---
title: Eguchi–Hanson Space Overview
url: https://www.emergentmind.com/topics/eguchi-hanson-space
type: topic
---

# Eguchi–Hanson Space Overview

Eguchi–Hanson space is the unique, up to overall scale, non-compact, complete, \(4\)-dimensional, Ricci-flat, hyperkähler ALE gravitational instanton asymptotic to \(\mathbb{R}^4/\mathbb{Z}_2\) of type \(A_1\). Geometrically, it resolves the orbifold singularity of \(\mathbb{C}^2/\mathbb{Z}_2\) by replacing the singular point with an exceptional \(2\)-sphere, the bolt, and it is diffeomorphic to \(T^*S^2\), equivalently to the holomorphic line bundle \(\mathcal{O}_{\mathbb{CP}^1}(-2)\) [2604.23410].

## 1. Geometric definition, topology, and asymptotics

Topologically, Eguchi–Hanson space is the minimal smooth resolution of \(\mathbb{C}^2/\mathbb{Z}_2\). The exceptional sphere has self-intersection \(-2\), and in the standard normalization with parameter \(a>0\) its area is
\[
\operatorname{Area}(S^2_{\mathrm{bolt}})=\pi a^2.
\]
Its volume growth is Euclidean, \(\operatorname{Vol}(B_r)\sim r^4\), and the asymptotic link is \(S^3/\mathbb{Z}_2\cong \mathbb{RP}^3\) [2604.23410].

An ALE space is a connected Riemannian orbifold \((M^n,g)\) for which, outside a compact set, there is a chart to \((\mathbb{R}^n\setminus B_R(0))/\Gamma\) with \(\Gamma\subset SO(n)\) finite and acting freely on \(S^{n-1}\), such that
\[
|\nabla^k(\Phi^*\bar g-g)|\le C r^{-\tau-k}
\]
for all \(k\in \mathbb{N}_0\). For Eguchi–Hanson, \(\Gamma=\mathbb{Z}_2\). A sharp decay theorem of Kröncke–Szabó, as adapted to orbifolds, implies that any Ricci-flat ALE space with nontrivial \(\Gamma\) has optimal order \(\tau=n\); hence in dimension \(4\),
\[
g-\bar g=O(r^{-4}),\qquad \nabla^k(g-\bar g)=O(r^{-4-k}),
\]
and Eguchi–Hanson attains this order-\(4\) decay [2604.23410].

Within Kronheimer’s ADE classification of simply-connected hyperkähler ALE \(4\)-manifolds with \(\Gamma\subset SU(2)\), Eguchi–Hanson is the unique \(A_1\) space. This places it simultaneously at the intersection of complex surface resolution theory, Ricci-flat Kähler geometry, and the structure theory of ALE gravitational instantons [2604.23410].

## 2. Explicit metric and hyperkähler structure

In a cohomogeneity-one description, let \(\sigma_1,\sigma_2,\sigma_3\) be the standard left-invariant \(1\)-forms on \(S^3\cong SU(2)\), normalized by
\[
d\sigma_i=-\frac12\sum_{j,k}\varepsilon_{ijk}\,\sigma_j\wedge \sigma_k.
\]
For \(r\in[a,\infty)\), \(a>0\), the metric is
\[
ds^2=\Big(1-\frac{a^4}{r^4}\Big)^{-1}dr^2+\frac{r^2}{4}\big(\sigma_1^2+\sigma_2^2\big)+\frac{r^2}{4}\Big(1-\frac{a^4}{r^4}\Big)\sigma_3^2.
\]
It is smooth and complete on the total space of \(\mathcal{O}_{\mathbb{CP}^1}(-2)\) after the circle in the \(\sigma_3\)-direction collapses at \(r=a\). The level sets \(r=\mathrm{const}\) are homogeneous \(S^3/\mathbb{Z}_2\cong \mathbb{RP}^3\), and the collapsing orbit at \(r=a\) is the \(2\)-sphere bolt [2604.23410].

The metric is Kähler and indeed hyperkähler. It is the crepant resolution of \(\mathbb{C}^2/\mathbb{Z}_2\), and one can exhibit a Kähler form \(\omega\) and a holomorphic symplectic form \(\Omega\) coming from the hyperkähler triple. In complex coordinates adapted to the resolution, a Kähler potential is
\[
\Phi(\rho)=\sqrt{\rho^2+a^4}+a^2\log\!\Big(\frac{\rho}{\sqrt{\rho^2+a^4}+a^2}\Big),\qquad \rho=r^2,
\]
with \(g=\partial\bar\partial \Phi\). The metric is Ricci-flat, and its Weyl tensor is \((\)anti-\()\)self-dual depending on orientation [2604.23410].

