---
abstract: 'We show that there exists a universal positive constant $\varepsilon_0 > 0$ with the following property: Let $g$ be a positive Einstein metric on $S^4$. If the Yamabe constant of the conformal class $[g]$ satisfies $$Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}])  - \varepsilon_0\,,$$ where $g_{\mathbb S}$ denotes the standard round metric on $S^4$, then, up to rescaling, $g$ is isometric to $g_{\mathbb S}$. This is an extension of Gursky’s gap theorem for positive Einstein metrics on the four-sphere.'
address:
- 'Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan'
- 'Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan'
- 'Mathematics Department, Indian Institute of Science, 560012 Bangalore, India'
author:
- 'Kazuo Akutagawa${}^*$'
- 'Hisaaki Endo${}^{**}$'
- Harish Seshadri
date: 'January, 2018;  February, 2018 (revised version).'
title: |
    A gap theorem for positive Einstein metrics\
    on the four-sphere
---

Introduction and main results
=============================

A smooth Riemannian metric $g$ is said to be [*Einstein*]{} if its Ricci tensor ${\rm Ric}_g$ is a constant multiple $\lambda$ of $g$: $${\rm Ric}_g = \lambda\,g\,.$$ When such a metric exists, it is natural to ask whether it is unique. However in dimension $n \geq 5$, there exist many examples of closed $n$-manifolds each of which has infinitely many non-homothetic Einstein metrics (cf.[@Besse]). In fact, there exists infinitely many non-homothetic Einstein metrics of positive sacalar curvature ([*positive Einstein*]{} for brevity) on $S^n$ when $5 \le n \le 9$ [@Bohm] (cf.[@Jensen], [@B-K]). There are no non-existence or uniqueness results known when $n \geq 5$.

When $n = 4$, there are necessary topological conditions for a closed $4$-manifold $M$ to admit an Einstein metric [@Thorpe], [@Hitchin-1], [@LeBrun-2]. Uniqueness is known in some special cases: when $M$ is a smooth compact quotient of real hyperbolic $4$-space ([resp.]{} complex-hyperbolic $4$-space), the standard negative Einstein metric is the unique Einstein metric (up to rescaling and isometry) [@BCG] ([resp.]{}[@LeBrun-1]). In the positive case, there are some partial rigidity results on the $4$-sphere $S^4$ and the complex projective plane $\mathbb{CP}^2$ [@GL], [@G], [@Y]. When $M = S^4$, the standard round metric $g_{\mathbb{S}}$ of constant curvature $1$ is, to date, the only known Einstein metric (up to rescaling and isometry). In this connection we have the following gap theorem due to M.Gursky (see [@ABKS] for the significance of the constant $\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}])$):

\[Gursky\] Let $g$ be a positive Einstein metric on $S^4$. If its Yamabe constant $Y(S^4, [g])$ satisfies the following inequality $$Y(S^4, [g]) \geq \frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}])$$ then, up to rescaling, $g$ is isometric to $g_{\mathbb{S}}$. Here, $[g]$ denotes the conformal class of $g$.

Note that $Y(S^4, [h]) \leq Y(S^4, [g_{\mathbb{S}}]) = 8\sqrt{6}\pi$ for any Riemannian metric $h$ and that $Y(S^4, [g]) = R_g \sqrt{V_g}$ for any Einstein metric $g$, where $R_g$ and $V_g = {\rm Vol}(S^4, g)$ denote respectively the scalar curvature of $g$ and the volume of $(S^4, g)$.

Our main result in this paper is an extension of Theorem\[Gursky\]:

\[MainThm1\] There exists a universal positive constant $\varepsilon_0 > 0$ with the following property$:$ If $g$ is a positive Einstein metric on $S^4$ with Yamabe constant $$Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}])  - \varepsilon_0,$$ then, up to rescaling, $g$ is isometric to $g_{\mathbb{S}}$.

This result can be restated in terms of the [*Weyl constant*]{} of $[g]$ (cf.[@ABKS]). Indeed, the Chern-Gauss-Bonnet theorem (see Remark\[ALE\]-(1)) implies that the lower bound on the Yamabe constant is equivalent to the following upper bound on the Weyl constant: $\int_M |W_g|^2 d  \mu_g < \frac{32}{3} \pi^2 + \widetilde{\varepsilon}_0$, where $\widetilde{\varepsilon}_0 := \frac{\varepsilon_0}{24}(16\sqrt{2}\pi - \varepsilon_0) > 0$.

