Rate of asymptotic convergence near isolated singularity of G$_2$ manifold

TL;DR

This paper constructs a G2 holonomy metric with a slow convergence rate to the cone metric near an isolated singularity on a specific geometric space.

Contribution

It introduces a new G2 holonomy metric with a unique slow convergence behavior near an isolated singularity.

Findings

Constructed a G2 metric with slow convergence rate

Analyzed the behavior near the isolated singularity

Provides insights into the geometry of G2 manifolds

Abstract

In this paper, a metric with G$_2$ holonomy and slow rate of convergence to the cone metric is constructed on a ball inside the cone over the flag manifold.

Equations

$${\pi }_{14}^{\varphi }({\ast }_{\varphi }d{\ast }_{\varphi +d\xi }(\varphi +d\xi )+\frac{3}{2}d{d}_{\varphi }^{\ast }\xi )=0,$$

$$L_X\varphi =d(X┘\varphi )\in d{\Omega }_7^{2,\varphi }.$$

$$d(\varphi +d\xi )=d{\ast }_{\varphi +d\xi }(\varphi +d\xi )=0,$$

$$d_{\varphi }^{\ast }\xi =0,$$

$$h_1=i{E}_{11},h_2=i{E}_{22},h_3=i{E}_{33},$$

$$e_1=E_{12}-E_{12},e_3=E_{31}-E_{13},e_5=E_{23}-E_{23}$$

$$e_2=i(E_{12}+E_{12}),e_4=i(E_{31}+E_{13}),e_6=i(E_{23}+E_{23}),$$

$$\omega =e^{12}+e^{34}+e^{56}$$

$$\Omega$$

$$=(e^{246}-e^{136}-e^{235}-e^{145})+i(e^{135}-e^{245}-e^{146}-e^{236}).$$

$$d\omega =3\mathrm{Re}\Omega ,d\mathrm{Im}\Omega =-2{\omega }^2,$$

$${\Lambda }^2{\mathbb{R}}^6={\Lambda }_1^2\oplus {\Lambda }_6^2\oplus {\Lambda }_8^2,$$

$$x_i(u)=h({\mathrm{Ad}}_{u^{-1}}\zeta ,e_i),v_i(u)=h({\mathrm{Ad}}_{u^{-1}}\zeta ,h_i).$$

$$\eta =v_1{e}^{56}+v_2{e}^{34}+v_3{e}^{12},$$

$$\begin{array}{r}d\eta =(x_4{e}^3-x_3{e}^4-x_2{e}^1+x_1{e}^2)\wedge e^{56}+(x_2{e}^1-x_1{e}^2-x_6{e}^5+x_5{e}^6)\wedge e^{34}\\ +(x_6{e}^5-x_5{e}^6-x_4{e}^3+x_3{e}^4)\wedge e^{12},\end{array}$$

$$\left\{\begin{array}[]{l}d^{*}K=d^{*}(JK)=0,\\ \alpha(dJK)=-6K,\end{array}\right.$$

$$dJK=-\frac{1}{3}(d^{\ast }K)\omega +\frac{1}{2}\alpha (dJK)┘\mathrm{Re}\Omega +{\pi }_8(dJK).$$

$$0=ddJK=-\frac{1}{3}(d{d}^{\ast }K)\wedge \omega -(d^{\ast }K)\mathrm{Re}\Omega +\frac{1}{2}d(\alpha (dJK)┘\mathrm{Re}\Omega )+d{\pi }_8(dJK).$$

$$0=-\frac{1}{3}(d{d}^{\ast }K)-\frac{1}{2}(\frac{1}{2}\alpha dJ\alpha dJK+3\alpha dJK)-\frac{1}{2}J{d}^{\ast }{\pi }_8(dJK).$$

$$18JK$$

$$=d^{\ast }d(JK)$$

$$=\frac{1}{3}\ast d\ast [(d^{\ast }K)\omega ]-\frac{1}{2}\ast d\ast [\alpha (dJK)┘\mathrm{Re}\Omega ]+d^{\ast }{\pi }_8(dJK)$$

