Einstein metrics of cohomogeneity one with $\mathbb{S}^{4m+3}$ as principal orbit

A Riemannian manifold $(M,g)$ is Einstein if its Ricci curvature satisfies $\mathrm{Ric}(g)=\Lambda g$ for some constant $\Lambda$. A Riemannian manifold $(M,g)$ is of cohomogeneity one if a Lie Group $G$ acts isometrically on $M$ with principal orbit $G/K$ of codimension one. The Einstein equations of a cohomogeneity one manifold is reduced to a dynamic system.

In this article we focus on constructing non-compact cohomogeneity one Einstein metrics. Known examples include the first inhomogeneous Einstein metric in [Cal75], which has Kähler holonomy. More non-compact Kähler–Einstein metrics of cohomogeneity one were constructed in [BB82][DW98][WW98][DS02]. Non-compact cohomogeneity one $G_{2}$ and $\mathrm{Spin}(7)$ metrics, which are motivations to this article, were constructed in [BS89][GPP90][CGLP04][FHN18]. Fixing the principal orbit $G/K=Sp(m+1)U(1)/Sp(m)\Delta U(1)$, we aim to look into the full dynamic system of cohomogeneity one Einstein metrics without imposing any special holonomy condition. Odd dimensional cohomogeneity one Einstein metrics with generic holonomy include those constructed in [BB82][WW98][Che11]. The case where the isotropy representation of the principal orbit consists of exactly two inequivalent irreducible summands was studied in [Böh99][Win17]. Examples where the principal orbit is a product of irreducible homogeneous spaces was constructed in [Böh99]. In [Chi19b], Ricci-flat metrics with Wallach spaces as principal orbits were constructed. The isotropy representation of Wallach spaces consists of three inequivalent irreducible summands, two of which are from the singular orbit, allowing the singular orbit to be squashed. In this article, the principal orbit also consists of three irreducible summands. Our main results are the following.

Some known Einstein metrics are recovered in this family. In the case where $m=1$, $\zeta_{(1,0,0)}$ is the $\mathrm{Spin}(7)$ metric in [BS89][GPP90]. The 1-parameter family of $\mathrm{Spin}(7)$ metrics $\zeta_{(s_{1},s_{2},0)}$ was constructed in [CGLP04]. For all $m\geq 1$, metrics $\zeta_{(s_{1},0,s_{3})}$ are of two summands type. They were constructed in [Böh99][Win17]. All the other metrics in $\zeta_{(s_{1},s_{2},s_{3})}$ are new to the author.

On $\mathbb{R}^{4m+4}$, we have the following.

Although not included in the theorem above, the parameter $(s_{1},s_{2},s_{3})$ can be the origin for $\gamma_{(s_{1},s_{2},s_{3})}$. The metric represented is the Euclidean metric on $\mathbb{R}^{4m+4}$, as shown in Section 3. Metrics $\gamma_{(0,s_{2},s_{3})}$ are of two summand type. They first appeared in [BB82]. Metrics $\gamma_{(s_{1},0,s_{3})}$ is also of two summands type. They were constructed in [Chi19a]. In the case where $m=1$, $\gamma_{\left(\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}},0\right)}$ is the $\mathrm{Spin}(7)$ metric with the opposite chirality to the metric $\mathbb{A}_{8}$ constructed in [CGLP04]. All the other metrics in $\gamma_{(s_{1},s_{2},s_{3})}$ are new to the author.

In some sense, the 2-dimensional parameter $(s_{1},s_{2},s_{3})\in\mathbb{S}^{2}$ in Theorem 1.1 and Theorem 1.2 controls the asymptotic limit of the metric represented. The non-vanishing of $s_{2}$ in $(s_{1},s_{2},0)$ gives the ALC asymptotics. The parameter also describes how the principal orbit is squashed near the singular orbit. More details are discussed in Section 3. The non-vanishing of $s_{3}$ gives the AH asymptotics. As discussed in Section 2, the dynamic system of the negative Einstein metrics has a subsystem that can represent the Ricci-flat system. Integral curves with $s_{3}=0$ are solutions of this subsystem.