More generally, we obtain the following (note that $8\sqrt{2}\pi = \frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}])$):

\[MainThm2\] For $c > 0$, let $\mathcal{E}_{\geq c}(S^4)$ denote the space of all unit-volume positive Einstein metrics $g$ on $S^4$ with $c \leq Y(S^4, [g]) < 8\sqrt{2}\pi$. Then the number of connected components of the moduli space $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ is finite. In particular, $\{ Y(S^4, [g]) \in [c, 8\sqrt{2}\pi)\ |\ g \in \mathcal{E}_{\geq c}(S^4) \}$ is a finite set $($possible empty$)$.

Here $ \mathcal{M}_1(S^4)/{\rm Diff}(S^4)$ has the $C^\infty$-topology and $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ is endowed with the subspace topology.

These theorems follow from the following crucial result:

\[MainProp\] Let $\{g_i\}$ be a sequence in $  \mathcal{E}_{\geq c}(S^4)$ for some positive constant $c > 0$. Then there exists a subsequence $\{j\} \subset \{i\}$, $\{\phi_j\} \subset {\rm Diff}(S^4)$ and a unit-volume positive Einstein metric $g_{\infty}$ on $S^4$ such that $\phi_j^*g_j$ converges to $g_{\infty}$ with respect to the $C^{\infty}$-topology on $\mathcal{M}_1(S^4)$.

[**Remark:**]{} TheoremD of [@Anderson] states that the same conclusion as the one in Proposition\[MainProp\] holds for any sequence $\{g_i\} \subset \mathcal{E}_{\geq c}(M)$ on any closed $4$-manifold $M$ with $1 \leq \chi(M) \leq 3$, where $\chi(M)$ denotes the Euler characteristic of $M$. Unfortunately, the proof appears to be incorrect. Specifically, Theorem D is based on Lemma 6.3, which asserts that a Ricci-flat ALE 4-space $X$ with $\chi(X)=1$ is necessarily isometric to the Euclidean $4$-space $({\mathbb R}^4, g_{\mathbb{E}})$. This is not true: the Ricci-flat ALE 4-space $X_1$ constructed by Eguchi-Hanson [@EH] has a free, isometric ${\mathbb Z}_2$-action whose quotient $X_2 = X_1/{\mathbb Z}_2$ is a Ricci-flat ALE $4$-space with $\chi(X_2)=1$. Note that $X_2$ is nonorientable. Even if we assume that $X$ is orientable in Lemma6.3, the topological argument in the proof still contains some gaps. Proposition3.10 of [@Anderson-GAFA] corrects a minor inaccuracy of Lemma6.3. However, the proof also contains some gaps in the topological argument (see Remark\[Counter\] in $\S$4 for details).

Gursky’s proof of Theorem\[Gursky\] involves a sophisticated Bochner technique, a modified scalar curvature and a conformal rescaling argument. The proof of Proposition1.4 is based on topological results about $S^3$-quotients embedded in $S^4$ and the convergence theory of Einstein metrics in four-dimensions. Given this proposition, we invoke Gursky’s result to prove Theorems\[MainThm1\] and\[MainThm2\].

In $\S$2, we recall some background material and prove Theorems\[MainThm1\] and\[MainThm2\], assuming Proposition1.4. In $\S$3, we review two key results needed for the proof of Proposition1.4. Finally, in $\S$4, we prove Proposition1.4.\
\
[**Acknowledgements.**]{} The authors would like to thank Anda Degeratu and Rafe Mazzeo for valuable discussions on the eta invariant, and Shouhei Honda for helpful discussions on convergence results of Riemannian manifolds with bounded Ricci curvature. They would also like to thank Matthew Gursky and Claude LeBrun for useful advices, and Gilles Carron for crucial comments.\

Preliminaries and proofs of Theorems 1.2 and 1.3
================================================

