Moduli Space of Einstein Metrics

• Moduli space of Einstein metrics is defined as the quotient of all Einstein metrics on a compact manifold by diffeomorphisms (and rescaling), capturing key geometric invariants.
• Its local analytic structure is governed by the Lichnerowicz Laplacian on transverse-traceless tensors, which controls infinitesimal deformations and stability conditions.
• Global behavior features isolated rigid points, smooth orbifold moduli, and diverse compactifications via Gromov–Hausdorff and algebraic techniques in varied geometric settings.

The moduli space of Einstein metrics on a compact manifold $M$ is obtained by quotienting Einstein metrics by diffeomorphisms and, depending on the normalization, by homothetic rescaling or by fixing the volume. In the unit-volume normalization one writes

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$

where $\mathcal M$ is the Fréchet manifold of all Riemannian metrics on $M$ and $\mathcal M_1\subset\mathcal M$ is the codimension-$1$ submanifold of unit-volume metrics. The local structure of $\mathcal E(M)$ is governed by the Lichnerowicz Laplacian on transverse-traceless tensors, while its global behavior ranges from smooth orbifold moduli in special-holonomy settings to isolated rigid points, connected components cut out by curvature positivity in dimension four, and Gromov–Hausdorff/algebro-geometric compactifications in the Kähler–Einstein Fano case (Schwahn et al., 24 Jul 2025, LeBrun, 2020, Li et al., 2015).

1. Definition, gauge fixing, and local analytic structure

For a compact smooth $n$-manifold $M$, $n\ge 3$, the Einstein equation is

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$0

An equivalent normalization, used frequently in dimension four, is

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$1

with the quotient topology induced by $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$2-convergence or by Gromov–Hausdorff distance on unit-volume metrics. The action of $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$3 accounts for gauge, while the passage to $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$4 removes homothetic scalings (Schwahn et al., 24 Jul 2025, LeBrun, 2014).

At an Einstein metric $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$5 with $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$6, the relevant linear operator is the Lichnerowicz Laplacian on symmetric $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$7-tensors,

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$8

The transverse-traceless sector

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$9

is the essential deformation space after gauge fixing. In that sector the linearization of the Einstein equation becomes

$\mathcal M$0

so the Zariski and analytic tangent space to the pre-moduli inside a slice is

$\mathcal M$1

This identifies infinitesimal Einstein deformations with eigentensors of $\mathcal M$2 at eigenvalue $\mathcal M$3 (Schwahn et al., 24 Jul 2025).

The nonlinear problem is encoded by the Einstein operator

$\mathcal M$4

Its linearization at an Einstein metric is not onto, because its cokernel is isomorphic to its kernel. Accordingly, the strict implicit function theorem does not directly apply. Instead one obtains a finite-dimensional Kuranishi model: there is a real-analytic submanifold $\mathcal M$5 with

$\mathcal M$6

and the true pre-moduli is a real-analytic subset of $\mathcal M$7. If $\mathcal M$8, then $\mathcal M$9 is rigid; more generally, $M$0 is locally a real-analytic space and, when $M$1 acts trivially on it, locally an orbifold (Schwahn et al., 24 Jul 2025).

2. Infinitesimal deformations, second-order obstructions, and stability

The same operator that governs deformations also governs variational stability. The Einstein–Hilbert action

$M$2

has as its critical points on $M$3 exactly the Einstein metrics. On transverse-traceless tensors one has

$M$4

Thus the spectrum of $M$5 on $M$6 determines linear stability. “Strictly stable” means

$M$7

“semistable” means no eigenvalue below $M$8, and instability is signalled by an eigenvector with $M$9. In particular, $\mathcal M_1\subset\mathcal M$0 is a local maximum of $\mathcal M_1\subset\mathcal M$1 if and only if it is strictly stable in this sense (Schwahn et al., 24 Jul 2025).