New Taub-NUT metrics on $\mathbb{R}^{4m+4}$ with conical singularity at the origin are also constructed.

Consider the holonomy of the Ricci-flat metrics in Theorems 1.1-1.3. Combining our Lemma 6.5 with Theorem 2.1 in [Hit74] and [Wan89], we obtain the following.

In particular for $m=1$, the continuous family of Ricci-flat metrics $\gamma_{(s_{1},s_{2},0)}$ has the $\mathrm{Spin}(7)$ metric $\mathbb{A}_{8}$ lies in the interior and all the other Ricci-flat metrics have generic holonomy. Hence the parallel spinor on $\mathbb{A}_{8}$ is not preserved under a continuous deformation of Ricci-flat metrics through the family $\gamma_{(s_{1},s_{2},0)}$. Such a phenomenon also occurs for $G_{2}$ holonomy [Chi19b]. Parallel spinors are preserved under a continuous deformation of Ricci-flat metrics if the manifold is compact. Please see Theorem A in [Wan91] for more details.

Principal orbit of manifolds studied in this article are from the group triple $(G,H,K)$ given by

$$(Sp(m+1)U(1),Sp(m)Sp(1)U(1),Sp(m)\Delta U(1)).$$

The principal orbit is the total space of quaternionic Hopf fibration

$$\mathbb{S}^{3}\hookrightarrow\mathbb{S}^{4m+3}\rightarrow\mathbb{HP}^{m}.$$

(1.1)

Take $\mathbb{S}^{4m+3}$ as the space of unit quaternionic vector in $\mathbb{H}^{m+1}$. The fibration $\mathbb{S}^{4m+3}\rightarrow\mathbb{HP}^{m}$ is given by $(q_{1},\dots,q_{m+1})\mapsto[q_{1}:\ldots:q_{m+1}]$. The transitive action of $G$ on $\mathbb{S}^{4m+3}$ is given by

$$(A,z)\cdot\mathbf{q}:=A\mathbf{q}\bar{z}$$

(1.2)

for each $(A,z)\in G$. The isotropy group for $(0,\dots,0,1)\in\mathbb{S}^{4m+3}$ is $K$. The action of $G$ passes down to the base. The isotropy group for $[0:\ldots:0:1]$ is $H$. Therefore, the quaternionic Hopf fibration is indeed the homogeneous fibration $H/K\hookrightarrow G/K\rightarrow G/H.$ More details of the isotropy representation are discussed in the next section.

Let $M$ be the cohomogeneity one manifold with principal orbit $G/K$ and singular orbit $G/H$. Then $M$ is an $\mathbb{R}^{4}$ bundle over $\mathbb{HP}^{m}$. A cohomogeneity one metric on $M$ has the form of $dt^{2}+g_{G/K}(t)$, where $g_{G/K}(t)$ is an invariant metric on each $\{t\}\times G/K$ with $t>0$ and it collapse to an invariant metric on $G/H$ as $t\to 0$. We also construct cohomogeneity one Einstein manifolds where the singular orbit for these manifolds is a singleton. In that scenario, the homogeneous part $g_{G/K}(t)$ vanishes as $t\to 0$. Since the principal orbit is $\mathbb{S}^{4m+3}$, the cohomogeneity one manifold is topologically $\mathbb{R}^{4m+4}$.

One feature of the case in this article that differs from the one in [Chi19b] is that the singular orbit is irreducible and the fiber is of two irreducible summands. Moreover, irreducible summands in $\mathfrak{g}/\mathfrak{k}$ all have different dimensions, as shown in Section 2. The cohomogeneity one dynamic systems have less symmetry than the one in [Chi19b]. It is worth mentioning that the cohomogeneity one equation in the article shares some degree of similarity with the one that appears in [Rei11]. The study may help shed some light on the global existence question of $\mathrm{Spin}(7)$ metric with an Aloff–Wallach space as the principal orbit.

The Einstein metrics constructed and recovered in this article have three kinds of asymptotic behaviors. We give definitions in the following.