We first review the definitions of Yamabe constants and Yamabe metrics. Let $M^n$ be a closed $n$-manifold with $n \geq 3$. It is well known that a Riemannian metric on $M$ is Einstein if and only if it is a critical point of the normalized Einstein-Hilbert functional $I$ on the space $\mathcal{M}(M)$ of all Riemannian metrics on $M$ $$I : \mathcal{M}(M) \rightarrow \mathbb{R},\quad g \mapsto I(g) := \frac{\int_MR_gd\mu_g}{{\rm Vol}(M, g)^{(n-2)/n}},$$ where $d\mu_g$ denotes the volume form of $g$. The restriction of $I$ to any conformal class $[g] := \{ e^{2f}\,g\ |\ f \in C^{\infty}(M) \}$ is always bounded from below. Hence, we can consider the following conformal invariant $$Y(M, [g]) := \inf_{\widetilde{g} \in [g]}I(\widetilde{g}),$$ which is called the [*Yamabe constant*]{} of $(M, [g])$. A remarkable theorem of H.Yamabe, N.Trudinger, T.Aubin and R.Schoen asserts that each conformal class $[g]$ contains metrics $\check{g}$, called [*Yambe metrics*]{}, which realize the minimum (cf.[@LP], [@Sc-1]) $$Y(M, [g]) = I(\check{g}).$$ These metrics must have constant scalar curvature $$R_{\check{g}} = Y(M, [g])\cdot V_{\check{g}}^{-2/n},$$ where $V_{\check{g}} = {\rm Vol}(M, \check{g})$. Aubin proved that $$Y(M^n, C) \leq Y(S^n, [g_{\mathbb{S}}]) = n(n-1) V_{g_{\mathbb{S}}}^{2/n}$$ for any conformal class $C$ on $M$. Obata’s Theorem[@Obata] implies that [*any Einstein metric is a Yamabe metric*]{}. When $n = 4$, $$Y(M^4, [g]) = R_{\widehat{g}} \sqrt{V_{\widehat{g}}} \leq Y(S^4, [g_{\mathbb{S}}]) = 8\sqrt{6}\pi$$ for any Einstein metric $\widehat{g} \in [g]$.

Assuming Proposition1.4, we can now prove Theorem\[MainThm1\].

Suppose that there exists a sequence $\{g_i\}$ of unit-volume Einstein metrics on $S^4$ satisfying $$Y(S^4, [g_i]) = R_{g_i} < 8\sqrt{2}\pi\ \ ({\rm for}\ \ \forall i),\quad Y(S^4, [g_i]) = R_{g_i} \nearrow 8\sqrt{2}\pi\ \ ({\rm as}\ \ i \to \infty).$$ By Proposition1.4, there exists a subsequence $\{j\} \subset \{i\}$, a sequence $\{\phi_j\} \subset {\rm Diff}(S^4)$ and a unit-volume positive Einstein metric $g_{\infty}$ on $S^4$ such that $\phi_j^*g_j$ converges to $g_{\infty}$ with respect to the $C^{\infty}$-topology on $S^4$. Then, we get $$\label{lll}
Y(S^4, [g_{\infty}]) = R_{g_{\infty}} = 8\sqrt{2}\pi.$$

On the other hand, Theorem\[Gursky\] implies that $(S^4, g_{\infty})$ is isometric to $(S^4, g_{\mathbb{S}})$. Hence, $$Y(S^4, [g_{\infty}]) = Y(S^4, [g_{\mathbb{S}}]) = 8\sqrt{6}\pi.$$ This contradicts (\[lll\]). Therefore, there exists a positive constant $\varepsilon_0 > 0$ such that any unit-volume positive Einstein metric $g$ on $S^4$ satisfying $$Y(S^4, [g]) > 8\sqrt{2}\pi - \varepsilon_0$$ is isometric to $g_{\mathbb{S}}$.

By the result of N. Koiso[@Koiso-2 Theorem3.1] and [@Besse Corollary12.52] (cf.[@Ebin Theorem7.1], [@Koiso-1 Theorem2.2]), we first remark that, for each $g \in \mathcal{E}_{\geq c}(S^4)$, the premoduli space $\mathcal{E}_{\geq c}(S^4)$ around $g$ is a real analytic subset of a finite dimensional real analytic submanifold in $\mathcal{M}_1(S^4) := \{ g \in \mathcal{M}(S^4)\ |\ V_g = 1 \}$, and the moduli space $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ is arcwise connected. Moreover, the Yamabe constant $Y(S^4, [\bullet])$ is a locally constant function and and it takes (at most) countably many values on $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$.

Suppose that there exist infinitely many connected components of the moduli space $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ (see [@Besse Chapters4 and 12] for the topology on it). Then, there exists a sequence $\{g_i\}$ in $\mathcal{E}_{\geq c}(S^4)$ such that the equivalence classes of any two $g_{i_1}$ and $g_{i_2}$ are contained in different connected components of $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ if $i_1 \ne i_2$. Similar to the proof of Theorem\[MainThm1\], there exists a subsequence $\{j\} \subset \{i\}$, a sequence $\{\phi_j\} \subset {\rm Diff}(S^4)$ and a unit-volume positive Einstein metric $g_{\infty}$ on $S^4$ such that $\phi_j^*g_j$ converges to $g_{\infty}$ with respect to the $C^{\infty}$-topology on $S^4$. We note here that the topology of the moduli space is induced from the one of the space $\mathcal{M}_1(S^4)$. Then, there exists a large positive integer $j_0$ such that the set $\{\phi_j^*g_j\}_{j \geq j_0}$ is contained in a connected component. This contradicts the choice of $\{g_i\}$. Hence, the number of connected components of the moduli space $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ is finite (possibly zero). In particular, the set of $\{ Y(S^4, [g]) \in [c, 8\sqrt{2}\pi)\ |\ g \in \mathcal{E}_{\geq c}(S^4) \}$ is a finite set $($possibly empty$)$.