A persistent misconception is that a nonzero space of infinitesimal Einstein deformations automatically yields a smooth local family of Einstein metrics. The obstruction theory shows that this is false. For the bi-invariant Einstein metric on $\mathcal M_1\subset\mathcal M$2, one has

$\mathcal M_1\subset\mathcal M$3

with

$\mathcal M_1\subset\mathcal M$4

but every $\mathcal M_1\subset\mathcal M$5 is obstructed at second order, and there are no other Einstein metrics in a neighborhood of $\mathcal M_1\subset\mathcal M$6. Equivalently, the metric is isolated in its Einstein moduli space despite the presence of essential infinitesimal Einstein deformations (Batat et al., 2021).

The obstruction is packaged by a quadratic Kuranishi map

$\mathcal M_1\subset\mathcal M$7

where $\mathcal M_1\subset\mathcal M$8 on TT-tensors and $\mathcal M_1\subset\mathcal M$9 is the obstruction space. In the $1$0 case, $1$1, $1$2, and $1$3. The local moduli therefore has dimension zero even though the Zariski tangent is large. This is the same formal phenomenon as Koiso’s example on $1$4, but in a non-Kähler, non-product setting (Batat et al., 2021).

3. Rigidity, unobstructedness, and model examples

The round metric on $1$5 is the basic rigid example. Let $1$6 be the unit-radius round sphere, so that $1$7. Any one-parameter family of Einstein metrics $1$8 on $1$9 with $\mathcal E(M)$0 and $\mathcal E(M)$1 must be trivial up to diffeomorphism and homothety. Equivalently, in a smooth neighborhood of $\mathcal E(M)$2 the only Einstein metrics are of the form

$\mathcal E(M)$3

with $\mathcal E(M)$4 and $\mathcal E(M)$5. In the volume-unfixed normalization the local moduli is one-dimensional, generated by scaling; after quotienting by homothety, the round sphere is isolated (Ho et al., 2023).

The rigidity proof is spectral. On $\mathcal E(M)$6, in de Donder gauge, the linearized Einstein equation on TT-tensors reduces to

$\mathcal E(M)$7

Using the $\mathcal E(M)$8-harmonic analysis of TT-eigentensors, the Lichnerowicz eigenvalues are

$\mathcal E(M)$9

so $n$0 has strictly positive spectrum on TT-tensors. There is therefore no TT zero-mode; the only infinitesimal Einstein deformations are constant rescalings and pure gauge (Ho et al., 2023).

At the opposite extreme lie unobstructed moduli spaces. For Calabi–Yau manifolds, where $n$1, the tangent space to moduli is

$n$2

of dimension

$n$3

All these infinitesimal deformations are unobstructed by Bogomolov–Tian–Todorov, and the moduli is a smooth orbifold of the expected dimension. For $n$4- and $n$5-holonomy metrics, again $n$6, the moduli has dimension $n$7 or $n$8, and all deformations are integrable, so the moduli is a smooth manifold. These examples show that the Einstein moduli space can be genuinely finite-dimensional and smooth when the holonomy enforces strong integrability properties (Schwahn et al., 24 Jul 2025).

Positive Einstein examples exhibit both rigidity and stability phenomena. The Fubini–Study metric on $n$9 is Einstein with $M$0, and

$M$1

on $M$2; it is therefore strictly stable under the Einstein–Hilbert functional. This contrasts with the round sphere, which is locally unique up to scaling but is not a strict maximum of $M$3 on all of $M$4 (Schwahn et al., 24 Jul 2025).

4. Four-dimensional components: Del Pezzo manifolds, Weyl positivity, and connectedness

In dimension four, the global topology of Einstein moduli spaces can be described under explicit curvature positivity hypotheses. Let $M$5 be the underlying smooth oriented $M$6-manifold of a Del Pezzo surface. Since $M$7, for any $M$8 there is a unique, up to nonzero scale, harmonic self-dual $M$9-form $n\ge 3$0. One defines the open “positive symplectic” region

$n\ge 3$1

The inequality forces $n\ge 3$2 to be everywhere nonzero and hence symplectic, and it shows that the conformal class is of positive symplectic type. If $n\ge 3$3 is compact, oriented, Einstein, $n\ge 3$4, and $n\ge 3$5 on all of $n\ge 3$6, then $n\ge 3$7 is diffeomorphic to one of the ten Del Pezzo $n\ge 3$8-manifolds

$n\ge 3$9

and, up to an overall constant rescaling, $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$00 is exactly one of the known positive-scalar-curvature Einstein metrics: a Kähler–Einstein metric when the complex structure admits one, the Page metric on $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$01, or the Chen–LeBrun–Weber metric on $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$02. Conversely, each of these metrics satisfies $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$03 everywhere (LeBrun, 2014).