Equivalent formulas occur in other normalizations. In Hopf coordinates and SU(2)-invariant form, one finds
\[
g_{\mathrm{EH},\epsilon}=\Big(1-\frac{\epsilon^4}{\xi^4}\Big)^{-1}d\xi^2+\xi^2\Big[\sigma_1^2+\sigma_2^2+\Big(1-\frac{\epsilon^4}{\xi^4}\Big)\sigma_3^2\Big],
\]
and in complex coordinates \((z_1,z_2)\) the Kähler form can be written as
\[
\omega_{\mathrm{EH},\epsilon}
=i\partial\bar\partial\!\Big[\sqrt{r^4+\epsilon^4}+\log r^2-\log\big(\epsilon^2+\sqrt{r^4+\epsilon^4}\big)\Big],
\]
which is the same family in a different normalization [2111.00652].

## 3. Linear analysis and analytical rigidity

On a Ricci-flat manifold, the Lichnerowicz Laplacian on symmetric \(2\)-tensors is
\[
\Delta_L h=\nabla^*\nabla h+2\,\mathrm{Rm}*h,
\]
or in components,
\[
(\Delta_L h)_{kl}=g^{ij}\nabla_i\nabla_j h_{kl}+2R_{kijl}h^{ij}.
\]
Modulo diffeomorphism gauge, \(\Delta_L\) is the linearization of the Ricci tensor, and its \(L^2\)-kernel describes normalizable infinitesimal Ricci-flat deformations [2604.23410].

For Eguchi–Hanson,
\[
\dim \ker_{L^2}(\Delta_L)=3.
\]
One generator is the scaling direction, and two come from deformations of the hyperkähler complex structure. By contrast,
\[
\ker_{L^2}(\Delta_L)=\{0\}
\]
on the flat orbifold \(\mathbb{R}^4/\mathbb{Z}_2\) [2604.23410].

A recent characterization theorem due to Law states: if \((M^4,g)\) is a complete Ricci-flat ALE orbifold with finitely many orbifold points, group at infinity \(\Gamma=\mathbb{Z}_2\), and
\[
\dim \ker_{L^2}(\Delta_L)\le 3,
\]
then \((M,g)\) is isometric to either Eguchi–Hanson space or the flat orbifold \(\mathbb{R}^4/\mathbb{Z}_2\) [2604.23410].

The proof combines several ingredients. First, sharp order-\(4\) ALE asymptotics and a CMC foliation of the end produce leaves \(\Sigma_\rho\approx S^3/\mathbb{Z}_2\) with mean curvature \(H=3/\rho\). Second, Killing fields on the round limit \(\mathbb{RP}^3\) are extended into the end and converted into \(L^2\)-harmonic Lie derivatives \(\mathcal{L}_Wg\); a small kernel forces some of these to vanish identically, producing global Killing fields. Third, a cohomogeneity analysis splits the argument. In the cohomogeneity-one case, Lock–Viaclovsky and Kronheimer imply the non-flat possibility is Eguchi–Hanson. In the cohomogeneity-\(\ge 2\) case, the end reduces locally to an Euclidean Schwarzschild form
\[
\tilde g=\frac{1}{V(\xi)}\,d\xi^2+V(\xi)\,dt^2+\xi^2 g_{S^2},\qquad V(\xi)=1-\frac{2\mu}{\xi},
\]
and the optimal ALE decay forces \(\mu=0\), hence flatness. The resulting picture is a rigidity statement driven by linear analysis: a sufficiently small obstruction space forces large symmetry, and large symmetry forces the metric into the Eguchi–Hanson or flat model [2604.23410].