In Definition 1.9, it can be checked that sectional curvature of $g_{M}$ approaches to $-\|df\|_{f^{2}g_{M}}$ near $\partial M$. If $(M,g_{M})$ is negative Einstein, then the sectional curvature must approach to a constant near $\partial M$. With normalization, we fix $\|df\|_{f^{2}g_{M}}=1$. Hence a conformally compact Einstein manifold is also called an asymptotically hyperbolic (AH) manifold.

This article is structured as the following. In Section 2, we derive the cohomogeneity one Einstein equation with principal orbit $G/K$. Then finding a cohomogeneity one Einstein metric is equivalent to finding an integral curve defined on $[0,\infty)$. Then we apply coordinate change inspired by the one in [DW09a][DW09b]. In the new coordinate, initial conditions and the asymptotic limits of the original system are transformed to critical points. Then the construction of Einstein metrics boils down to finding integral curves that emanate from one critical point and tend to the other. Proving the completeness of the metric is equivalent to showing that the new integral curve is defined on $\mathbb{R}$.

In Section 3, we compute linearizations of some critical points with geometric significance of the new system. There are three critical points that represents different initial conditions. One of them gives the smooth extension of the metric to $G/H$; one gives the smooth extension of the metric to the origin of $\mathbb{R}^{4m+4}$; and third one gives the singular extension to the origin of $\mathbb{R}^{4m+4}$. There are two types of critical points that represent different asymptotic limits. One of them represents the ALC limit and the other type serves as the AH limit for the integral curves.

In Section 4, we construct a compact invariant set that contains sellected critical points in the previous section on its boundary. Linearization in the previous section helps to prove that some integral curves that emanate from these points are in the compact invariant set initially. Hence the completeness of the represented metrics follows. The technique we use is very similar to the one in [Chi19a].

In Section 5, we give a rigorous proof for the asymptotic behaviour of the complete integral curves. We prove that all the new Ricci-flat metrics constructed are ALC, generalizing the conclusion in [CGLP04] and [Baz07]. We also prove that all the new negative Einstein metrics constructed are AH.

In Section 6, we recover some well-known examples, including the Kähler–Einstein metrics in [BB82], the quaternionic Kähler metric in [Swa91], the $G_{2}$ and $\mathrm{Spin}(7)$ metrics in [BS89][GPP90] and the $\mathrm{Spin}(7)$ metric in [CGLP04]. Then we show that all the ALC Ricci-flat metrics constructed have generic holonomy.

We derive the Einstein equation in this section. We first introduce some notation regarding representation theory. Let $\mathbb{I}$ be the trivial representation. Let $\mu_{m}$ be the matrix multiplication representation (over complex numbers) of $Sp(m)$. Let $\sigma^{l}$ denote the $l$-th symmetric tensor power of the matrix multiplication representation (over complex numbers) of $Sp(1)$ (hence $\mu_{1}=\sigma^{1}$). Let $\mathfrak{t}^{l}$ denote the complex representation of $U(1)$ of weight $l$. Define the inner product for $\mathfrak{g}$ as $Q(X,Y)=-\mathrm{tr}(XY)$. Note that $Q$ is non-degenerate on $\mathfrak{g}$ and equal to a multiple of the Killing form of $\mathfrak{sp}(m+1)$ when restricted to $Sp(m+1)$.

The action of $G$ on $T_{eK}(G/K)$ is equivalent to the adjoint action of $G$ on $\mathfrak{g}/\mathfrak{k}$. Let $(G,H,K)=(Sp(m+1)U(1),Sp(m)Sp(1)U(1),Sp(m)\Delta U(1))$. We have the following $Q$-orthogonal decomposition for $\mathfrak{g}$.

