A note on cohomogeneity one Einstein metrics on closed 4-manifolds

Introduction

Broder [2 Theorem 1.1] states that if $(M^4,g)$ is a connected closed Einstein 4-manifold with $\dim\operatorname{Isom}(M,g) \ge3$, then $g$ is flat or its universal cover is homothetic to one of

First, [3 Theorem 3] (see also [Bér82]) implies that only cohomogeneity one actions with 3-dimensional isometry group need to be considered. These actions are by $\mathrm{SU}(2)$ and are known, see [8, 6]. In this setting, every invariant Einstein metric is diagonal (c.f. [4 Theorem 3.1]); along a normal geodesic it can be uniquely given in the form

The topologies and smoothness conditions depend on which $f_i$ vanishes at each singular orbit, and the possibilities are listed in [4 Table 4] (Table 1 below).

Let $M_n$ be the space whose group diagram (see §2) is $\mathbb{Z}_n \subset\{\mathrm{U}(1),\mathrm{U}(1)\} \subset\mathrm{SU}(2)$, ($M_{\mathrm{odd}} \cong\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ and $M_{\mathrm{even}} \cong S^2 \times S^2$). For $n \in\{1,2\}$, $M_n$ appears twice in Table 1 as there are two configurations: either the same warping function vanishes at both ends (row (R5)), or different functions do (row (R6)). In [2 proof of Theorem 5.1] row (R6) on $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ is reduced to row (R5) by changing the group diagram by conjugating one singular orbit by an element of $N(H)_0$, where $H$ is a principal isotropy. In [2 §6], on $S^2 \times S^2$, only row (R6) is treated. This does not suffice since rows (R5) and (R6) genuinely must be treated separately; see §3. In this note, we close the remaining cases.

We only need to consider $n \in\{1,2,4\}$, as otherwise the metric is $\mathrm{U}(2)$-invariant and thus classified in [Bér82]. One should note that only techniques from [2] are required to prove Theorem 3.5.

We introduce the required notation and preliminary results in § 3. In § 4, we show that the group diagram read off along a normal geodesic is determined by the metric up to interchanging $K_{\pm}$ and simultaneous conjugation; see Proposition 3.2. Finally, we prove Theorem 3.5 in § 5 and § 6, the former dealing with $\mathrm{S}^{2} \times\mathrm{S}^{2}$ and the latter with $\mathbb{CP}^{2}\#\overline{\mathbb{CP}}^{2}$.

Acknowledgements

I discussed these issues with Kyle Broder after [2] appeared on arXiv. While working on a related project, I realised that the remaining cases could be resolved quite simply using the techniques of [2], which led to this short note. I would also like to thank Qiu Shi Wang for interesting conversations related to this topic.

Preliminaries

One familiar with [2], or cohomogeneity one actions more generally, should skip this section.

Suppose $(M^4,g)$ is a closed simply-connected 4-manifold on which $G = \mathrm{SU}(2)$ acts isometrically and by cohomogeneity one. The orbit space is homeomorphic to a closed interval. Let $c : [0,T] \to M$ be a unit speed geodesic between the singular orbits $S_{-} = G \cdot c(0)$ and $S_{+} = G \cdot c(T)$ that intersects each orbit orthogonally and exactly once; we call such geodesics normal. The principal isotropy is $H = G_{c(t)}$ for $t \in(0,T)$, and the singular isotropies are $K_{-} = G_{c(0)}$, $K_{+} = G_{c(T)}$. The diagram $H \subset\{K_{-},K_{+}\} \subset G$ determines $M$. The diagrams obtained by interchanging $K_{\pm}$, simultaneous conjugation of $H,K_{\pm}$ by an element of $G$, and replacing $K_{+}$ by $aK_{+}a^{-1}$ with $a \in N(H)_0$ (c.f. [5]) also determine $M$, up to equivariant diffeomorphism.

We will use notation introduced in [2 §2]. Fix an $\operatorname{Ad}$-invariant inner product $Q$ on $\mathfrak{su}(2)$ and a $Q$-orthonormal basis $\{X_i\}_{i=1}^{3}$ with $[X_i,X_{i+1}] = 2X_{i+2}$, with $\{\sigma_i\}$ a dual coframe of the action fields $\{X_i^*\}$. A diagonal metric is given by $g = \mathrm{d}t^2 + \sum_i f_i(t)^2 \sigma_i^2$ on the principal part. It is known that any Einstein metric on $g$ can be written in a diagonal form, and that the functions $f_i$ are uniquely determined up to permutation (c.f. [4 Theorem 3.1]).

