally self-adjoint, fourth-order, conformally covariant tridifferential operator $D_4$ associated to the conformally variational invariant $\sigma_2 = \frac12(J^2 - |P|^2)$, with $P$ the Schouten tensor and $J = \mathrm{tr} P$. $D_4$ is constructed by polarization of an explicit scalar formula for $L_4(u)$ involving divergence, Laplacian, the first Newton tensor $T_1 = Jg - P$, and $\sigma_2$:

$$L_4(u) := \frac{1}{2}\,\delta ( \nabla(u^2) du ) - \frac{n-4}{16}[u \Delta (\nabla(u^2)) - \delta( (\Delta u^2) du )] - \frac{1}{2}\left(\frac{n-4}{4}\right)^2 u\,\delta( T_1 (\nabla(u^2)) ) + \left( \frac{n-4}{4}\right)^2 \sigma_2 u^3.$$

The trilinear operator is

$$D_4(u,v,w) = \frac{1}{3!} \sum_{\sigma\in S_3} L_4\big( \sqrt{u_{\sigma(1)} v_{\sigma(2)} w_{\sigma(3)}} \big),$$

with monomials symmetrized as appropriate. This operator satisfies conformal covariance with bidegree

$a = \frac{n-4}{4}, \quad b = \frac{3n+4}{4}$

and is the simplest fully nonlinear analogue of the Paneitz operator, arising from the second variation of the $\sigma_2$-curvature functional. Its existence and uniqueness are consequences of the general correspondence between homogeneous conformally variational invariants (CVI) and formally self-adjoint, conformally covariant polydifferential operators (Case et al., 2020).

5. Representation-Theoretic and Spherical Constructions

On the sphere $S^{n-1}$, conformally covariant tridifferential operators arise as singular residues of $G$-invariant trilinear forms, where $G=SO_0(1,n)$ is the conformal group. For generic complex parameters $(\alpha_1,\alpha_2,\alpha_3)$, the trilinear form

$$T_{\alpha_1,\alpha_2,\alpha_3}(f_1,f_2,f_3) = \int_{S^3} |x_2 - x_3|^{-\rho+\alpha_1} |x_3 - x_1|^{-\rho+\alpha_2} |x_1-x_2|^{-\rho+\alpha_3} f_1(x_1) f_2(x_2) f_3(x_3) dx_1 dx_2 dx_3$$

($\rho = (n-1)/2$) is unique up to scale and $G$-invariant when the $\alpha_j$ and representation weights $\lambda_j$ are suitably related. Meromorphic extension in $\vec{\alpha}$ features singular hyperplanes (poles), and taking residues along all three produces distributions supported on the diagonal $x_1 = x_2 = x_3$, equivalent to tridifferential operators $D_{k_1, k_2, k_3}$ intertwining three different principal series. These are unique up to scaling, with principal symbol determined by the orders in each argument, and obey explicit covariance relations

$$D_{k_1,k_2,k_3}\circ (\pi_{-\rho-k_1} \otimes \pi_{-\rho-k_2} \otimes \pi_{-\rho-k_3}) = \pi_{\rho+k_1+k_2+k_3} \circ D_{k_1,k_2,k_3}.$$

Examples include the product ($k_1=k_2=k_3=0$), the Yamabe operator in one slot, and higher-order fully symmetric trilinear operators (Clerc, 2011).

6. Comparison with Linear and Bilinear Constructions

The linear GJMS operators $P_{2k}$ and the curved Ovsienko–Redou bidifferential operators have unique or low-dimensional families in their respective settings. For the tridifferential case, the ambient construction always yields a $(k+1)$-dimensional family of order $2k$ operators for generic weights, with formal self-adjointness realized only at the special conformal weight. On the sphere, the tridifferential family is never unique but always $(k+1)$-dimensional, in contrast to the uniqueness of linear GJMS on the sphere. The symmetrization to enforce self-adjointness is also distinctively new in the rank-3 case (Case et al., 13 Nov 2025).

7. Applications and Analytic Implications

These operators serve as canonical nonlinear analogues of the Paneitz operator for fourth-order equations, giving rise to variational structures controlling nonlinear energies associated to $\sigma_2$-geometry. They also control optimal fourth-order boundary inequalities and appear in fully nonlinear generalizations of Q-curvature theory. For boundary value problems, tridifferential analogues offer tools for formulating fully nonlinear boundary operators and energy functionals with sharp Sobolev trace inequalities, generalizing the role of the Paneitz operator and its boundary companions. The multilinear setting enlarges the analytic toolkit available for attacking nonlinear conformally invariant PDEs and for realizing representation-theoretic objects as explicit differential operators (Case et al., 2020, Case, 2015).

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References:

• "Some constructions of formally self-adjoint conformally covariant polydifferential operators" (Case et al., 2020)
• "A general construction of conformally covariant tridifferential operators" (Case et al., 13 Nov 2025)
• "Singular conformally invariant trilinear forms and covariant differential operators on the sphere" (Clerc, 2011)
• "Boundary operators associated to the Paneitz operator" (Case, 2015)