$$\begin{split}\mathfrak{g}&=\mathfrak{h}\oplus[\mu_{m}\otimes\sigma^{1}]_{\mathbb{R}}\quad\text{as a $H$-module}\\ &=\mathfrak{l}\oplus[\mathfrak{t}^{2}]_{\mathbb{R}}\oplus[\mu_{m}\otimes\mathfrak{t}^{1}]_{\mathbb{R}}\quad\text{as an $L$-module}\\ &=\mathfrak{k}\oplus\mathbb{I}\oplus[\mathfrak{t}^{2}]_{\mathbb{R}}\oplus[\mu_{m}\otimes\mathfrak{t}^{1}]_{\mathbb{R}}\quad\text{as a $K$-module}\end{split}.$$

(2.1)

Consider $Sp(m+1)=U(2m+2)\cap Sp(2m+2;\mathbb{C})$ and embed $G$ in $Sp(2m+4;\mathbb{C})$. Identify $\mathbb{H}^{m+1}$ with $\mathbb{C}^{2m+2}=\mathbb{C}^{m+1}\oplus j\mathbb{C}^{m+1}$. The isotropy representation of $G/K$ hence has a $Q$-orthonormal basis $\{E_{1},E_{2},E_{3},E_{l_{1}l_{2}}\mid l_{1}=1,\dots,m,\quad l_{2}=1,2,3,4\}$, where

$$\begin{split}&E_{1}=\frac{1}{2}\begin{bmatrix}O&&&&&&\\ &\ddots&&&&&\\ &&O&&&&\\ &&&-i&0&&\\ &&&0&i&&\\ &&&&&i&0\\ &&&&&0&-i\\ \end{bmatrix},\\ &E_{2}=\frac{1}{\sqrt{2}}\begin{bmatrix}O&&&\\ &\ddots&&&&\\ &&O&&&\\ &&&0&1&\\ &&&-1&0&\\ &&&&&O\\ \end{bmatrix},\quad E_{3}=\frac{1}{\sqrt{2}}\begin{bmatrix}O&&&&&\\ &\ddots&&&&\\ &&O&&&\\ &&&0&i&\\ &&&i&0&\\ &&&&&O\\ \end{bmatrix},\end{split}$$

(2.2)

and each $E_{l_{1}l_{2}}$ is given by

$$E_{l_{1}l_{2}}=\frac{1}{2}\begin{bmatrix}O&&&&O\\ &\ddots&&&\vdots\\ &&\ddots&&A_{l_{2}}\\ &&&\ddots&\vdots\\ O&\dots&-A_{l_{2}}^{*}&\dots&O\\ &&&&&O\\ \end{bmatrix},\quad l_{1}=1,\dots,m,\quad l_{2}=1,2,3,4,$$

with

$$O=\begin{bmatrix}0&\\ &0\end{bmatrix},\quad A_{1}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix},\quad A_{2}=\begin{bmatrix}i&0\\ 0&-i\end{bmatrix},\quad A_{3}=\begin{bmatrix}0&i\\ i&0\end{bmatrix},\quad A_{4}=\begin{bmatrix}0&-1\\ 1&0\end{bmatrix}.$$

The trivial representation $\mathbb{I}$ is spanned by $E_{1}$, which is orthogonal to $\mathfrak{k}$. Note that $Q=-\frac{1}{4}B_{1}=-\frac{1}{2m+4}B_{2}$, where $B_{1}$ and $B_{2}$ are respectively the Killing form for $Sp(2;\mathbb{C})$ and $Sp(2m+2;\mathbb{C})$. We abuse the notation by using $Q$ to denote the invariant metric on $G$ that is induced by the inner product. Take $Q$ as the background metric. By Schur’s Lemma, an invariant metric on $G/K$ has the form of

$$g_{G/K}=a^{2}\left.Q\right|_{\mathbb{I}}+b^{2}\left.Q\right|_{[\mathfrak{t}^{2}]_{\mathbb{R}}}+c^{2}\left.Q\right|_{\mathfrak{g}/\mathfrak{h}}.$$

(2.3)

By Corollary 7.39 in [Bes08], the formula of the scalar curvature for $g_{G/K}$ is

$$R_{s}=\frac{4}{b^{2}}+\frac{4m(m+2)}{c^{2}}-\frac{1}{2}\frac{a^{2}}{b^{4}}-\frac{m}{4}\dfrac{a^{2}}{c^{4}}-m\dfrac{b^{2}}{c^{4}}.$$

