Einstein-Scalar Field Conformal Constraints

• Einstein–Scalar Field Conformal Constraint Equations are the reformulated initial-value constraints in Einstein gravity coupled with a scalar field, using a conformal factor and vector potential.
• They are derived via the conformal method, employing techniques such as the Einstein–Lichnerowicz equation, modified Yamabe geometry, and drift formulations to manage near-CMC and far-from-CMC regimes.
• The framework connects with Jordan/Einstein frame equivalence and accommodates explicit radial models, offering insights into stability, mass analysis, and applications in asymptotically flat and hyperbolic spaces.

Einstein–scalar field conformal constraint equations are the conformally reformulated initial-value constraints for Einstein gravity coupled to a scalar field, together with closely related conformal-frame and conformal-compactification formulations in which scalar fields interact with the geometry through conformal rescalings, conformally invariant wave equations, or canonically selected conformal representatives. In the standard constraint-theoretic setting, the unknowns are a conformal factor and a vector potential, while the free data include a background metric, mean curvature, scalar field, scalar momentum, and transverse–traceless data; the resulting system consists of an Einstein–Lichnerowicz equation coupled to a vector equation (Premoselli, 2013). More broadly, the topic also encompasses Jordan/Einstein frame equivalence for scalar–tensor theories (Morris, 2014), drift-based reformulations of the conformal method (Holst et al., 2017), conformal compactification with scalar radiation at null infinity (Mädler et al., 26 Apr 2025), and conformal geometric constructions such as Einstein’s preferred metric and weight‑0 scalar invariants (Sánchez-Rodríguez, 2017).

1. Constraint system and conformal reduction

On an Einstein–scalar field initial data set $(M,g,K,\phi,\pi,V)$, the Hamiltonian and momentum constraints are written in the data as

$\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$

$$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$

Equivalently, in the notation used for a spacelike hypersurface $M^n\subset \tilde M^{n+1}$, one has

$$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$

with $\psi=\tilde\psi|_M$ and $\tilde\pi=(\partial_t\tilde\psi)|_M$ (Premoselli, 2013).

The conformal method rewrites these constraints by introducing a background metric $g$, a positive conformal factor $\varphi$, a prescribed mean curvature $\tau$, a symmetric traceless divergence-free tensor $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$0 or $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$1, a scalar field $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$2, and a scalar momentum variable $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$3. The physical variables are reconstructed through

$\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$4

in one formulation (Premoselli, 2013), and equivalently

$\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$5

with $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$6 in another (Gicquaud et al., 2015).

Substitution of this ansatz yields a determined elliptic system. In the notation of Premoselli, the Einstein–Lichnerowicz equation and the momentum constraint are

$\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$7

$\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$8

where $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$9,

$$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$0

$$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$1

The same structure appears in Maxwell’s notation as

$$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$2

$$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$3

with

$$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$4

(Gicquaud et al., 2015).

These equations encode the scalar field in three distinct ways. The background scalar field gradient modifies the conformal Laplacian through $$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$5 or $$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$6; the scalar potential modifies the positive-power term through $$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$7 or $$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$8; and the scalar momentum enters both the negative-power term and the vector equation through $$\operatorname{div}_{g}\big( K - (\operatorname{tr}_{g}K)\,g \big) = \pi\, d\phi.$$9 (Premoselli, 2013).

2. Free data, modified Yamabe geometry, and drift formulations

The conformal method separates the problem into free data and determined data. One standard choice of free data is $M^n\subset \tilde M^{n+1}$0, with unknowns $M^n\subset \tilde M^{n+1}$1 (Premoselli, 2013). Another uses seed data $M^n\subset \tilde M^{n+1}$2, again with $M^n\subset \tilde M^{n+1}$3 as unknowns (Gicquaud et al., 2015). The central geometric operator is not the usual conformal Laplacian but the modified conformal Laplacian

$M^n\subset \tilde M^{n+1}$4

and the basic positivity assumption is coercivity: $M^n\subset \tilde M^{n+1}$5 for some $M^n\subset \tilde M^{n+1}$6 (Gicquaud et al., 2015). In the terminology used there, this is the scalar-field analogue of positivity of the Yamabe invariant.

A different reformulation is the drift method, which replaces direct prescription of the mean curvature by a decomposition into a constant volumetric component and a divergence term: $M^n\subset \tilde M^{n+1}$7 Here $M^n\subset \tilde M^{n+1}$8 is the volumetric momentum, $M^n\subset \tilde M^{n+1}$9 is a vector field representing a drift, and $$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$0 is a conformal Killing field. The associated drift space is

$$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$1

and the vacuum momentum constraint becomes an equation between conformal drift and volumetric drift: $$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$2 (Holst et al., 2017).

This reformulation is designed to address the conformal Killing field obstruction. If $$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$3 is a conformal Killing field, then a solution of the momentum constraint implies the CKF compatibility condition

$$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$4

or equivalently, in conformal variables,

$$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$5

The drift method absorbs this obstruction by solving simultaneously for $$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$6 and the conformal factor. In the presence of matter, the compatibility condition becomes

$$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$7

for all conformal Killing fields $$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$8 (Holst et al., 2017). This is directly relevant to scalar fields because the matter momentum density in the conformal framework is the scalar contribution $$R(g) + (\operatorname{tr}_g K)^2 - |K|_g^2 = \tilde\pi^2 + |\nabla\psi|_g^2 + 2V(\psi), \qquad \partial_i(\operatorname{tr}_g K) - K_{i,j}^j = \tilde\pi\,\partial_i\psi,$$9, and the data explicitly note that the CMC condition requires orthogonality of $\psi=\tilde\psi|_M$0 to all conformal Killing fields.

The drift paper also states that scalar fields do not satisfy the energy scaling condition used there for “scaling sources” (Holst et al., 2017). This indicates that the drift geometry extends naturally to the Einstein–scalar field system, while the scalar-specific analysis must be carried out separately for the Hamiltonian equation.

3. Existence theory, near-CMC and far-from-CMC regimes

The best-developed existence theory in the supplied material concerns compact manifolds in the positive case and far-from-CMC solutions with small TT-tensor. Premoselli studies the case

$\psi=\tilde\psi|_M$1

with strict positivity somewhere, on compact manifolds of positive Yamabe type and with $\psi=\tilde\psi|_M$2 coercive. The main theorem asserts that if

$\psi=\tilde\psi|_M$3

strictly somewhere, and

$\psi=\tilde\psi|_M$4

with $\psi=\tilde\psi|_M$5, then the conformal Einstein–scalar constraint system admits a smooth solution $\psi=\tilde\psi|_M$6 (Premoselli, 2013). This covers both the vacuum case with positive cosmological constant and the massive Klein–Gordon setting $\psi=\tilde\psi|_M$7.

A complementary far-from-CMC result is obtained by Maxwell for compact manifolds with positive modified Yamabe invariant. Under coercivity of $\psi=\tilde\psi|_M$8, absence of conformal Killing fields, and a smallness condition

$\psi=\tilde\psi|_M$9

being sufficiently small, the Einstein–scalar field conformal constraint equations admit a solution $\tilde\pi=(\partial_t\tilde\psi)|_M$0 without any smallness restriction on $\tilde\pi=(\partial_t\tilde\psi)|_M$1 (Gicquaud et al., 2015). The proof combines variational analysis of the Lichnerowicz equation with a Schauder fixed-point argument for the coupled system. A key intermediate result is that, for sufficiently small

$\tilde\pi=(\partial_t\tilde\psi)|_M$2

the Lichnerowicz equation admits a stable minimizer for the functional

$\tilde\pi=(\partial_t\tilde\psi)|_M$3

(Gicquaud et al., 2015).

These two analyses display two distinct mechanisms. In the positive case, the sign condition on $\tilde\pi=(\partial_t\tilde\psi)|_M$4 and coercivity of $\tilde\pi=(\partial_t\tilde\psi)|_M$5 control the high-power and linear terms (Premoselli, 2013). In the small-TT far-from-CMC setting, control arises from a small-data regime for $\tilde\pi=(\partial_t\tilde\psi)|_M$6, even when $\tilde\pi=(\partial_t\tilde\psi)|_M$7 is arbitrary (Gicquaud et al., 2015).

The material also records a nonexistence mechanism. In the positive case, if $\tilde\pi=(\partial_t\tilde\psi)|_M$8 everywhere and $\tilde\pi=(\partial_t\tilde\psi)|_M$9 is too large in $g$0, then the system has no solution (Premoselli, 2013). This shows that scalar momentum is not merely a perturbative parameter; it can force obstruction.

4. Conformal frames, Jordan–Einstein equivalence, and multiple scalar fields

A distinct but related line of work concerns conformal frame transformations. In four-dimensional scalar–tensor theory, Morris considers Jordan-frame and Einstein-frame actions connected by

$g$1

The corresponding equations of motion are shown to be mathematically equivalent: when Jordan-frame equations are rewritten in Einstein-frame variables, they agree with the Einstein-frame equations provided two consistency conditions hold, and these conditions are identities following from the transformation of the stress-energy tensor and matter coupling (Morris, 2014). In particular,

$g$2

A solution set obtained in one conformal frame therefore maps reliably to a solution set in the other (Morris, 2014).

Although this analysis is carried out at the level of covariant equations of motion rather than 3+1 constraints, it has a direct implication for constraint equations: if the full Einstein and scalar equations are equivalent in Jordan and Einstein frames, then their Hamiltonian and momentum projections are equivalent as well, provided the 3+1 data are transformed consistently. The text explicitly states that this implies equivalence of the associated constraint equations in both frames for Einstein–scalar systems (Morris, 2014).

For multiple scalar fields, the Einstein-frame kinetic sector becomes a nonlinear sigma model with field-space metric

$g$3

after the conformal transformation

$g$4

The kinetic sector is “quasi-canonical” precisely when this field-space metric is conformally flat (Tang et al., 2021). For $g$5 scalar fields, this requires vanishing of the field-space Weyl tensor; the paper gives explicit families of non-minimal couplings $g$6 for which this occurs, including

$g$7

(Tang et al., 2021). It also states that in modified gravity models $g$8, including Starobinsky-type models, the Einstein-frame field space is always conformally flat in a suitable extended sense (Tang et al., 2021).

This suggests that in conformal constraint problems with several scalars, the tractability of the Einstein-frame system depends not only on the conformal geometry of the physical metric but also on the intrinsic geometry of scalar target space. The supplied data present this as a geometric condition on $g$9, not as a separate constraint theorem.

5. Conformal geometry beyond the standard conformal method

The supplied papers describe several broader conformal formulations in which scalar fields enter through conformally regularized equations rather than only through the standard Lichnerowicz–York reduction.

One such formulation is conformal compactification in spherical symmetry with a massless scalar field. With physical and unphysical fields related by

$\varphi$0

and an affine-null coordinate compactification

$\varphi$1

the Einstein–scalar equations reduce to a regular hierarchical system for the unphysical fields $\varphi$2. After introducing auxiliary fields $\varphi$3 and $\varphi$4, the conformal system takes the form

$\varphi$5

$\varphi$6

$\varphi$7

$\varphi$8

(Mädler et al., 26 Apr 2025). The paper shows that this conformal system is identical to the system obtained from a compactified coordinate in physical spacetime plus suitable regularized fields. At null infinity, it recovers the Bondi mass loss law for a massless scalar field,

$\varphi$9

with $\tau$0 the scalar monopole coefficient (Mädler et al., 26 Apr 2025).

A different conformal regularization appears in the tracefree matter setting for the conformally invariant scalar field. There, the unphysical scalar $\tau$1 satisfies the regular equation

$\tau$2

and the unphysical energy–momentum tensor is

$\tau$3

To control derivatives of $\tau$4, auxiliary fields

$\tau$5

are added, and both the evolution equations and the subsidiary system are closed, allowing propagation of the conformal constraints (Carranza et al., 2019). This provides a conformal Einstein–scalar framework in which the scalar is tracefree matter rather than minimally coupled matter.

The conformal-geometric paper “Scalar conformal invariants of weight zero” adds yet another direction. For a generic conformal structure $\tau$6 in dimension $\tau$7, with

$\tau$8

there is a preferred metric

$\tau$9

that is independent of the representative $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$00, and satisfies $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$01 (Sánchez-Rodríguez, 2017). The Ricci scalar of this preferred metric,

$\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$02

is a conformal scalar curvature of weight $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$03 (Sánchez-Rodríguez, 2017). Einstein’s 1921 proposal of an additional scalar equation

$\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$04

is presented there as a scalar differential condition naturally attached to the conformal class (Sánchez-Rodríguez, 2017). This does not yield a standard initial-data constraint system by itself, but it provides a geometric mechanism for introducing additional scalar conformal conditions. A plausible implication is that such weight‑0 scalar invariants can be combined with Einstein–scalar constraint frameworks when one seeks conformally invariant auxiliary conditions.

6. Spherical symmetry and explicit radial models

A recent radial analysis gives the most explicit treatment in the supplied material. On harmonic manifolds and for radial seed data, the Einstein–scalar field conformal constraint system reduces to a single nonlinear scalar equation and is “completely resolved in the standard cases” (Castillon et al., 23 Feb 2026). For space forms $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$05, $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$06, and $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$07, the vector equation can be solved explicitly using a radial primitive $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$08, and the full problem reduces to one equation for the conformal factor $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$09.

The results exhibit a sharp compact/noncompact contrast. On the sphere, the paper proves nonexistence of radial solutions in the near-CMC regime for monotone $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$10, and nonexistence of radial vacuum TT-free solutions for any radial $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$11 (Castillon et al., 23 Feb 2026). It also proves instability when $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$12 changes sign: there exist sequences $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$13 with $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$14 in $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$15 while $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$16 (Castillon et al., 23 Feb 2026). By contrast, on Euclidean and hyperbolic manifolds, the radial equations are always solvable in the standard radial classes considered there, and the expected compactness or stability properties hold (Castillon et al., 23 Feb 2026).

The same paper also analyzes mass. In the asymptotically hyperbolic case, if the conformal factor decays like $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$17, then the asymptotically hyperbolic mass is zero for $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$18, can have arbitrary sign at the critical rate $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$19, and can be infinite of either sign for $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$20 (Castillon et al., 23 Feb 2026). In the asymptotically flat case, for radial $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$21, the ADM mass is $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$22 for $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$23, finite and negative at the critical decay $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$24, and zero for $\Scal_{g} - |K|_{g}^2 + (\operatorname{tr}_{g}K)^2 = \pi^2 + |d\phi|_{g}^2 + 2 V(\phi),$25 (Castillon et al., 23 Feb 2026). These statements are presented there as evidence that the usual decay thresholds in positive mass results are sharp with respect to the decay of the extrinsic curvature.

The radial theory therefore functions as a diagnostic model for the conformal method itself. It confirms that compact manifolds with conformal Killing fields can display pathologies absent from noncompact geometries, while asymptotically flat and asymptotically hyperbolic settings remain comparatively well behaved (Castillon et al., 23 Feb 2026). This supports the view, stated explicitly in the abstract, that despite drawbacks on compact manifolds, the conformal method remains a promising tool for parametrizing Einstein–scalar field initial data on asymptotically flat and hyperbolic manifolds in arbitrary mean-curvature regimes (Castillon et al., 23 Feb 2026).