The proof combines conformal geometry and almost-Kähler geometry. One rescales $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$04 so that $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$05 is almost-Kähler with $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$06. Since $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$07 is Einstein, $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$08, and this becomes a Weitzenböck-type equation for $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$09. Paired with algebraic identities for an almost-Kähler metric and integrated by parts, the argument shows that either $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$10 or $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$11 is Kähler; the positivity hypothesis forces the Kähler alternative. The classification of conformally Kähler Einstein $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$12-manifolds then identifies the metric (LeBrun, 2014).

The corresponding moduli component is explicit. If

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$13

then $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$14 is connected, and if $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$15, equivalently $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$16 blow-ups of $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$17, then $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$18 is a single point. The connectedness comes from the connectedness of the moduli of Del Pezzo complex structures; for $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$19 this moduli is a single point because the blow-up points can be moved by $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$20 into standard position (LeBrun, 2014).

A broader four-dimensional classification arises from the stronger condition $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$21. Up to oriented diffeomorphism there are exactly fifteen compact $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$22-manifolds admitting an Einstein metric with $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$23 everywhere. Besides the ten simply connected Del Pezzo manifolds, there are five non-simply-connected quotients: if $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$24 is the antipodal involution and $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$25 the “mirror” involution on $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$26, define

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$27

and then $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$28 for $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$29. On each of these fifteen diffeotypes, the Einstein metrics satisfying $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$30 fill out exactly one connected component of the full Einstein-metric moduli space. In the simply connected cases, the real dimensions are $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$31 for $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$32, while $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$33 and $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$34 give a single point; among the non-simply-connected cases, $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$35 has real moduli dimension $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$36 and $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$37 has real moduli dimension $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$38 (LeBrun, 2020).

5. Kähler–Einstein Fano moduli, Gromov–Hausdorff compactification, and the CM line bundle

For smooth Del Pezzo surfaces of degree $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$39, the Kähler–Einstein moduli

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$40

admits both a metric compactification and an algebro-geometric one. The Gromov–Hausdorff compactification $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$41 is compact Hausdorff, and in degrees $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$42 there is a homeomorphism

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$43

to an explicit algebraic compactification parametrizing $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$44-Gorenstein smoothable log Del Pezzo surfaces. In each degree the algebraic compactification is described by GIT on an anticanonical embedding or by a global K-moduli construction. The complex dimensions are

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$45

This realizes, in dimension two, the identification of the Gromov–Hausdorff compactification with the natural algebraic compactification (Odaka et al., 2012).

The broader smoothable Fano setting is formulated in terms of K-polystable $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$46-Fano varieties and $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$47-Gorenstein smoothings. Fixing the anticanonical Hilbert polynomial $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$48, the differential-geometric KE moduli

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$49

is pre-compact and Hausdorff in the Gromov–Hausdorff topology, while the algebraic side is

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$50

The Q-smoothable condition ensures that every boundary point under Gromov–Hausdorff limits corresponds to a singular $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$51-Fano admitting a weak Kähler–Einstein metric, so the Gromov–Hausdorff compactification of $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$52 is identified with the algebraic compactification $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$53 (Spotti, 2017).