Compute the first variation of the Hilbert–Einstein functional on $G/K$. The Ricci endomorphism is given by

$$\begin{split}r_{a}&=\frac{1}{2}\frac{a^{2}}{b^{4}}+\frac{m}{4}\frac{a^{2}}{c^{4}}\\ r_{b}&=\frac{2}{b^{2}}-\frac{1}{2}\frac{a^{2}}{b^{4}}+\frac{m}{2}\frac{b^{2}}{c^{4}}\\ r_{c}&=\frac{m+2}{c^{2}}-\frac{1}{8}\frac{a^{2}}{c^{4}}-\frac{1}{2}\frac{b^{2}}{c^{4}}\end{split}.$$

(2.4)

Note that $M\backslash(G/H)$ and $\mathbb{R}^{4m+4}\backslash\{0\}$ are both $G$-diffeomorphic to $(0,\infty)\times G/K$. We construct Einstein metrics $g=dt^{2}+g_{G/K}(t)$ by setting $(0,\infty)$ as a geodesic and assign $G$-invariant metric $g_{G/K}$ on each $\{t\}\times G/K$. Then (2.3) becomes a $S^{2}(\mathfrak{g}/\mathfrak{k})^{K}$-valued function on $t$, where $S^{2}(\mathfrak{g}/\mathfrak{k})^{K}$ is the space of $K$-invariant symmetric 2-tensor. By [EW00], the cohomogeneity one Einstein system is

$$\begin{split}\frac{\ddot{a}}{a}-\left(\frac{\dot{a}}{a}\right)^{2}&=-\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}\right)\frac{\dot{a}}{a}+\frac{1}{2}\frac{a^{2}}{b^{4}}+\frac{m}{4}\frac{a^{2}}{c^{4}}-\Lambda\\ \frac{\ddot{b}}{b}-\left(\frac{\dot{b}}{b}\right)^{2}&=-\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}\right)\frac{\dot{b}}{b}+\frac{2}{b^{2}}-\frac{1}{2}\frac{a^{2}}{b^{4}}+\frac{m}{2}\frac{b^{2}}{c^{4}}-\Lambda\\ \frac{\ddot{c}}{c}-\left(\frac{\dot{c}}{c}\right)^{2}&=-\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}\right)\frac{\dot{c}}{c}+\frac{m+2}{c^{2}}-\frac{1}{8}\frac{a^{2}}{c^{4}}-\frac{1}{2}\frac{b^{2}}{c^{4}}-\Lambda\\ \end{split}$$

(2.5)

with conservation law

$$\begin{split}&\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}\right)^{2}-\left(\frac{\dot{a}}{a}\right)^{2}-2\left(\frac{\dot{b}}{b}\right)^{2}-4m\left(\frac{\dot{c}}{c}\right)^{2}=R_{s}-(4m+2)\Lambda\end{split}.$$

(2.6)

There are three possible initial conditions for (2.5). The first possibility is having $G/H$ as the singular orbit. The cohomogeneity one manifold $M$ is an $\mathbb{R}^{4}$-bundle over $\mathbb{HP}^{m}$. The principal orbit $G/K$ becomes the zero section $G/H$ as $t\to 0$. In order to smoothly extend the metric on the tubular neighbourhood around $G/H$, we have the following proposition.

Another possible initial condition is $G/K$ collapsing to a singleton as $t\to 0$. Since $G/K=\mathbb{S}^{4m+3}$, the cohomogeneity one manifold is topologically $\mathbb{R}^{4m+4}$. In order to extend the metric on the neighborhood of the origin of $\mathbb{R}^{4m+4}$, we have the following proposition.