A review of two key results
===========================

An embedding theorem:
---------------------

It will be necessary to know which quotients $S^3/\Gamma$ of $S^3$ embed smoothly in $S^4$. The theorem below gives a complete answer, which is one of the two key results for the proof of Proposition1.4.

\[Key-2\] Let $\Gamma \subset SO(4)$ be a finite subgroup such that $S^3/\Gamma$ is a smooth quotient of $S^3$. If $S^3/\Gamma$ can be smoothly embedded in $S^4$, then either $\Gamma = \{1\}$ or $\Gamma = Q_8$. Here, $Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$ denotes the quaternion group.

Convergence of Einstein metrics:
--------------------------------

We first review the definition of the energy of metrics on $4$-manifolds.

$(1)$  For a closed Riemannian $4$-manifold $(M, g)$, the [*energy*]{} $\mathscr{E}(g)$ of $g$ (or $(M, g)$) is defined by $$\mathscr{E}(g) := \frac{1}{8\pi^2}\int_M|\mathscr{R}_g|^2d\mu_g,$$ where $\mathscr{R}_g = (R^i_{\ jk\ell})$ denotes the curvature tensor of $g$ and $|\mathscr{R}_g|^2 = \frac{1}{4}R^i_{\ jk\ell}R_i^{\ jk\ell}$.

$(2)$  If $(X,h)$ is an [*asymptotically locally Euclidean*]{} $4$-manifold of order $\tau > 0$ (ALE $4$-space for brevity, cf.[@BKN]), the energy $\mathscr{E}(h)$ of $h$ (or $(X, h)$) is again defined by $$\mathscr{E}(h) := \frac{1}{8\pi^2}\int_X|\mathscr{R}_h|^2d\mu_h < \infty.$$

\[ALE\] $(1)$  By the Chern-Gauss-Bonnet formula, $\mathscr{E}(g) = \chi(M)$ for any Einstein metric $g$ on a closed $4$-manifold $M$. Indeed, $$\begin{aligned}
\mathscr{E}(g) &= \frac{1}{8\pi^2}\int_M|\mathscr{R}_g|^2d\mu_g = \frac{1}{8\pi^2}\int_M(|W_g|^2 + \frac{1}{24}R_g^2 + \frac{1}{2}|E_g|^2)d\mu_g\\ 
&= \frac{1}{8\pi^2}\int_M(|W_g|^2 + \frac{1}{24}R_g^2 - \frac{1}{2}|E_g|^2)d\mu_g = \chi(M),\end{aligned}$$ where $W_g = (W^i_{\ jk\ell})$ and $E_g = (E_{ij})$ denote respectively the Weyl tensor and the trace-free Ricci tensor ${\rm Ric}_g - \frac{R_g}{4}g$ of $g$, and $|W_g|^2 = \frac{1}{4}W^i_{\ jk\ell}W_i^{\ jk\ell}$. In particular, $\mathscr{E}(g) = 2$ if $M = S^4$.\
$(2)$  The Chern-Gauss-Bonnet formula for $4$-manifolds with boundary implies the following (cf.[@Hitchin-2 formula(7)]): any Ricci-flat ALE $4$-space $(X, h)$ with end $S^3/\Gamma$ satisfies $$\chi(X) = \mathscr{E}(h) + \frac{1}{|\Gamma|},$$ where $\Gamma$ is a finite subgroup of $O(4)$ acting freely on $\mathbb{R}^4 - \{0\}$ and $|\Gamma|$ is the order of $\Gamma$. If $\chi(X) = 1$, we get, in particular, the following: $$\mathscr{E}(h) = 1 - \frac{1}{|\Gamma|}.$$ $(3)$  Bando-Kasue-Nakajima[@BKN] proved that any Ricci-flat ALE $4$-space $(X, h)$ is an ALE $4$-space of order of $4$. Moreover, when $(X, h)$ is [*asymptotically flat*]{} (AF for brevity, cf.[@Bartnik]), that is, $\Gamma = \{1\}$, this combined with a result of R. Bartnik[@Bartnik Theorem4.3] implies that the mass of $(X, h)$ is zero. The Positive Mass Theorem[@Sc-1 Theorem4.3] for AF manifolds then implies that $(X, h)$ is isometric to $(\mathbb{R}^4, g_{\mathbb{E}})$. Note that $\mathscr{E}(h) = \mathscr{E}(g_{\mathbb{E}}) = 0$.

