Solvability of the Conformal Constraint Equations

The conformal method turns the vacuum constraints into a coupled scalar–vector elliptic system. It is most complete at constant mean curvature, where the equations decouple and solvability is classified by the Yamabe class and degeneracies of the free data. Away from that regime, existence, uniqueness, and nonuniqueness depend on quantitative hypotheses; “choose free conformal data” is not a general parametrization theorem.

Required background. Analysis for Mathematical Relativity supplies semilinear elliptic theory, The Einstein Equation supplies the constraints, Hypersurface Normals and Tangential Projectors supplies $h_{ij}$ and $K_{ij}$, and Differential Equations for Relativity supplies boundary-value methods.

Helpful background. Causality and Global Structure explains developments, while PDEs and Green Functions for Gravitation reviews elliptic operators.

Conformal decomposition

Work on a smooth closed three-manifold first, in vacuum with $\Lambda=0$. Decompose

$$K_{ij}=A_{ij}+\frac13\tau h_{ij}, \qquad \tau=K,$$

using $K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu$. Choose free data consisting of a conformal metric $\widetilde h_{ij}$, a transverse-traceless tensor $\widetilde\sigma^{ij}$, and a mean-curvature function $\tau$. Seek

$$h_{ij}=\phi^4\widetilde h_{ij}, \qquad A^{ij}=\phi^{-10} \left(\widetilde\sigma^{ij}+(\widetilde L W)^{ij}\right),$$

where $\phi>0$ and

$$(\widetilde L W)_{ij} =\widetilde D_iW_j+\widetilde D_jW_i -\frac23\widetilde h_{ij}\widetilde D_kW^k.$$

The constraints become

$$\widetilde D_j(\widetilde L W)^{ij} =\frac23\phi^6\widetilde D^i\tau,$$ $$-8\widetilde\Delta\phi +\widetilde R\phi -\left|\widetilde\sigma+\widetilde L W\right|_{\widetilde h}^2\phi^{-7} +\frac23\tau^2\phi^5=0.$$

Here $\widetilde\Delta=\widetilde D^i\widetilde D_i$ and transverse-traceless means $\widetilde D_j\widetilde\sigma^{ij}=0$ and $\widetilde h_{ij}\widetilde\sigma^{ij}=0$. Two identities explain the exponents rather than requiring them to be memorized:

$$\begin{aligned} R(h)&=\phi^{-5}(-8\widetilde\Delta\phi+\widetilde R\phi),\\ |A|_h^2&=\phi^{-12}|\widetilde\sigma+\widetilde LW|_{\widetilde h}^2. \end{aligned}$$

Insert these into $R(h)-|A|_h^2+\tfrac23\tau^2=0$ and multiply by $\phi^5$. This gives exactly the scalar equation above. The powers are specific to three spatial dimensions; the vector equation follows from conformal covariance of the divergence of a trace-free tensor. See Bartnik and Isenberg 2004, §4.1, equations (47)–(51).

Constant mean curvature theorem

Assume $\tau$ is constant. Then the right-hand side of the vector equation vanishes. On the closed manifold, multiplying by $W_i$ and integrating gives

$$0=\int_\Sigma W_i\widetilde D_j(\widetilde LW)^{ij}d\mu_{\widetilde h} =-\frac12\int_\Sigma|\widetilde LW|^2d\mu_{\widetilde h}.$$

Consequently $\widetilde LW=0$, even if conformal Killing fields exist. The vector $W$ may be nonunique, but its contribution to the physical data is zero. In the non-CMC problem a nonzero vector source must instead be orthogonal to every conformal Killing field.

The remaining Lichnerowicz equation is a scalar semilinear elliptic equation. A representative CMC result is:

Let $\Sigma$ be a smooth, connected, closed three-manifold, $\widetilde h$ a smooth Riemannian metric, $\widetilde\sigma$ a smooth transverse-traceless tensor, and $\tau$ a constant. The vacuum $\Lambda=0$ CMC system has a smooth positive solution exactly in the cases listed below. For fixed free data, $\phi$ is unique except in the Yamabe-zero case $\widetilde\sigma=0=\tau$, where there is a scaling family. The vector ambiguity is only a conformal Killing field and does not change $(h,K)$.

Why the qualifications matter is visible without memorizing a table. If $\widetilde\sigma=0$ and $\tau=0$, the equation reduces to

$$-8\widetilde\Delta\phi+\widetilde R\phi=0,$$

which asks for a scalar-flat metric in the conformal class. On a closed manifold this is possible only in the Yamabe-zero class; constant multiples of one solution give the familiar scaling degeneracy. Conversely, for negative Yamabe type the $\tau^2\phi^5$ term is needed to balance the negative scalar-curvature contribution, while positive Yamabe type requires a nonzero transverse-traceless term in the standard vacuum classification. Carlotto reviews the precise CMC cases and their geometric meaning in Carlotto 2021, §2.