Note that $G/K$ admits two homogeneous Einstein metrics. Hence for a cohomogeneity one metric of Taub-NUT type, $G/K$ can also degenerate to a point as a Jensen sphere [Jen73]. Then the corresponding initial condition is given by

$$\lim_{t\to 0}(a,b,c,\dot{a},\dot{b},\dot{c})=\left(0,0,0,\beta,\frac{\sqrt{2}}{2}\beta,\frac{\sqrt{2m+3}}{2}\beta\right),$$

(2.9)

where $(4m+3)(4m+2)\beta^{2}=6+\frac{16m(m+2)(2m+3)-12m}{(2m+3)^{2}}$. In other words, if

$$g_{G/K}(t)=\beta^{2}t^{2}\left(\left.Q\right|_{\mathbb{I}}+\frac{1}{2}\left.Q\right|_{[\mathfrak{t}^{2}]_{\mathbb{R}}}+\frac{2m+3}{4}\left.Q\right|_{\mathfrak{g}/\mathfrak{h}}\right),$$

then $dt^{2}+g_{G/K}(t)$ is a singular cone metric on $\mathbb{R}^{4m+4}$ with the Jensen sphere $\mathbb{S}^{4m+3}$ as its base.

As pointed out in Remark 2.9 in [Chi19b], in the Ricci-flat case, changing $h$ in (2.7) is essentially the homothetic change of the solution around $G/H$. Moreover, (2.7) does not fully determine the metric in a tubular neighborhood of $G/H$. This is also the case for (2.8). Using Lemma 1.1 in [EW00], we can prove that there exists a free parameter for $a-b$ of order 3 for (2.7) and (2.8). We consider (2.7) bellow. Statements concerning (2.8) can be obtained without substantial change of the argument.

We first rephrase Lemma 1.1 in [EW00] for $M$ below.

Since $\chi=\left[\sigma^{1}\otimes\mathfrak{t}^{1}\right]_{\mathbb{R}}$ and $\mathfrak{g}/\mathfrak{h}=\left[\mu_{m}\otimes\sigma^{1}\right]_{\mathbb{R}}$ are inequivalent, we have decomposition

$$W_{i}=W_{i}^{+}\oplus W_{i}^{-}:=\mathrm{Hom}(S^{i}(\chi),S^{2}(\chi))^{H}\oplus\mathrm{Hom}(S^{i}(\chi),S^{2}(\mathfrak{g}/\mathfrak{h}))^{H}$$

By induction, we have

$$\begin{split}S^{2k}(\chi)\otimes\mathbb{C}&=\sum_{l=0}^{k-1}\sum_{j=0}^{l}\left(\sigma^{2k-2j}\otimes\mathfrak{t}^{2k-2l}+\sigma^{2k-2j}\otimes\mathfrak{t}^{-(2k-2l)}\right)+\sum_{l=0}^{k}\sigma^{2k-2l}\\ S^{2k+1}(\chi)\otimes\mathbb{C}&=\sum_{l=0}^{k}\sum_{j=0}^{l}\left(\sigma^{2k+1-2j}\otimes\mathfrak{t}^{2k+1-2l}+\sigma^{2k+1-2j}\otimes\mathfrak{t}^{-(2k+1-2l)}\right)\end{split}$$

(2.10)

as $H$-modules. In particular, we have

$$S^{2}(\chi)=[\sigma^{2}\otimes\mathfrak{t}^{2}]_{\mathbb{R}}+[\sigma^{2}]_{\mathbb{R}}+\mathbb{I}.$$

We also have

$$S^{2}(\mathfrak{g}/\mathfrak{h})=\left\{\begin{array}[]{ll}\left[\mu_{m}^{2}\otimes\sigma^{2}\right]_{\mathbb{R}}+\left[\mu_{m}\mathring{\wedge}\mu_{m}\right]_{\mathbb{R}}+\mathbb{I}&m\neq 1\\ \left[\sigma^{2}\otimes\sigma^{2}\right]_{\mathbb{R}}+\mathbb{I}&m=1\\ \end{array}\right.,$$