Recall again that any Einstein metric $g$ on a closed $4$-manifold $M$ satisfies that $Y(M, [g]) = R_g\sqrt{V_g}$. Moreover, if $g$ is a unit-volume Einstein metric with $Y(M, [g]) \geq c\ (c > 0)$, then ${\rm Ric}_g \geq \frac{c}{4}g$. Hence, Myers’ diameter estimate gives $${\rm diam}(M, g) \leq \frac{2\sqrt{3}\pi}{\sqrt{c}}.$$

Using this fact and Remark3.3-(1), we can now state a modified version of the convergence theorem for Einstein metrics due to M. Anderson[@Anderson], H. Nakajima[@Nakajima-1] and Bando-Kasue-Nakajima[@BKN], which is the other of the two key results for the proof of Proposition1.4.

\[Key-1\] Let $M$ and $\{g_i\}$ be respectively a closed $4$-manifold and a sequence of unit-volume positive Einstein metrics on $M$ with $Y(M, [g]) \geq c$ for a fixed $c > 0$. Then, there exist a subsequence $\{j\} \subset \{i\}$ and a compact Einstein $4$-orbifold $(M_{\infty}, g_{\infty})$ with a finite singular points $\mathcal{S} = \{p_1, p_2, \cdots, p_{\ell}\} \subset M_{\infty}$ $($possibly empty$)$ and an orbifold structure group $\Gamma_a \subset O(4)$ around $p_a$ for which the following assertions hold$:$\
$(1)$  $(M, g_j)$ converges to $(M_{\infty}, g_{\infty})$ in the Gromov-Hausdorff distance.\
$(2)$  There exists a smooth embedding $\phi_j : M_{\infty} - \mathcal{S} \rightarrow M$ for each $j$ such that $\phi_j^*g_j$ converges to $g_{\infty}$ in the $C^{\infty}$-topology on $M_{\infty} - \mathcal{S}$. If $\mathcal{S}$ is empty, then each $\phi_j$ is a diffeomorphism from $M_{\infty}$ onto $M$.\
$(3)$  For each $p_a \in \mathcal{S}$ and $j$, there exists $p_{a, j} \in M$ and a positive number $r_j$ such that\
$(3.1)$  $B_{\delta}(p_{a, j}; g_j)$ converges to $B_{\delta}(p_a;g_{\infty})$ in the pointed Gromov-Hausdorff distance for all $\delta > 0$, where $B_{\delta}(p_{a, j}; g_j)$ denotes the geodesic ball of radius $\delta > 0$ centered at $p_{a, j}$ with respect to $g_j$.\
$(3.2)$  $\lim_{j \to \infty}r_j = 0$.\
$(3.3)$  $((M, r_j^{-2}g_j), p_{a, j})$ converges to $((X_a, h_a), x_{a, \infty})$ in the pointed Gromov-Hausdorff distance, where $(X_a, h_a)$ is a complete, non-compact, Ricci-flat, non-flat ALE $4$-space of order $4$ with $$0 < \int_{X_a}|\mathscr{R}_{h_a}|^2d\mu_{h_a} < \infty,$$ and $x_{a, \infty} \in X_a$.\
$(3.4)$  There exists smooth embeddings $\Phi_j : X_a \rightarrow M$ such that $\Phi_j^*(r_j^{-2}g_j)$ converges to $h_a$ in the $C^{\infty}$-topology on $X_a$.\
$(4)$  It holds that $$\lim_{j \to \infty}\int_M|\mathscr{R}_{g_j}|^2d\mu_{g_j} \geq \int_{M_{\infty}}|\mathscr{R}_{g_{\infty}}|^2d\mu_{g_{\infty}} + \sum_a\int_{X_a}|\mathscr{R}_{h_a}|^2d\mu_{h_a}.$$

\[Tree\] Since $S^3/\Gamma_a$ is smoothly embedded in $M_{\infty}$ around $p_a$ for each $a$, it is also smoothly embedded in $M$ and it separates $M$ into two components $V_a, W_a$, which are compact $4$-manifolds with boundary. More precisely, $M = V_a \cup W_a,\ S^3/\Gamma_a = \partial V_a = \partial W_a = V_a \cap W_a$. Here, we choose $V_a$ satisfying $V_a \subset M_{\infty}$. The infinity $X_a(\infty) \cong S^3/\widetilde{\Gamma}_a$ of $X_a$ is also smoothly embedded in $M$. By the existence of intermediate Ricci-flat ALE $4$-orbifolds in the bubbling tree arising from each singular point $p_a$, $\Gamma_a \ne \widetilde{\Gamma}_a$ in general (cf.[@Bando], [@Nakajima-3]).