For $\{i,j,k\} = \{1,2,3\}$ define

The Einstein equations, $\operatorname{Ric}(g) = \lambda g$, imply the following equations [2]:

Only one smoothness condition is needed in this work: a diagonal metric $g$ on $M_n$ is smooth about $S_-$ if and only if there are smooth functions $\phi_i:(-\varepsilon,\varepsilon)\to\mathbb{R}$, such that [9]

where $\phi_2$ is positive and $\phi_3$ necessarily vanishes unless $n\in\{1,2,4\}$. The smoothness condition at $S_+$ is given similarly; substitute $t\mapsto T-t$ and permute indices if required. Table 1 lists all possible smoothness conditions:

Action	$t\to0$	$t\to T$	Einstein metrics
(R1) $\operatorname{SO}(3)$ on $S^4$	(2.4), $n=4$	(2.4), $n=4$, $f_1\leftrightarrow f_3$	round [2 Thm. 3.2]
(R2) $\operatorname{SO}(3)$ on $\mathbb{CP}^2$	(2.4), $n=4$	(2.4), $n=2$, $f_1\leftrightarrow f_3$	Fubini–Study [2 Thm. 4.2]
(R3) $\operatorname{SU}(2)$ on $S^4$	fixed point	fixed point	round [2 Thm. 3.1]
(R4) $\operatorname{SU}(2)$ on $\mathbb{CP}^2$	fixed point	(2.4), $n=1$	Fubini–Study [2 Thm. 4.1]
(R5) $M_n$, $n$ odd	(2.4)	(2.4)	Page [2 Thm. 5.1]
(R5) $M_n$, $n$ even	(2.4)	(2.4)	none, Theorem 4.1
(R6) $M_2$	(2.4)	(2.4), $f_1\leftrightarrow f_3$	product [2 Thm. 6.1]
(R6) $M_1$	(2.4)	(2.4), $f_1\leftrightarrow f_3$	none, Theorem 5.4

Equivariant diffeomorphisms and the smoothness conditions

This section is a clarification of [4 Remark 2.9].

There may exist metrics compatible with some group diagram that are not compatible with others representing the same action. In particular, a metric $g$ being compatible with a given group diagram is only preserved by interchanging $K_{\pm}$ and simultaneous conjugation. That is, the diagram obtained by replacing $K_{\pm}$ by $aK_{\pm}a^{-1}$ with $a \in N(H)_0$ (c.f. [5]) has the same space of invariant metrics, but need not have the same space of compatible metrics.

Proof. Let $c$ be a normal geodesic of $g$ realising $H \subset\{K_{-},K_{+}\} \subset G$.

(a): If the new diagram is of this form, then $b\cdot c$ (or $t\mapsto b\cdot c(T-t)$ if $K_{\pm}$ are interchanged) is a normal geodesic of $g$ realising the new diagram. Conversely, suppose $g$ is compatible with a group diagram that is realised by $\widetilde c$ a normal geodesic of $g$. Reversing $\widetilde c$ if necessary, we may assume that $\widetilde c(0)\in S_{-}$, say $\widetilde c(0)=h\cdot c(0)$ with $h\in G$. Then $h^{-1}\widetilde c$ is a normal geodesic starting at $c(0)$. By the slice theorem, $K_{-}=G_{c(0)}$ acts transitively on $\partial B(T_{c(0)}S_{-}^{\perp})$, i.e., the space of unit vectors perpendicular to $S_{-}$ (c.f. [4 proof of Lemma 2.6]). Thus, there is a $k\in K_{-}$ that maps the initial velocity of $c$ to that of $h^{-1}\widetilde c$, i.e.,

By uniqueness of geodesics, $\widetilde c=b\cdot c$ with $b=hk$, and

for all $t\in[0,T]$.

(b): The map $\varphi$ is an isometry from $(M',\varphi^{*}g)$ to $(M,g)$ mapping orbits to orbits, so $\varphi^{-1}\circ c$ is a normal geodesic of $\varphi^{*}g$. By equivariance, $G_{\varphi(p)}=G_p$, so $\varphi^{-1}\circ c$ realises the same group diagram as $c$.

Suppose $G$ acts via cohomogeneity one on $(M,g)$ and let $c$ be a normal geodesic. Let $\mathfrak{g}$ be the Lie algebra of $G$ and let $\mathfrak{h}$ be the Lie algebra of $H$, a principal isotropy. Let $\mathfrak{n}$ be an $\operatorname{Ad}_{H}$-invariant subspace of $\mathfrak{g}$ such that $\mathfrak{n}\oplus\mathfrak{h}=\mathfrak{g}$. Identifying on the principal part of $M$ with $(G/H)\times(0,T)$ via $(h,t)\mapsto h\cdot c(t)$, the metric takes the form $g=dt^{2}+g_{t}$, where $g_{t}(X,Y)=g(X^{*},Y^{*})|_{c(t)}$ is an inner product on $\mathfrak{n}$ for each $t\in(0,T)$. The following well known and simple result specifies to which group diagram a specific metric belongs along $c$.

Proof. We consider $t=0$; the argument at $t=T$ is the same. $g_t$ extends continuously to a positive semi-definite form $g_0$ since for any $X\in\mathfrak n$, $g_t(X,X)=\lvert X^*(c(t))\rvert^2 \mathop{\longrightarrow}_{t\to0}\lvert X^*(c(0))\rvert^2$. An element $X\in\mathfrak n$ also lies in $\mathfrak k_-$ precisely when $X^*(c(0))=0$, i.e., $X\in\ker g_0$. That is, $\mathfrak n\cap\mathfrak k_-=\ker g_0$ and the result now follows since $\mathfrak h\subset\mathfrak k_-$.