## 4. Constructions and equivalent descriptions

Eguchi–Hanson space admits several equivalent constructions. In Calabi’s ansatz, it is the Ricci-flat Kähler metric on the canonical line bundle \(K_{\mathbb{CP}^{n-1}}\cong \mathcal{O}_{\mathbb{CP}^{n-1}}(-n)\), with the Eguchi–Hanson case corresponding to \(n=2\). In rotationally symmetric coordinates on \(\mathbb{C}^n\setminus\{0\}\), if \(u=r^2\) then Ricci-flatness reduces to
\[
f'(u)=\Big(1+\frac{a^n}{u^n}\Big)^{1/n},
\]
and for \(n=2\) this yields the Eguchi–Hanson metric on \(\mathcal{O}_{\mathbb{CP}^1}(-2)\) [2201.07295].

The same paper proves
\[
\mathcal{O}_{\mathbb{CP}^{n-1}}(-k)\cong \mathrm{Bl}_0(\mathbb{C}^n)/\mu_k,
\]
so for \(k=n\) one obtains a smooth resolution of \(\mathbb{C}^n/\mu_n\); in particular \(\mathcal{O}_{\mathbb{CP}^1}(-2)\) is the minimal resolution of \(\mathbb{C}^2/\mathbb{Z}_2\) [2201.07295]. A complementary analytic viewpoint shows that Eguchi–Hanson is the Tian–Yau Ricci-flat Kähler metric on
\[
(\mathbb{P}^1\times \mathbb{P}^1)\setminus \Delta,
\]
where \(\Delta\) is the diagonal and
\[
-K_{\mathbb{P}^1\times \mathbb{P}^1}=2[\Delta]
\]
[1301.4727].

There are also gauge-theoretic and twistor-theoretic descriptions. One recent account shows explicitly that Eguchi–Hanson is isometric to a suitable two-center Gibbons–Hawking ansatz, with harmonic function \(V=\frac{1}{R_1}+\frac{1}{R_2}\) and parameter relation \(a^2=2\alpha\) [2605.02046]. Another presents it as a hyperkähler quotient of \(\mathbb{H}\cong \mathbb{C}^2\oplus \mathbb{C}^2j\), where the distinguished \(U(1)\) connection arising from the quotient has \(L^2\)-normalizable curvature equal to the unique \(L^2\) harmonic \(2\)-form [2309.08453]. An orbit-theoretic formulation identifies \(T^*\mathbb{CP}^1\) with a complex adjoint orbit of \(SL(2,\mathbb{C})\) fibered over \(\mathbb{CP}^1\) with fibers diffeomorphic to \(\mathbb{H}^2\), and shows that the complex structure induced on each \(\mathbb{H}^2\) fiber by the hyperkähler extension differs from the natural complex structure of the unit disc [2504.19945].

## 5. Spinors, Ricci flow, and gluing theory

Spinorially, Eguchi–Hanson has a \(2\)-complex-dimensional space of parallel spinors. In the frame used by Cai and Zhang, these are constant complex spinors with two independent components, reflecting the reduction of holonomy to \(SU(2)\) [2305.18344]. A different spin-\(c\) treatment revisits the Dirac operator and shows that the untwisted Dirac operator has no \(L^2\)-normalizable zero modes, whereas twisting by a \(U(1)\) connection with \(L^2\)-normalizable curvature produces explicit zero modes; for integer flux \(\ell\), the total number of zero modes is
\[
\frac{\ell(\ell+1)}{2}
\]
[2309.08453].

Eguchi–Hanson also appears as a singularity model in Ricci flow. For a class of asymptotically cylindrical \(U(2)\)-invariant initial metrics on \(T^*S^2\), a finite-time Type II singularity modeled on Eguchi–Hanson develops. In the \(k=2\) case, the only blow-up limits are: the stationary Eguchi–Hanson space, the flat orbifold \(\mathbb{R}^4/\mathbb{Z}_2\), the \(4\)-dimensional Bryant steady soliton quotiented by \(\mathbb{Z}_2\), and the shrinking cylinder \(\mathbb{R}\times \mathbb{RP}^3\) [1903.09936].

In gluing problems, Eguchi–Hanson is the local model replacing \(A_1\) orbifold points. A recent construction of a Ricci-flat metric on the Kummer \(K3\) surface follows Donaldson’s gluing strategy and uses \(16\) Eguchi–Hanson pieces to desingularize \(T^4/\{\pm 1\}\). That work also compares the SU(2)-invariant, Kähler-potential, and Gibbons–Hawking descriptions of the space in detail [2605.02046]. In compact \(G_2\) geometry, families of Eguchi–Hanson spaces parameterized by a nonvanishing closed and coclosed \(1\)-form on an associative \(3\)-fold are glued into orbifold singularities of type \(\mathbb{R}^3\times (\mathbb{R}^4/\{\pm 1\})\) to produce compact torsion-free \(G_2\)-manifolds [1707.09325].