Proof of Proposition1.4
=======================

Let $\{g_i\}$ be a sequence of positive Einstein metrics on $S^4$ with $\{g_i\} \subset \mathcal{E}_{\geq c}(S^4)$ for some $c > 0$. We apply Theorem\[Key-1\], with $M=S^4$, for the sequence $\{g_i\}$. Then, in order to prove Proposition1.4, by Theorem\[Key-1\]-(2), it is enough to show that the singular set $\mathcal{S}$ is empty.

From now on, , that is, $\ell \geq 1$. By a similar reason to Remark\[Tree\], the case of $\Gamma_a = \{1\}$ for some $a\ (1 \leq a \leq \ell)$ may be possible logically for a general closed $4$-manifold, particularly $\Gamma_a \ = \{1\}$ for all $a = 1, 2, \cdots, \ell$. However, at least in the case of $M = S^4$, the following holds. We use the notation of Theorem\[Key-1\] and Remark\[Tree\].

\[Subkey\] $\Gamma_{a_0} \ne \{1\}$ for some $a_0\ (1 \leq a_0 \leq \ell)$.

Suppose that $\Gamma_a = \{1\}$ for all $a = 1, 2, \cdots, \ell$. As mentioned in Remark\[Tree\], a smooth embedded $S^3$ around $p_1 \in S^4$ separates $S^4$ into two components $V_1, W_1$ of compact $4$-manifolds with boundary satisfying $$S^4 = V_1 \cup W_1,\quad S^3 = \partial V_1 = \partial W_1 = V_1 \cap W_1,\quad V_1 \subset S^4_{\infty}.$$ By the Mayer-Vietoris exact sequence of homology groups for $(S^4; V_1, W_1)$, one can get $$H_i(V_1; \mathbb{R}) = H_i(W_1; \mathbb{R}) = 0\qquad {\rm for}\ \ i = 1, 2, 3,$$ and hence $\chi(V_1) = \chi(W_1) = 1$. Let $S^4_1 := V_1 \cup_{S^3} \overline{D^4}$ be a closed smooth $4$-manifold obtained by gluing along $S^3 = \partial V_1 = \partial\overline{D^4}$, where $\overline{D^4}$ denotes the closed $4$-ball in $\mathbb{R}^4$. Note that $\chi(S^4_1) = 2$. Similar to the above, a smooth embedded $S^3$ around $p_2 \in S^4_1$ separates $S^4_1$ into two components $V'_2, W'_2$. Then, the closed smooth $4$-manifold $S^4_2 := V'_2 \cup_{S^3} \overline{D^4}$ also satisfies that $\chi(S^4_2) = 2$. Repeating a similar procedure up to $a = \ell$, we get finally a closed smooth $4$-manifold $S^4_{\ell} := V'_{\ell} \cup_{S^3} \overline{D^4}$ with $\chi(S^4_{\ell}) = 2$. By construction, $S^4_{\ell}$ is homeomorphic to $S^4_{\infty}$ which implies that $\chi(S^4_{\infty}) = 2$.

By the removable singularities theorem for Einstein metrics [@BKN Theorem5.1], we note that $(S^4_{\infty}, g_{\infty})$ is a closed [*smooth*]{} Einstein $4$-manifold. Combining this with $\chi(S^4_{\infty}) = 2$, we get that $\mathscr{E}(g_{\infty}) = 2$. However, each Ricci-flat ALE $4$-space $(X_a, h_a)$ bubbling out from $p_a$ has a positive energy $\mathscr{E}(h_a) > 0$. This, combined with Theorem\[Key-1\]-(4) leads to a contradiction: $$2 = \lim_{j \to \infty}\mathscr{E}(g_j) \geq \mathscr{E}(g_{\infty}) + \sum_a\mathscr{E}(h_a) > 2.$$ Therefore, $\Gamma_{a_0} \ne \{1\}$ for some $a_0\ (1 \leq a_0 \leq \ell)$.

We can now prove Proposition1.4.