We now specialise to the case of cohomogeneity one actions on 4 manifolds. In particular, due to the Einstein condition, we may assume that the metric is diagonal. Recall that we only consider the case where $G=\mathrm{SU}(2)$, $K_\pm$ are $\mathrm{U}(1)$ subgroups of $\mathrm{SU}(2)$, and $H=\mathbb{Z}_n$.

Proof. By Lemma 3.3, $\mathfrak k_-=\ker g_0$ is spanned by the $X_k$ for which $f_k(0)=0$.

We say a $G$-invariant metric $g=dt^2+g_t$ on $M$ is diagonalisable if there exists a single choice of basis of $\mathfrak n$ as in §2 such that $g_t$ is diagonal for all $t\in(0,T)$.

Proof. Note that the row to which each metric belongs follows from Corollary 3.4. The latter also implies that $\mathfrak k_-=\mathfrak k_+$ for some group diagram if and only if $g\in M_{n,s}^G$. Since the condition $\mathfrak k_-=\mathfrak k_+$ is preserved under conjugation, the result follows immediately from Proposition 3.2.

Remark 3.6. In general, to study the space of all cohomogeneity one metrics on $M$ of the form $dt^2+g_t$, Proposition 3.2 shows that one must consider many different group diagrams. In particular, given an action with some group diagram $H\subset\{K_-,K_+\}\subset G$, one must check the space of metrics compatible with the diagram $H\subset\{K_-,aK_+a^{-1}\}\subset G$ for $a\in N(H)_0$. It suffices to let $a$ range over a set of representatives of the double cosets of

This formulation is essentially given in [7]. If $K_-$ and $K_+$ are not conjugate to each other by an element of $N(H)$, then each double coset represents a different class of compatible invariant metrics. If $K_-$ and $K_+$ are conjugate to each other by an element $c\in N(H)$, then one could also account for reversing the normal geodesic. That is, the diagrams $H\subset $\{K,cKc^{-1}\}\subset G$ and $H\subset\{K,bKb^{-1}\}\subset G$ for $b\in N(H)_0c$ or $b\in N(H)_0c^{-1}$ have the same space of compatible metrics if and only if

or

respectively.

$S^2\times S^2$ in row (R5)

This follows using the exact same treatment as in Theorem 5.1 (see also Theorem 3.1 and Theorem 4.1) of [2] (which we repeat below for the convenience of the reader).

Proof. Suppose $f_1$ vanishes at both ends. By (2.4), as $t\to0$ and $t\to T$, $f_2/f_3\to1$ and $f_1/f_3\to0$. Suppose some ratio $f_i/f_3$ with $i\in\{1,2\}$ exceeds 1 somewhere, and hence is attained at an interior point $t_0$. Let $j\in\{1,2\}$, $j\ne i$ be the remaining index. At $t_0$, we have that $f_j(t_0)\le f_i(t_0)$, $0<f_3(t_0)<f_i(t_0)$ and $u_{i3}=\log(f_i/f_3)$ reaches an interior maximum, but (2.3) gives

a contradiction. Thus the ratio $f_2/f_3$ does not exceed 1, i.e., $f_2\le f_3$. Similarly, $f_3\le f_2$, so $f_2\equiv f_3$. This implies that the metric is U(2)-invariant and we can appeal to the classification of U(2)-invariant compact Einstein manifolds [Bér82].

$\mathbb{CP}^2\#\mathbb{CP}^2$ in row (R6)

We now rule out Einstein metrics in row (R6). This is achieved by using the conditions at the endpoints to contradict the forward invariant cone $\{a_1-a_2,a_2-a_3>0\}$. To show that this cone is forward invariant, we use the maximum principal from the previous section and the positivity of $A_i$. This follows along the lines of the proof of Theorem 3.2 of [2].

Let $g=dt^2+\sum_{i=1}^{3}f_i(t)^2\sigma_i^2$ be an SU(2)-invariant Einstein metric on $M_1$ in row (R6), i.e., with $f_1$ vanishing at $t=0$ and $f_3$ at $t=T$. Recall that a compact Einstein manifold with a non-abelian identity component of the isometry group implies the Einstein constant is positive. As in [4], define the constants $\beta_- = 1/f_2(0)^2 = 1/f_3(0)^2$ and $\beta_+ = 1/f_2(T)^2 = 1/f_1(T)^2$, and also parametrise the reverse direction by $s$; $s=T-t$. Solving (2.1) order by order about $t=0$ using (2.4) gives,

and hence around $s=0$,

In particular, about $t=0$,

and about $s=0$,

Note that (2.2) implies that

Proof. By (2.2),

Lemma 5.3 implies that the cone $a_1-a_2,a_2-a_3>0$ is forward invariant on $(0,T)$. The initial conditions and Lemma 5.2 implies that $(a_1-a_2)(0)=\frac{18\beta_{-}-\lambda}{2}>0$ and $(a_2-a_3)(T)=\frac{\lambda-6\beta_{+}}{2}<0$. Thus, for a contradiction, it suffices to show that $a_2-a_3$ is positive at some time. As before, we integrate (5.2) to see that for positive $t_0$: