$\mathrm{Spin}(7)$ metrics of cohomogeneity one with Aloff–Wallach spaces as principal orbits

Metrics with $\mathrm{Spin}(7)$ holonomy are interesting subjects in differential geometry and theoretical physics. The first example was constructed in [Bry87] on the cone over Berger space $SO(5)/SO(3)$. The first complete example was constructed in [BS89] and [GPP90] on the spinor bundle over $\mathbb{S}^{4}$. The first compact example was constructed in [Joy96]. In recent years, $\mathrm{Spin}(7)$ metrics with an Aloff–Wallach space as principal orbit became an active field of research. It is expected that a cohomogeneity one space with an Aloff–Wallach space as its principal orbit admits a continuous 1-parameter family of $\mathrm{Spin}(7)$ metrics with asymptotically locally conical (ALC) asymptotics: each metric’s asymptotic limit is a product between a 7-dimensional Ricci-flat cone and a circle with fixed radius $l>0$. Moreover, with $l\to\infty$, the metric converges to an asymptotically conical (AC) one: its asymptotic limit is an 8-dimensional cone. In this way, one has a continuous 1-parameter family of metrics with non-maximal volume growth and its boundary is a metric with maximal volume growth.

An Aloff–Wallach space is a homogeneous space $N_{k,l}:=SU(3)/U(1)_{k,l}$, where $U(1)_{k,l}$ is embedded in $SU(3)$ as $\mathrm{diag}\left(e^{k\sqrt{-1}t},e^{l\sqrt{-1}t},e^{-(k+l)\sqrt{-1}t}\right)$ with integers $k$ and $l$. Without loss of generality, we assume that $k$ and $l$ are coprime. Using outer automorphism and Weyl group of $SU(3)$ to normalize $(k,l)$, we assume $k\geq l\geq 0$ in this article. Due to the geometric differences, we call $N_{k,l}$ an exceptional Aloff–Wallach space if $kl(k-l)=0$ and a generic Aloff–Wallach space if otherwise. With an Aloff–Wallach space as the principal orbit, one can reduce the Einstein equations to a system of nonlinear second order ODEs. The $\mathrm{Spin}(7)$ condition, being Ricci-flat, is reduced to a system of first order ODEs. The isolated explicit solution of cohomogeneity one $\mathrm{Spin}(7)$ metric with $N_{1,0}$ as principal orbit and $\mathbb{CP}^{2}$ as singular orbit was given in [GS02]. In [Rei11], the author proved that singular orbit of a cohomogeneity one $\mathrm{Spin}(7)$ metric with a fixed $N_{k,l}$ can only be either $\mathbb{S}^{5}$ or $\mathbb{CP}^{2}$. The singular orbit $\mathbb{S}^{5}$ appears exclusively for the case $N_{1,0}$. Recently, some ALC $\mathrm{Spin}(7)$ metrics with $N_{1,0}$ as principal orbit and $\mathbb{S}^{5}$ as singular orbit were constructed in [Fos21]. The setting with $N_{1,0}$ as the principal orbit was further studied in [Leh20], in which examples in [GS02] and [Fos21] were extended to two continuous families of ALC $\mathrm{Spin}(7)$ metrics. The boundary of each family is an AC $\mathrm{Spin}(7)$ metric. Explicit solutions with $N_{1,1}$ as principal orbit were given in [CGLP01] and [KY02a]. In [Baz07] and [Baz08], it is observed that formulas for $\mathrm{Spin}(7)$ condition in [CGLP02a] and [CGLP02b] can be applied in a broader setting where the cohomogeneity one manifold is deformed from a HyperKähler cone. Specifically, by using the 3-Sasakian structure on $N_{1,1}$, a 2-parameter family of $\mathrm{Spin}(7)$ metrics was constructed on a complex line bundle over $SU(3)/T^{2}$ in [Baz07] and a 2-parameter family of $\mathrm{Spin}(7)$ metrics was constructed on a quaternionic line bundle over $\mathbb{CP}^{2}$ in [Baz08]. Explicit isolated solutions with a generic $N_{k,l}$ as principal orbit are included in [KY02a] and[CGLP02a].