The closed vacuum CMC pattern

The Yamabe sign is the sign of the constant scalar curvature of a constant-scalar-curvature representative of $[\widetilde h]$. In the theorem above the complete pattern is:

Yamabe class	$\widetilde\sigma\equiv0$, $\tau=0$	$\widetilde\sigma\not\equiv0$, $\tau=0$	$\widetilde\sigma\equiv0$, $\tau\ne0$	$\widetilde\sigma\not\equiv0$, $\tau\ne0$
Positive	No solution	Unique	No solution	Unique
Zero	Scaling family	No solution	No solution	Unique
Negative	No solution	No solution	Unique	Unique

The condition $\widetilde\sigma\not\equiv0$ means not identically zero; it does not require the tensor to be nonzero at every point. The classification and its maximum-principle and barrier proof are described in Bartnik and Isenberg 2004, §4.1, equation (56) and the discussion of closed CMC data. For a direct nonexistence check, choose a positive-Yamabe representative with $\widetilde R\gt0$ and $\widetilde\sigma=0$. Multiply the scalar equation by $\phi$ and integrate:

$$8\int|\widetilde D\phi|^2d\mu +\int\widetilde R\phi^2d\mu +\frac23\tau^2\int\phi^6d\mu=0.$$

Every term is nonnegative and the middle one is strictly positive for $\phi\gt0$, a contradiction. A claimed positive-Yamabe solution with these free data therefore fails before any numerical solve is attempted.

The table is confined to closed vacuum CMC data. Boundaries, asymptotically Euclidean ends, matter sources, conformal Killing fields, or nonconstant $\tau$ alter the Fredholm problem. That limitation is exactly why each application must state its domain rather than quote “the conformal method is solvable.”

Worked example on a flat three-torus

Take a flat torus, a constant trace-free tensor $\widetilde\sigma$ with constant squared norm $S^2\gt0$, and constant $\tau\ne0$. A constant tensor is divergence-free in the flat coordinates. Seeking a constant conformal factor reduces the equation to

$$-S^2\phi^{-7}+\frac23\tau^2\phi^5=0, \qquad \phi=\left(\frac{3S^2}{2\tau^2}\right)^{1/12}.$$

The physical metric is still flat, while $|A|_h^2=\phi^{-12}S^2=2\tau^2/3$. Thus $R(h)-|A|_h^2+2\tau^2/3=0$ directly, and the momentum constraint holds because all data are constant. Setting either $S$ or $\tau$ to zero while keeping the other nonzero destroys this balance, in agreement with the Yamabe-zero row. This example shows why the mean curvature and transverse-traceless data play different, complementary roles.

Nonconstant mean curvature

When $d\tau\ne0$, the vector equation contains $\phi^6d\tau$ and the two equations couple. Near-CMC theorems treat $d\tau$ as controlled relative to the remaining data and obtain solutions through fixed-point, sub/supersolution, or implicit-function arguments. Far from CMC there are existence results in selected regimes, but also nonexistence and multiple solutions. No condition stated only as “$\tau$ is smooth” yields a universal result.

Other geometries change the theorem again. Asymptotically Euclidean data require weighted spaces and $\phi\to1$ on each chosen end. Manifolds with apparent-horizon boundaries impose nonlinear Robin conditions. Matter terms carry their own conformal weights, and an unfortunate scaling can destroy elliptic decoupling or energy inequalities.

The conformal method therefore provides a construction, not a proof that its free data cover the entire constraint surface injectively. Its rigorous solvability results are the theorem-level counterpart of the formulation described in Conformal-Method Initial Data.

Check your understanding

Set $\widetilde\sigma=0$, $\tau=0$, and $W=0$ on a closed manifold. What does a positive solution $\phi$ imply about the physical metric $h=\phi^4\widetilde h$?

Solution

The Lichnerowicz equation becomes the conformal-Laplacian equation $-8\widetilde\Delta\phi+\widetilde R\phi=0$. The scalar-curvature transformation then gives $R(h)=0$. Thus the conformal class must be Yamabe zero. A positive- or negative-Yamabe class cannot solve these completely vanishing CMC free data on a closed manifold.

References

• Bartnik, Robert, and Jim Isenberg. “The Constraint Equations.” In The Einstein Equations and the Large Scale Behavior of Gravitational Fields, 1–38. Birkhäuser, 2004. arXiv:gr-qc/0405092.
• Carlotto, Alessandro. “The General Relativistic Constraint Equations.” Living Reviews in Relativity 24 (2021): 2. DOI.