For simplicity, we assume that $a_0 = 1$. It then follows from Theorem\[Key-2\] that $\Gamma_1 = Q_8$. By Remark\[ALE\]-(3), we also obtain that $\widetilde{\Gamma}_1 = Q_8$. Even if $\widetilde{\Gamma}_a = Q_8$ for some $a$, a similar Mayer-Vietoris argument to that in the proof of Lemma\[Subkey\] still holds, and so $\chi(X_1) = 1$. It then follows from Remark\[ALE\]-(2) that $$\mathscr{E}(h_1) = \chi(X_1) - \frac{1}{|Q_8|} = 1- \frac{1}{8} = \frac{7}{8}.$$

By the signature theorem for compact $4$-orbifolds (cf.[@Nakajima-2 (4.5)]) and the calculation of eta invariant $\eta_S(S^3/\Gamma)$ for the signature operator [@Hitchin-2 Section3], the compact Einstein $4$-orbifold $(S^4_{\infty}, g_{\infty})$ satisfies that $$\tau(S^4_{\infty}) = \frac{1}{12\pi^2}\int_{S^4_{\infty}}\Big{(} |W_{g_{\infty}}^+|^2 - |W_{g_{\infty}}^-|^2 \Big{)}d\mu_{g_{\infty}} 
- \sum_{a=1}^{\ell}\eta_S(S^3/\Gamma_a),\quad \eta_S(S^3/Q_8) = \frac{3}{4},$$ where $\tau(S^4_{\infty})$ denotes the signature of $S^4_{\infty}$. Since $H_2(S^4_{\infty}; \mathbb{R}) = 0$, we have that $H^2(S^4_{\infty}; \mathbb{R}) = 0$, and so $\tau(S^4_{\infty}) = 0$. Combining that $R_{g_{\infty}} \geq c > 0$ and $\eta_S(S^3/\Gamma_a) \geq 0$ with the above and Theorem\[Key-1\]-(4), we then obtain that $$\frac{9}{8} = 2 - \frac{7}{8} \geq \mathcal{E}(g_{\infty}) = \frac{1}{8\pi^2}\int_{S^4_{\infty}}\Big{(} |W_{g_{\infty}}|^2 + \frac{R_{g_{\infty}}^2}{24} \Big{)}d\mu_{g_{\infty}} 
> \frac{1}{8\pi^2}\int_{S^4_{\infty}}|W_{g_{\infty}}|^2d\mu_{g_{\infty}}$$ $$\qquad \quad \geq \frac{1}{8\pi^2}\int_{S^4_{\infty}}|W_{g_{\infty}}^+|^2d\mu_{g_{\infty}} \geq \frac{3}{2}\Big{(} \frac{3}{4} 
+ \frac{1}{12\pi^2}\int_{S^4_{\infty}}|W_{g_{\infty}}^-|^2d\mu_{g_{\infty}} \Big{)} \geq \frac{9}{8},$$ and hence it leads a contradiction. Therefore, $\mathcal{S} = \emptyset$.

As mentioned in Remark of $\S$1, we describe some details on the topological argument.

\[Counter\] Let $N_2$ be the nonorientable disk bundle over the the real projective plane $\mathbb{RP}^2$ with Euler number $2$. Let $T_4$ be the disk bundle of the complex line bundle over $S^2$ of degree $4$. Then, the natural double cover $T_4 \rightarrow N_2$ is the universal cover of $N_2$. Note that $S^4 = N_2 \cup_{\partial N_2}N_2$ and $\partial N_2 = S^3/Q_8$ (see [@Lawson] for details). Note also that $N_2$ is orientable since $N_2$ can be smoothly embedded in $S^4$ as a compact $4$-submanifold. Moreover, we have the following: $$\ H_1(N_2; \mathbb{Z}) = \mathbb{Z}_2,\quad H_i(N_2; \mathbb{Z}) = 0\ \ (i = 2, 3, 4),$$ $$H_2(T_4; \mathbb{Z}) = \mathbb{Z},\quad \ \ H_i(T_4; \mathbb{Z}) = 0\ \  (i = 1, 3, 4).$$ We do not know whether the orientable open $4$-manifold ${\rm Int}(N_2)$ admits a Ricci-flat ALE metric or not. (We have proved here only that such a metric never appears as a bubbling off Ricci-flat ALE metric from a sequence in $\mathscr{E}_{\geq c}(S^4)$.) However, $N_2$ becomes an orientable counterexample to the topological arguments in the proofs of [@Anderson Lemma6.3] and [@Anderson-GAFA Proposition3.10].

[99]{} K.Akutagawa, B.Botvinnik, O.Kobayashi and H.Seshadri, [*The Weyl functional near the Yamabe invariant*]{}, [**J. Geom. Anal. 13**]{} (2003), 1–20. M.Anderson, [*Ricci curvature bounds and Einstein metrics on compact manifolds*]{}, [**J. Amer. Math.Soc. 2**]{} (1989), 455-490. M.Anderson, [*Einstein metrics with prescribed conformal infinity on $4$-manifolds*]{}, [**Geom. Fanct. Anal. 18**]{} (2008), 305–366. S.Bando, [*Bubbling out of Einstein manifolds*]{}, [**Tohoku Math. J. 42**]{} (1990), 205–216; Correction and addition, [**Tohoku Math. J. 42**]{} (1990), 587-588. S.Bando, A.Kasue and H.Nakajima, [*On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth*]{}, [**Invent. Math. 97**]{} (1989), 313–349. R.Bartnik, [*The mass of an asymptotically flat manifold*]{}, [**Comm. Pure Appl. Math. 39**]{} (1986), 661–693. A.Besse, [*Einstein Manifolds*]{}, Springer, 1987. G.Besson, G.Courtois and S.Gallot, [*Entropies et rigidités des espaces localement symétriques de courbure strictement négative*]{}, [**Geom. Funct. Anal. 5**]{} (1995), 731–799. C.Böhm, [*Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces*]{}, [**Invent. Mtah. 134**]{} (1998), 145–176. J.-P.Bourguignon and H.Karcher, [*Curvature operators: pinching estimates and geometric examples*]{}, [**Ann. Sci. École Norm. Sup. 11**]{} (1978), 71–92. J.S.Crisp and J.A.Hillman, [*Embedding Seifert fibred and $3$-manifolds and ${\rm Sol}^3$-manifolds in $4$-space*]{}, [**Proc. London Math. Soc. 76**]{} (1998), 685–710. D.Ebin, [*The manifold of Riemannian metrics*]{}, Global Analysis, [**Proc. Symp. Pure Math. 15**]{} (1968), 11–40. T.Eguchi and A.J.Hanson, [*Self-dual solutions to Euclidean gravity*]{}, [**Ann. of Phys. 120**]{} (1979), 82–106. M.Gursky, [*Four-manifolds with $\delta W^+ = 0$ and Einstein constants on the sphere*]{}, [**Math. Ann. 318**]{} (2000), 417–43. M.Gursky and C.LeBrun, [*On Einstein manifolds of positive sectional curvature*]{}, [**Ann. Global Anal. Geom. 17**]{} (1999), 315–328. N.Hitchin, [*Compact four-dimensional Einstein manifolds*]{}, [**J. Differential Geom. 9**]{} (1974), 435–441. N.Hitchin, [*Einstein metrics and the eta-invariant*]{}, [**Bullettino U. M. I. 11-B**]{} (1997), 92–105. G.R.Jensen, [*Einstein metrics on principal fibre bundles*]{}, [**J. Differential Geom. 8**]{} (1973), 599–614. N.Koiso, [*Non-deformability of Einstein metrics*]{}, [**Osaka J. Math. 15**]{} (1978), 419–433. N.Koiso, [*Einstein metrics and complex structures*]{}, [**Invent. Math. 73**]{} (1983), 71–106. T.Lawson, [*Splitting $S^4$ on $\mathbb{RP}^2$ via the branched cover of $\mathbb{CP}^2$ over $S^4$*]{}, [**Proc. Amer. Math. Soc. 86**]{} (1982), 328–330. C.LeBrun, [*Einstein metrics and Mostow rigidity*]{}, [**Math. Res. Lett. 2**]{} (1995), 1–8. C.LeBrun, [*Four-manifolds without Einstein metrics*]{}, [**Math. Res. Lett. 3**]{} (1996), 133–147. J.Lee and T.Parker, [*The Yamabe problem*]{}, [**Bull. Amer. Math. Soc. 17**]{} (1987), 37–81. H.Nakajima, [*Hausdorff convergence of Einstein $4$-manifolds*]{}, [**J. Fac. Sci. Univ. Tokyo 35**]{} (1988), 411–424. H.Nakajima, [*Self-duality of ALE Ricci-flat $4$-manifolds and positive mass theorem*]{}, in Recent Topics in Differential and Analytic Geometry, [**Advanced Studies in Pure Math. 18-I**]{} (1990), 313–349. H.Nakajima, [*A convergence theorem for Einstein metrics and the ALE spaces*]{}, [**Amer. Math. Soc. Transl. 160**]{} (1994) 79–94. M.Obata, [*The conjectures on conformal transformations of Riemannian manifolds*]{}, [**J. Differential Geom. 6**]{} (1972), 247–258. R.Schoen, [*Variational theory for the total scalar curvature functional for Riemannian metrics and related topics*]{}, Topics in Calculus of Variations, [**Lect. Notes in Math. 1365**]{}, 121–154, Springer, 1989. J.Thorpe, [*Some remarks on the Gauss-Bonnet integral*]{}, [**J. Math. Mech. 18**]{} (1969), 779–786. D.Yang, [*Rigidity of Einstein $4$-manifolds with positive curvature*]{}, [**Invent. Math. 142**]{} (2000),435–450.