This article aims to prove the global existence of two continuous 1-parameter families of $\mathrm{Spin}(7)$ metrics of cohomogeneity one with any Aloff–Walach spaces $N_{k,l}$ as principal orbit. Past examples mentioned above are partially recovered in the construction. Specifically, a fixed $N_{k,l}$ can be viewed as fiber bundles over $\mathbb{CP}^{2}$ with lens spaces $L(1,|k+l|)$, $L(1,|k|)$ and $L(1,|l|)$ as fibers, respectively. Let $i\in\{k+l,k,l\}$ and $i\neq 0$, define $M_{k,l}^{(i)}$ to be the $\mathbb{R}^{4}/\mathbb{Z}_{|i|}$ bundle over $\mathbb{CP}^{2}$ with $N_{k,l}$ as principal orbit. We prove the following theorems.

Note that for a generic $N_{k,l}$ as the principal orbit, metrics $\gamma_{(s_{1},s_{2})}^{(k+l)}$ on $M_{k,l}^{(k+l)}$ do not include the explicit ALC $\mathrm{Spin}(7)$ metrics in [CGLP02a] and [KY02a]. Hence it is likely to extend the continuous family of ALC metrics in Theorem 1.1 to a larger family using other methods. Metrics $\gamma_{(s_{1},s_{2})}^{(k)}$ on $M_{k,l}^{(k)}$ are new to the best knowledge of the author.

It is worth mentioning that our method for proving Theorem 1.1 also applies for the case of exceptional $N_{1,0}$. Only ALC metrics are constructed in this way. For each $(s_{1},s_{2})$, the obtained metrics $\gamma_{(s_{1},s_{2})}^{(1)}$ with both chiralities on $M_{1,0}^{(1)}$ can be identified by a global $\mathbb{Z}_{2}$ symmetry. They are part of the results obtained in [Leh20]. They have smooth extension to the singular orbit $\mathbb{CP}^{2}$. For a more complete study on the $\mathrm{Spin}(7)$ metrics with $N_{1,0}$ as principal orbit, please see [Fos21] and [Leh20] for more details.

We also consider the case where the principal orbit is the exceptional $N_{1,1}$

Metrics $\gamma_{(s_{1},s_{2})}^{(2)}$ on the orbifold $M_{1,1}^{(2)}$ are the ALC $\mathrm{Spin}(7)$ metrics in [CGLP01] and some of the solutions in [Baz07]. It is worth mentioning that $M_{1,1}^{(1)}$ is in fact $T^{*}\mathbb{CP}^{2}$. Metrics $\gamma_{(s_{1},s_{2})}^{(1)}$ on $M_{1,1}^{(1)}$ are the ones that were conjectured in [KY02b]. For most of the cases, we do not have metrics that shows geometric transition from AC $\mathrm{Spin}(7)$ metrics to ALC $\mathrm{Spin}(7)$ metrics like the ones that occur in [CGLP02a] and [Leh20]. However, if the principal orbit is $N_{1,1}$, we do have two continuous 1-parameter families of $\mathrm{Spin}(7)$ metrics that have $AC$ metrics on their respective boundaries. In particular, the AC metric on $M_{1,1}^{(1)}$ is the Calabi HyperKähler metric in [Cal79]. In other words, the 1-parameter family of ALC $\mathrm{Spin}(7)$ metrics on $M_{1,1}^{(1)}$ can be desingularized with Calabi’s AC metric.

This article is structured as the following. In Section 2, we study the Ricci-flat system with an $N_{k,l}$ as the principal orbit and apply coordinate change that is similar to the one in [Chi19b] and [Chi21]. The coordinate change transforms the singular orbit $\mathbb{CP}^{2}$, the AC asymptotics, and the ALC asymptotics to critical points of a polynomial system. The singular orbit $\mathbb{S}^{5}$, uniquely appears in the $N_{1,0}$ case, is blown up to the infinity in the new coordinate. Although being more complicated than the $\mathrm{Spin}(7)$ subsystem, the Ricci-flat system has some important estimate that is not obvious in the $\mathrm{Spin}(7)$ subsystem.

In Section 3, we study critical points of the $\mathrm{Spin}(7)$ subsystem and recover the locally existing result in [Rei11]. The cohomogeneity one $\mathrm{Spin}(7)$ metrics are represented by integral curves that emanates from various critical points.

In Section 4, we prove the global existence and the asymptotic limit by constructing compact invariant sets. The construction boils down to a classical algebraic geometry problem, which requires one to show non-negativity of a resultant polynomial. For readability, the very technical computation and formula are presented in the Appendix.

Acknowledgements. The author would like to thank NSFC for partial support under grants No. 11521101 and NO. 12071489. Thanks also go to Michael Baker, Jesse Madnick and McKenzie Wang for useful discussions.

In this section, we study the cohomogeneity one Ricci-flat system on $M^{(i)}_{k,l}$ derived in [Rei11]. We apply a coordinate change so that the system becomes a system of first order polynomial ODEs. The $\mathrm{Spin}(7)$ condition, originally a first order subsystem, is then transformed to a set of algebraic equation in the new coordinate.

Consider an Aloff–Wallach space $N_{k,l}$, we fix the basis for $\mathfrak{su}(3)$ as the one in [Rei11]. Specifically, we work with the basis

$$\begin{split}&e_{1}:=\begin{bmatrix}0&1&0\\ -1&0&0\\ 0&0&0\end{bmatrix},\quad e_{2}:=\sqrt{-1}\begin{bmatrix}0&1&0\\ 1&0&0\\ 0&0&0\end{bmatrix},\quad e_{3}:=\begin{bmatrix}0&0&1\\ 0&0&0\\ -1&0&0\end{bmatrix},\quad e_{4}:=\sqrt{-1}\begin{bmatrix}0&0&1\\ 0&0&0\\ 1&0&0\end{bmatrix}\\ &e_{5}:=\begin{bmatrix}0&0&0\\ 0&0&1\\ 0&-1&0\end{bmatrix},\quad e_{6}:=\sqrt{-1}\begin{bmatrix}0&0&0\\ 0&0&1\\ 0&1&0\end{bmatrix},\\ &e_{7}:=\sqrt{-1}\begin{bmatrix}2l+k&0&0\\ 0&-(2k+l)&0\\ 0&0&k-l\end{bmatrix},\quad e_{8}:=\sqrt{-1}\begin{bmatrix}k&0&0\\ 0&l&0\\ 0&0&-(k+l)\end{bmatrix},\\ \end{split}$$

(2.1)

where $e_{8}$ generates $\mathfrak{u}(1)_{k,l}$. The isotropy representation $\mathfrak{su}(3)/\mathfrak{u}(1)_{k,l}$ is decomposed as

$$\mathfrak{su}(3)/\mathfrak{u}(1)_{k,l}=\mathfrak{i}\oplus\mathfrak{t}^{k-l}\oplus\mathfrak{t}^{2k+l}\oplus\mathfrak{t}^{k+2l},$$

where

$$\mathfrak{t}^{k-l}=\mathrm{span}\{e_{1},e_{2}\},\quad\mathfrak{t}^{2k+l}=\mathrm{span}\{e_{3},e_{4}\},\quad\mathfrak{t}^{k+2l}=\mathrm{span}\{e_{5},e_{6}\},\quad\mathfrak{i}=\mathrm{span}\{e_{7}\}.$$

Recall our convention of Aloff–Wallach spaces, we set $k$ and $l$ to be coprime and $k\geq l\geq 0$ without loss of generality. Under this setting, if $N_{k,l}$ is generic, i.e., $kl(k-l)\neq 0$, one can conclude that all $\mathfrak{t}$’s are non-trivial and have different weights. The isotropy representation then consists of 4 inequivalent irreducible summands. The group action of $SU(3)$ on (2.1) generates a frame for an $SU(3)$-invariant Einstein metric $g_{N_{k,l}}$. The matrix representation of $g_{N_{k,l}}$ is hence

$$\begin{bmatrix}a^{2}&0&&&&&\\ 0&a^{2}&&&&&&\\ &&b^{2}&0&&&\\ &&0&b^{2}&&&\\ &&&&c^{2}&0&\\ &&&&0&c^{2}&\\ &&&&&&f^{2}\end{bmatrix}$$

(2.2)

for some $a,b,c,f\neq 0$. Consider the cohomogeneity one metric $g=dt^{2}+g_{N_{k,l}}(t)$, where each component in (2.2) is a function of $t$. Methods in [EW00] can be applied to derive the cohomogeneity one Einstein system. Let $\Delta=k^{2}+kl+l^{2}$. As shown in [Rei11], the cohomogeneity one Ricci-flat system is

$$\begin{split}\frac{\ddot{a}}{a}-\left(\frac{\dot{a}}{a}\right)^{2}&=-\left(2\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+2\frac{\dot{c}}{c}+\frac{\dot{f}}{f}\right)\frac{\dot{a}}{a}+\frac{6}{a^{2}}+\frac{a^{2}}{b^{2}c^{2}}-\frac{b^{2}}{a^{2}c^{2}}-\frac{c^{2}}{a^{2}b^{2}}-\frac{1}{2}\frac{(k+l)^{2}}{\Delta^{2}}\frac{f^{2}}{a^{4}}\\ \frac{\ddot{b}}{b}-\left(\frac{\dot{b}}{b}\right)^{2}&=-\left(2\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+2\frac{\dot{c}}{c}+\frac{\dot{f}}{f}\right)\frac{\dot{b}}{b}+\frac{6}{b^{2}}+\frac{b^{2}}{a^{2}c^{2}}-\frac{c^{2}}{a^{2}b^{2}}-\frac{a^{2}}{b^{2}c^{2}}-\frac{1}{2}\frac{l^{2}}{\Delta^{2}}\frac{f^{2}}{b^{4}}\\ \frac{\ddot{c}}{c}-\left(\frac{\dot{c}}{c}\right)^{2}&=-\left(2\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+2\frac{\dot{c}}{c}+\frac{\dot{f}}{f}\right)\frac{\dot{c}}{c}+\frac{6}{c^{2}}+\frac{c^{2}}{a^{2}b^{2}}-\frac{a^{2}}{b^{2}c^{2}}-\frac{b^{2}}{a^{2}c^{2}}-\frac{1}{2}\frac{k^{2}}{\Delta^{2}}\frac{f^{2}}{c^{4}}\\ \frac{\ddot{f}}{f}-\left(\frac{\dot{f}}{f}\right)^{2}&=-\left(2\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+2\frac{\dot{c}}{c}+\frac{\dot{f}}{f}\right)\frac{\dot{f}}{f}+\frac{1}{2}\frac{(k+l)^{2}}{\Delta^{2}}\frac{f^{2}}{a^{4}}+\frac{1}{2}\frac{l^{2}}{\Delta^{2}}\frac{f^{2}}{b^{4}}+\frac{1}{2}\frac{k^{2}}{\Delta^{2}}\frac{f^{2}}{c^{4}}\end{split}$$

(2.3)

with conservation law

$$\begin{split}&\left(2\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}+2\frac{\dot{c}}{c}+\frac{\dot{f}}{f}\right)^{2}-2\left(\frac{\dot{a}}{a}\right)^{2}-2\left(\frac{\dot{b}}{b}\right)^{2}-2\left(\frac{\dot{c}}{c}\right)^{2}-\left(\frac{\dot{f}}{f}\right)^{2}\\ &=12\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\right)-2\left(\frac{a^{2}}{b^{2}c^{2}}+\frac{b^{2}}{a^{2}c^{2}}+\frac{c^{2}}{a^{2}b^{2}}\right)-\frac{1}{2}\frac{(k+l)^{2}}{\Delta^{2}}\frac{f^{2}}{a^{4}}-\frac{1}{2}\frac{l^{2}}{\Delta^{2}}\frac{f^{2}}{b^{4}}-\frac{1}{2}\frac{k^{2}}{\Delta^{2}}\frac{f^{2}}{c^{4}}.\end{split}$$

(2.4)

The conserved quantity (2.4) is essentially equivalent to $g$ having zero scalar curvature. Note that the RHS of (2.4) is the scalar curvature of $(N_{k,l},g_{N_{k,l}}(t))$.

If $kl(k-l)=0$, then by our convention, we either have $(k,l)=(1,0)$ or $(k,l)=(1,1)$. In the first case, there are two equivalent isotropy summands $\mathfrak{t}^{1}$ in $\mathfrak{su}(3)/\mathfrak{u}_{1,0}$. In the latter case, the isotropy representation $\mathfrak{su}(3)/\mathfrak{u}_{1,1}$ consists of three trivial representations and two equivalent isotropy summands $\mathfrak{t}^{3}$. Therefore, $SU(3)$ invariant metrics on $N_{1,1}$ and $N_{1,0}$ are not necessarily diagonal.

For $N_{1,0}$, the matrix representation of an $SU(3)$-invariant $g_{N_{1,0}}$ is

$$\begin{bmatrix}a^{2}&0&&&A_{1}&A_{2}\\ 0&a^{2}&&&-A_{2}&A_{1}\\ &&b^{2}&0&&&\\ &&0&b^{2}&&&\\ A_{1}&-A_{2}&&&c^{2}&0&\\ A_{2}&A_{1}&&&0&c^{2}&\\ &&&&&&f^{2}\end{bmatrix}.$$

(2.5)

From [Rei08] we learn that $N_{SU(3)}(U_{1,0}(1))=U(1)^{2}$. One can further reduce the number of functions from 6 to 5, since the residual action of $U(1)$ on $g_{N_{1,0}}$ changes $A_{1}$ and $A_{2}$ while leaving diagonal entries unchanged. The metric $g_{N_{1,0}}$ is $SU(3)\times U(1)$ invariant if and only if its diagonal. In other words, with the enhanced $SU(3)\times U(1)$ symmetry, the dynamic system is simply $\eqref{eqn: original Einstein equation}$ and $\eqref{eqn: original conservation}$ with $(k,l)=(1,0).$

For $N_{1,1}$, the matrix representation of an $SU(3)$-invariant $g_{N_{1,1}}$ is

$$\begin{bmatrix}a_{1}^{2}&A_{1}&&&&&A_{2}\\ A_{1}&a_{2}^{2}&&&&&A_{3}\\ &&b^{2}&0&B_{1}&B_{2}&\\ &&0&b^{2}&-B_{2}&B_{1}&\\ &&B_{1}&-B_{2}&c^{2}&0&\\ &&B_{2}&B_{1}&0&c^{2}&\\ A_{2}&A_{3}&&&&&f^{2}\end{bmatrix}.$$

(2.6)

The normalizer $N_{SU(3)}(U_{1,1}(1))$ is isomorphic to $U(2)$ and $N_{SU(3)}(U_{1,1}(1))/U_{1,1}(1)\cong SU(2)$. The residual action of $SU(2)$ acts as $SO(3)$ on the three trivial representations [Rei08]. Therefore, one can use the $SU(2)$ to partially diagonalize $g_{N_{1,1}}$ so that all $A_{i}$’s vanish and the number of functions is reduced from 10 to 7. A $SU(3)\times SU(2)$-invariant $g_{N_{1,1}}$ may still not be diagonal. Nevertheless, one can consider the case with diagonal $g_{N_{1,1}}$, as shown in $(4.10)$ in [Rei11]. In this article, we further impose the condition $a_{1}=a_{2}$ and study (2.3) and (2.4) with $(k,l)=(1,1)$. One can check that such a dynamic system is a subsystem of $(4.10)$ in [Rei11]. For the case where $a_{1}$ and $a_{2}$ are unequal, please see [Baz07] for more details.

Singular orbits of cohomogeneity one space $M_{k,l}^{(k+l)}$, $M_{k,l}^{(l)}$ and $M_{k,l}^{(k)}$ are generated by $\mathfrak{t}^{2k+l}\oplus\mathfrak{t}^{k+2l}$, $\mathfrak{t}^{k-l}\oplus\mathfrak{t}^{k+2l}$ and $\mathfrak{t}^{k-l}\oplus\mathfrak{t}^{2k+l}$, respectively. Hence according to [EW00] for the case $M_{k,l}^{(1)}$ and the power series in [Rei11] for the rest of the orbifolds, the initial conditions for theses three types of $\mathbb{R}^{4}/\mathbb{Z}_{|i|}$ bundles, are respectively given by

$$\begin{split}&\lim_{t\to 0}(a,b,c,f,\dot{a},\dot{b},\dot{c},\dot{f})=\left(0,a_{0},a_{0},0,1,0,0,\frac{2\Delta}{k+l}\right),\\ &\lim_{t\to 0}(a,b,c,f,\dot{a},\dot{b},\dot{c},\dot{f})=\left(b_{0},0,b_{0},0,0,1,0,\frac{2\Delta}{l}\right),\\ &\lim_{t\to 0}(a,b,c,f,\dot{a},\dot{b},\dot{c},\dot{f})=\left(c_{0},c_{0},0,0,0,0,1,\frac{2\Delta}{k}\right).\end{split}$$

(2.7)

For exceptional Aloff–Wallach spaces, recent development in [VZ20] can also be applied to derive the initial condition even with equivalent isotropy summands.

The $\mathrm{Spin}(7)$ condition is derived in [Rei11] as the following

$$\begin{split}\frac{\dot{a}}{a}&=\frac{b}{ac}+\frac{c}{ab}-\frac{a}{bc}-\frac{k+l}{2\Delta}\frac{f}{a^{2}}\\ \frac{\dot{b}}{b}&=\frac{c}{ab}+\frac{a}{bc}-\frac{b}{ac}+\frac{l}{2\Delta}\frac{f}{b^{2}}\\ \frac{\dot{c}}{c}&=\frac{a}{bc}+\frac{b}{ac}-\frac{c}{ab}+\frac{k}{2\Delta}\frac{f}{c^{2}}\\ \frac{\dot{f}}{f}&=\frac{k+l}{2\Delta}\frac{f}{a^{2}}-\frac{l}{2\Delta}\frac{f}{b^{2}}-\frac{k}{2\Delta}\frac{f}{c^{2}}\end{split}.$$

(2.8)

Change the sign of $f$ in (2.8). We then obtain the $\mathrm{Spin}(7)$ condition with the opposite chirality:

$$\begin{split}\frac{\dot{a}}{a}&=\frac{b}{ac}+\frac{c}{ab}-\frac{a}{bc}+\frac{k+l}{2\Delta}\frac{f}{a^{2}}\\ \frac{\dot{b}}{b}&=\frac{c}{ab}+\frac{a}{bc}-\frac{b}{ac}-\frac{l}{2\Delta}\frac{f}{b^{2}}\\ \frac{\dot{c}}{c}&=\frac{a}{bc}+\frac{b}{ac}-\frac{c}{ab}-\frac{k}{2\Delta}\frac{f}{c^{2}}\\ \frac{\dot{f}}{f}&=-\frac{k+l}{2\Delta}\frac{f}{a^{2}}+\frac{l}{2\Delta}\frac{f}{b^{2}}+\frac{k}{2\Delta}\frac{f}{c^{2}}\end{split}.$$

(2.9)

Although we mainly consider the construction of $\mathrm{Spin}(7)$ metrics in this article, we start with the Ricci-flat system. As shown in the following, with a coordinate change, some important estimates can be quickly derived from the new Ricci-flat system while it is not obvious in the new $\mathrm{Spin}(7)$ subsystem.

It is clear that

$$L:=\begin{bmatrix}\frac{\dot{a}}{a}&0&&&&&\\ 0&\frac{\dot{a}}{a}&&&&&&\\ &&\frac{\dot{b}}{b}&0&&&\\ &&0&\frac{\dot{b}}{b}&&&\\ &&&&\frac{\dot{c}}{c}&0&\\ &&&&0&\frac{\dot{c}}{c}&\\ &&&&&&\frac{\dot{f}}{f}\\ \end{bmatrix}$$

is the second fundamental form of $N_{k,l}$ in $M^{(i)}_{k,l}$ at time $t$. The quantity $\mathrm{tr}{L}$ in (2.3) is hence the mean curvature. In many works on the construction of cohomogeneity one Einstein metrics, the coordinate change $d\eta=\mathrm{tr}{L}dt$ can help to simplify the original Einstein system[DW09][BDW15][Chi19a]. This case is no exception. Define functions

$$\begin{split}&X_{1}=\frac{\frac{\dot{a}}{a}}{\mathrm{tr}{L}},\quad X_{2}=\frac{\frac{\dot{b}}{b}}{\mathrm{tr}{L}},\quad X_{3}=\frac{\frac{\dot{c}}{c}}{\mathrm{tr}{L}},\quad X_{4}=\frac{\frac{\dot{f}}{f}}{\mathrm{tr}{L}},\\ &Z_{1}=\frac{\frac{a}{bc}}{\mathrm{tr}{L}},\quad Z_{2}=\frac{\frac{b}{ac}}{\mathrm{tr}{L}},\quad Z_{3}=\frac{\frac{c}{ab}}{\mathrm{tr}{L}},\quad Z_{4}=f\mathrm{tr}{L}.\end{split}$$

(2.10)

And define functions

$$\begin{split}&\mathcal{G}=2X_{1}^{2}+2X_{2}^{2}+2X_{3}^{2}+X_{4}^{2}\\ &\mathcal{R}_{1}=6Z_{2}Z_{3}+Z_{1}^{2}-Z_{2}^{2}-Z_{3}^{2}-\frac{1}{2}\frac{(k+l)^{2}}{\Delta^{2}}Z_{2}^{2}Z_{3}^{2}Z_{4}^{2}\\ &\mathcal{R}_{2}=6Z_{1}Z_{3}+Z_{2}^{2}-Z_{3}^{2}-Z_{1}^{2}-\frac{1}{2}\frac{l^{2}}{\Delta^{2}}Z_{1}^{2}Z_{3}^{2}Z_{4}^{2}\\ &\mathcal{R}_{3}=6Z_{1}Z_{2}+Z_{3}^{2}-Z_{1}^{2}-Z_{2}^{2}-\frac{1}{2}\frac{k^{2}}{\Delta^{2}}Z_{1}^{2}Z_{2}^{2}Z_{4}^{2}\\ &\mathcal{R}_{4}=\frac{1}{2}\frac{(k+l)^{2}}{\Delta^{2}}Z_{2}^{2}Z_{3}^{2}Z_{4}^{2}+\frac{1}{2}\frac{l^{2}}{\Delta^{2}}Z_{1}^{2}Z_{3}^{2}Z_{4}^{2}+\frac{1}{2}\frac{k^{2}}{\Delta^{2}}Z_{1}^{2}Z_{2}^{2}Z_{4}^{2}\\ &\mathcal{R}_{s}=2\mathcal{R}_{1}+2\mathcal{R}_{2}+2\mathcal{R}_{3}+\mathcal{R}_{4}\end{split}$$

(2.11)

Use ^′ to denote the derivative with respect to $\eta$. (2.3) is transformed to

$$\begin{bmatrix}X_{1}\\ X_{2}\\ X_{3}\\ X_{4}\\ Z_{1}\\ Z_{2}\\ Z_{3}\\ Z_{4}\end{bmatrix}^{\prime}=V(X_{i},Z_{i})\colon=\begin{bmatrix}X_{1}(\mathcal{G}-1)+\mathcal{R}_{1}\\ X_{2}(\mathcal{G}-1)+\mathcal{R}_{2}\\ X_{3}(\mathcal{G}-1)+\mathcal{R}_{3}\\ X_{4}(\mathcal{G}-1)+\mathcal{R}_{4}\\ Z_{1}(\mathcal{G}+X_{1}-X_{2}-X_{3})\\ Z_{2}(\mathcal{G}+X_{2}-X_{3}-X_{1})\\ Z_{3}(\mathcal{G}+X_{3}-X_{1}-X_{2})\\ Z_{4}(-\mathcal{G}+X_{4})\\ \end{bmatrix}$$

(2.12)

The conservation law (2.4) becomes

$$\mathcal{G}-1+\mathcal{R}_{s}=0.$$

(2.13)

Note that $\left(\frac{1}{\mathrm{tr}(L)}\right)^{\prime}=\frac{1}{\mathrm{tr}(L)}\mathcal{G}$. Therefore, $\frac{1}{\mathrm{tr}(L)}$ can be treated as function of $\eta$ by

$$\frac{1}{\mathrm{tr}(L)}=\exp\left(\int_{\eta^{*}}^{\eta}\mathcal{G}d\tilde{\eta}+C\right).$$

To recover the original coordinate, we simply compute

$$t=\int_{\eta^{*}}^{\eta}\frac{1}{\mathrm{tr}(L)}d\tilde{\eta}=\int_{\eta^{*}}^{\eta}\exp\left(\int_{\eta^{**}}^{\eta^{*}}\mathcal{G}d\tilde{\tilde{\eta}}+C\right)d\tilde{\eta}+t_{0}$$

and

$$a=\frac{1}{\mathrm{tr}(L)}\frac{1}{\sqrt{Z_{2}Z_{3}}},\quad b=\frac{1}{\mathrm{tr}(L)}\frac{1}{\sqrt{Z_{1}Z_{3}}},\quad c=\frac{1}{\mathrm{tr}(L)}\frac{1}{\sqrt{Z_{1}Z_{2}}},\quad f=\frac{1}{\mathrm{tr}(L)}Z_{4}$$

From the definition of $X_{i}$’s in (2.10), one expects $2X_{1}+2X_{2}+2X_{3}+X_{4}=1$ is preserved by the new dynamic system. Indeed, since

$$\begin{split}(2X_{1}+2X_{2}+2X_{3}+X_{4})^{\prime}&=(2X_{1}+2X_{2}+2X_{3}+X_{4})(\mathcal{G}-1)+\mathcal{R}_{s}\\ &=(2X_{1}+2X_{2}+2X_{3}+X_{4})(\mathcal{G}-1)+1-\mathcal{G}\quad\text{by \eqref{eqn: new conservation}}\\ &=(2X_{1}+2X_{2}+2X_{3}+X_{4}-1)(\mathcal{G}-1)\end{split},$$

(2.14)

it is clear that

$$\mathcal{H}:=\{2X_{1}+2X_{2}+2X_{3}+X_{4}=1\}$$

is invariant. It is worth mentioning that in many works on constructing cohomogeneity one steady Ricci solitons[DW09][BDW15][Win17], coordinate change $d\eta=(-\dot{u}+\mathrm{tr}{L})dt$ is applied, where $u$ is the potential function. For our case, one obtains the same polynomial system as (2.12) with

$$\mathcal{G}-1+\mathcal{R}_{s}\leq 0,$$

where the equality is needed for (2.14) to hold. From this perspective, one can treat the invariance of $\mathcal{H}$ as the outcome of setting the potential of cohomogeneity one steady Ricci solitons to be a constant. From (2.12), it is clear that we can assume $Z_{i}$’s be non-negative without loss of generality. In fact, the set $\{Z_{1},Z_{2},Z_{3},Z_{4}\geq 0\}$ is invariant. A straightforward observation also gives the following proposition.

Therefore, cohomogeneity one Ricci-flat metrics with an Aloff–Wallach space $N_{k,l}$ as the principal orbit is represented by an integral curve to (2.12) on the following subset of $\mathbb{R}^{8}$:

$$\begin{split}&\mathcal{C}_{RF}\\ &:=\left\{2X_{1}+2X_{2}+2X_{3}+X_{4}=1,\quad\mathcal{G}-1+\mathcal{R}_{s}=0,\quad X_{4}\geq 0,\quad Z_{1},Z_{2},Z_{3},Z_{4}\geq 0\right\},\end{split}$$

(2.16)

a $6$-dimensional algebraic surface with boundary. By Lemma 5.1 in [BDW15], we know that $\lim\limits_{\eta\to\infty}t=\infty$. Therefore, if an integral curve is defined on $\mathbb{R}$, then the Ricci-flat metric represented is forward complete.

We now consider the $\mathrm{Spin}(7)$ condition (2.8) and (2.9) in the new coordinate. The $\mathrm{Spin}(7)$ condition form an invariant subset of $\mathcal{C}_{RF}$ with a lower dimension. Specifically, (2.8) turns to

$$\begin{split}F_{1}&:=X_{1}+Z_{1}-Z_{2}-Z_{3}+\frac{k+l}{2\Delta}Z_{2}Z_{3}Z_{4}=0\\ F_{2}&:=X_{2}+Z_{2}-Z_{3}-Z_{1}-\frac{l}{2\Delta}Z_{1}Z_{3}Z_{4}=0\\ F_{3}&:=X_{3}+Z_{3}-Z_{1}-Z_{2}-\frac{k}{2\Delta}Z_{1}Z_{2}Z_{4}=0\\ F_{4}&:=X_{4}-\frac{k+l}{2\Delta}Z_{2}Z_{3}Z_{4}+\frac{l}{2\Delta}Z_{1}Z_{3}Z_{4}+\frac{k}{2\Delta}Z_{1}Z_{2}Z_{4}=0\end{split}$$

(2.17)

It is known that $\mathrm{Spin}(7)$ metrics are Ricci-flat. Hence (2.8) is a first order subsystem of (2.3). This is can also be shown using the new coordinates. Define $\mathcal{C}^{+}_{\mathrm{Spin}(7)}:=\mathcal{C}_{RF}\cap\left(\bigcap\limits_{i=1}^{4}\left\{F_{i}=0\right\}\right)$. We claim the following.