## 6. Higher-dimensional analogs and related geometries

Eguchi–Hanson is the \(m=2\) member of Calabi’s higher-dimensional Ricci-flat Kähler ALE metrics on
\[
\mathcal{O}_{\mathbb{CP}^{m-1}}(-m).
\]
In the Calabi ansatz, with Hopf fibration \(\pi:S^{2m-1}\to \mathbb{CP}^{m-1}\), one writes
\[
\omega=u(r)\,r\,dr\wedge \sigma+v(r)\,\pi^*\omega_{\mathrm{FS}},
\]
and the Kähler and Ricci-flat equations reduce to
\[
v'(r)=2ru(r),\qquad u(r)v(r)^{m-1}=2.
\]
The general solution is
\[
u(r)=2\big(2m(r^2+\epsilon)\big)^{-(m-1)/m},\qquad
v(r)=\big(2m(r^2+\epsilon)\big)^{1/m}.
\]
For \(\epsilon>0\), this closes smoothly at \(r=0\); for \(m=2\) it is precisely Eguchi–Hanson, while for \(m\ge 3\) it is not hyperkähler [2604.23410].

Law proves a higher-dimensional analog of the \(4\)-dimensional rigidity theorem: if \((M^{2m},g,J)\) is a complete Ricci-flat Kähler ALE space with \(\Gamma=\mathbb{Z}_m\), singular set of real codimension at least \(4\), and
\[
\dim \ker_{L^2}(\Delta_L)\le 2m-3\quad (m\neq 4),\qquad
\dim \ker_{L^2}(\Delta_L)\le 2\quad (m=4),
\]
then \((M,g,J)\) is biholomorphically isometric to either Calabi’s metric on \(\mathcal{O}_{\mathbb{CP}^{m-1}}(-m)\) or the flat orbifold \(\mathbb{C}^m/\mathbb{Z}_m\). For \(m\ge 3\), Morteza–Viaclovsky showed that the Calabi spaces have
\[
\dim \ker_{L^2}(\Delta_L)=1,
\]
namely the scaling mode [2604.23410].

Eguchi–Hanson also arises as a geometric limit. On Calabi–Hirzebruch manifolds \(\mathcal{F}_{n,k}\), a family of Kähler–Einstein edge metrics \(\eta_{\beta_1}\) with cone angles along the zero and infinity sections converges, as \(\beta_1\uparrow n/k\), to a pointed Gromov–Hausdorff limit on \(-kH_{\mathbb{CP}^{n-1}}\) carrying the Ricci-flat metric \(\omega_{\mathrm{eh},n,k}\). In the special case \(n=k=2\), this limit is exactly the Eguchi–Hanson metric with \(\epsilon=1\) [2111.00652].

Several distinct Lorentzian or higher-curvature analogs inherit part of the Eguchi–Hanson pattern without reproducing the classical geometry. The Eguchi–Hanson–AdS\(_5\) family consists of static, geodesically complete, asymptotically locally AdS\(_5\) solitons with smooth \(S^2\) bolt, boundary \(L(p,1)\), \(p\ge 3\), and negative mass
\[
M=-\frac{\pi a^4}{8\ell^2 p},
\]
with the \(4\)-dimensional Eguchi–Hanson geometry appearing as the spatial section or as a formal \(p\to 2\) limit [2212.12685]. In higher-curvature and \(F(R)\) gravity, one finds “Eguchi–Hanson-type” metrics built from circle fibrations over Kähler–Einstein bases or from deformed cohomogeneity-one ansätze; in these families the classical Ricci-flat, self-dual, hyperkähler structure is generally lost and is recovered only in the Einstein or GR limit [2207.04014] [1210.3629] [2509.08033].

These developments suggest a precise distinction. The classical Eguchi–Hanson space is the rigid \(A_1\) Ricci-flat hyperkähler ALE manifold; many later constructions retain its bolt structure, asymptotic quotient, or cohomogeneity-one form, but only the classical metric on \(\mathcal{O}_{\mathbb{CP}^1}(-2)\) carries the full package of properties summarized above.

Source: https://www.emergentmind.com/topics/eguchi-hanson-space