(2.11)

where $\left[\mu_{m}\mathring{\wedge}\mu_{m}\right]_{\mathbb{R}}+\mathbb{I}=\left[\mu_{m}\wedge\mu_{m}\right]_{\mathbb{R}}$. Hence it is clear that

$$W_{2k}^{+}=\left\{\begin{array}[]{ll}\mathbb{R}&k=0\\ \mathbb{R}^{3}&k\geq 1\end{array}\right.\quad W_{2k+1}^{+}=0\quad W_{2k}^{-}=\mathbb{R}\quad W_{2k+1}^{-}=0.$$

Inspired by [DW09a][DW09b], we apply coordinate change $d\eta=\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}\right)dt$. The quantity $\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}$ is the trace of the shape operator of the hypersurface orbit. Define

$$\begin{split}&X_{1}=\frac{\frac{\dot{a}}{a}}{\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}},\quad X_{2}=\frac{\frac{\dot{b}}{b}}{\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}},\quad X_{3}=\frac{\frac{\dot{c}}{c}}{\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}},\\ &Y_{1}=\frac{a}{b},\quad Y_{2}=\frac{\frac{1}{b}}{\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}},\quad Y_{3}=\frac{\frac{b}{c^{2}}}{\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}},\quad\tilde{W}=\frac{1}{\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+4m\frac{\dot{c}}{c}}.\end{split}$$

(2.14)

Define functions on $\eta$

$$\begin{split}&R_{1}=\frac{1}{2}Y_{1}^{2}Y_{2}^{2}+\frac{m}{4}Y_{1}^{2}Y_{3}^{2}\\ &R_{2}=2Y_{2}^{2}-\frac{1}{2}Y_{1}^{2}Y_{2}^{2}+\frac{m}{2}Y_{3}^{2}\\ &R_{3}=(m+2)Y_{2}Y_{3}-\frac{1}{8}Y_{1}^{2}Y_{3}^{2}-\frac{1}{2}Y_{3}^{2}\\ &R_{s}=R_{1}+2R_{2}+4mR_{3},\quad G=X_{1}^{2}+2X_{2}^{2}+4mX_{3}^{2}\end{split}.$$

(2.15)

Let ^′ denote the derivative with respect to $\eta$. The Einstein equations (2.5) become a polynomial system

$$\begin{split}\begin{bmatrix}X_{1}\\ X_{2}\\ X_{3}\\ Y_{1}\\ Y_{2}\\ Y_{3}\\ \tilde{W}\end{bmatrix}^{\prime}&=V(X_{1},X_{2},X_{3},Y_{1},Y_{2},Y_{3},\tilde{W})=\begin{bmatrix}X_{1}(G+\Lambda\tilde{W}^{2}-1)+R_{1}-\Lambda\tilde{W}^{2}\\ X_{2}(G+\Lambda\tilde{W}^{2}-1)+R_{2}-\Lambda\tilde{W}^{2}\\ X_{3}(G+\Lambda\tilde{W}^{2}-1)+R_{3}-\Lambda\tilde{W}^{2}\\ Y_{1}(X_{1}-X_{2})\\ Y_{2}(G+\Lambda\tilde{W}^{2}-X_{2})\\ Y_{3}(G+\Lambda\tilde{W}^{2}+X_{2}-2X_{3})\\ \tilde{W}(G+\Lambda\tilde{W}^{2})\end{bmatrix}\\ \end{split},$$

(2.16)

with conservation law (2.6) becomes

$$\begin{split}\mathcal{C}:1-G&=R_{s}-(4m+2)\Lambda\tilde{W}^{2}\end{split}$$

(2.17)

It is clear that $X_{1}+2X_{2}+4mX_{3}\equiv 1$ from the definition of coordinate change. In fact, let

$$\mathcal{H}=\{(X_{1},X_{2},X_{3},Y_{1},Y_{2},Y_{3},\tilde{W})\mid X_{1}+2X_{2}+4mX_{3}=1\},$$

one can check that $\mathcal{C}\cap\mathcal{H}\cap\{\tilde{W}\geq 0\}$ is a flow-invariant $5$-dimensional manifold in $\mathbb{R}^{7}$ with a $4$-dimensional boundary $\mathcal{C}\cap\mathcal{H}\cap\{\tilde{W}\equiv 0\}$.