The CM line bundle supplies the polarization on moduli. Over the proper moduli space of smoothable Kähler–Einstein Fano varieties, Li–Wang–Xu prove that the CM line bundle descends to a $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$54-line bundle and carries a canonically defined continuous Hermitian metric whose Chern curvature current equals the Weil–Petersson current on all of the moduli space. The CM line bundle is nef and big on the compactified moduli space, and its restriction to the smooth locus is ample. Consequently, for $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$55, the linear system $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$56 defines a birational embedding of the moduli space into projective space; in particular, the moduli space of smooth Kähler–Einstein Fano manifolds is quasi-projective. The proof uses Deligne pairings, Bergman/Fubini–Study metrics, partial $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$57-estimates, pluripotential theory, and descent from Luna-slice charts on the Hilbert scheme (Li et al., 2015).

These results show that, in the Fano case, Einstein moduli can be studied simultaneously as a differential-geometric Gromov–Hausdorff space, as a real-analytic quotient near a fixed metric, and as an algebraic moduli space polarized by the CM line bundle. This suggests a higher-dimensional analogue of the Deligne–Mumford picture for curves, but with K-stability and weak Kähler–Einstein metrics replacing stable curves (Odaka et al., 2012, Spotti, 2017).

6. Homogeneous, Sasaki–Einstein, and noncompact variants

Homogeneous Einstein moduli admit an algebraic compactification by a Newton polytope. Let $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$58 be a compact simply connected homogeneous space whose isotropy representation is multiplicity-free. The space $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$59 of invariant unit-volume metrics is identified by a moment map with the interior of a compact convex polytope $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$60,

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$61

The Einstein equation becomes a system of Laurent-polynomial equations whose Newton polytope is again $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$62. Boundary points of $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$63 correspond to contracted homogeneous geometries, and any Einstein solution on the boundary is Ricci-flat, hence locally Euclidean, by the Alekseevsky–Kimel’fel'd theorem. The subset $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$64 of flat limits is a union of faces of the standard simplex, with a natural simplicial structure given by quasi-toral subalgebras. Using this compactification, one gets an algebraic proof that the set $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$65 of invariant Einstein metrics is bounded, hence compact (Graev, 2012).

Sasaki–Einstein moduli provide a related but not identical picture. For a quasi-smooth Fano hypersurface $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$66 of degree $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$67, Theorem 2.2.1 of Boyer–Galicki–Nakamaye gives

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$68

and when $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$69 admits a Kähler–Einstein metric this yields a real $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$70-times-that-many-parameters family of orbifold Kähler–Einstein metrics. Pulling back to the link $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$71 gives the local moduli of quasi-regular Sasaki–Einstein structures. For the rational homology $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$72-sphere links arising from invertible polynomials in the Johnson–Kollár list, all cycle-type links have no nontrivial local Sasaki–Einstein deformations, whereas chain–cycle links have

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$73

and therefore real dimension twice that (Valle et al., 24 Mar 2025).

Noncompact Einstein moduli can be genuinely one-dimensional after quotienting by homothety. For each $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$74, the $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$75-invariant Einstein metrics on the interior of $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$76 obtained from Calabi’s ansatz and Derdziński duality depend only on the scale-invariant parameter

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$77

so the quotient by overall scale is a real $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$78-dimensional moduli line

$\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$79

The distinguished values $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$80 and $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$81 divide the line into three regions corresponding to Poincaré–Einstein fillings, cone-angle Einstein metrics compactifying on the Hirzebruch surface $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$82, and incomplete Einstein metrics on $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$83. The moduli line is connected, and at $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$84 the family meets scalar-flat ALE or ALF limits such as Eguchi–Hanson for $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$85 and Taub–bolt for $\mathcal E(M)=\{\,g\in\mathcal M_1\mid \Ric_g=\lambda\,g\text{ for some }\lambda\}/\Diff(M),$86 (Oliveira et al., 2023).

Taken together, these variants show that “moduli space of Einstein metrics” does not refer to a single universal geometric behavior. On compact manifolds it is locally a real-analytic quotient and may be rigid, obstructed, orbifold-smooth, or globally compactified by algebro-geometric or polyhedral constructions; in Sasaki–Einstein and noncompact settings, closely related deformation theories yield explicit local dimensions or connected one-parameter families (Schwahn et al., 24 Jul 2025, Graev, 2012)