$$=\frac{1}{3}\ast d[(d^{\ast }K)\frac{{\omega }^2}{2}]+\frac{1}{2}\ast d[J\alpha (dJK)\wedge \mathrm{Re}\Omega ]+d^{\ast }{\pi }_8(dJK)$$

$$=\frac{1}{3}\ast [(d{d}^{\ast }K)\wedge \frac{{\omega }^2}{2}]+\frac{1}{2}\ast [dJ\alpha (dJK)\wedge \mathrm{Re}\Omega ]+d^{\ast }{\pi }_8(dJK)$$

$$=\frac{1}{3}Jd{d}^{\ast }K+\frac{1}{2}J\alpha dJ\alpha dJK+d^{\ast }{\pi }_8(dJK)$$

$$=Jd{d}^{\ast }K+J\alpha dJ\alpha dJK+3J\alpha dJK.$$

$$18d^{\ast }K=-d^{\ast }J(18JK)=d^{\ast }d{d}^{\ast }K+4d^{\ast }K,$$

$$d^{\ast }\alpha dJ=d^{\ast }(J\alpha dJ+4J)J=-4d^{\ast },$$

$$\alpha dJ\alpha dJK+3\alpha dJK=18K.$$

$$d^{\ast }{K}_3=-\frac{1}{4}{d}^{\ast }\alpha dJ{K}_3=-\frac{3}{4}{d}^{\ast }{K}_3,$$

$$d^{\ast }J{K}_3=\frac{1}{3}{d}^{\ast }J\alpha dJ{K}_3=0.$$

$$\overline{△}J{K}_3=△J{K}_3+J\alpha dJ{K}_3=21J{K}_3.$$

$$-\frac{1}{3}(d{d}^{\ast }A)-\frac{1}{2}(\frac{1}{2}\alpha dJ\alpha dJA+3\alpha dJA)=\frac{1}{2}J{d}^{\ast }{\pi }_8(dJA)=0.$$

$$0=-\frac{1}{3}{d}^{\ast }d{d}^{\ast }A+2d^{\ast }A$$

Full text

1 Introduction

There are many papers about isolated conical singularities with special holonomy. Many of them require the isolated singularity to have polynomial rate of convergence to a cone [15]. In general, there is a well-developed theory for the analysis on isolated singularity with polynomial rate of convergence to a cone including [2, 4, 14, 17, 20, 21]. See also [5, 6, 7, 10, 11, 13, 16, 18, 22] for related topics.

Therefore, it is natural to ask, whether every isolated conical singularity has polynomial rate of convergence. Such kind of question was studied by Adams and Simon [1]. Roughly speaking, a geometric object with isolated singularity has polynomial rate of convergence to the cone if and only if the deformation of the links of the cone is unobstructed. It is well known that a cone metric is G2 if and only if the link is nearly-Kähler. As Foscolo [9] proved that the deformation of nearly-Kähler structures on the flag manifold $M$ is obstructed, it would seems trivial to prove that there exists a metric with G2 holonomy modelled on a cone with slow convergence rate. However, this is not the case and there are still several problems to study:

First of all, after investigating the paper of Adams and Simon, a metric $g$ with G2 holonomy is a “cone” in the sense of Adams-Simon if $L_{r\frac{\partial}{\partial r}}g=2g$. On the contrary, $g$ is called a cone metric in the usual sense if in addition $g(\frac{\partial}{\partial r},\frac{\partial}{\partial r})=1$. Therefore, the deformation space of links of G2 “cones” in the sense of Adams-Simon is larger than the deformation space of nearly-Kähler structures. Secondly, there exist infinite-dimensional symmetries induced by the diffeomorphism group.

In order to solve the problems, this paper will start from the G2 cone metric induced by $\phi$ on the cone $CM$ over the flag manifold $M$ and then use the method of Adams and Simon [1] to solve the equation

[TABLE]

with boundary conditions $d^{*}_{\phi}\xi|_{r=1}=0$ and $r_{0}^{-3}||\xi||_{C^{k,\alpha}_{r^{-2}dr^{2}+h}(\{r_{0}/e<r<r_{0}\})}\rightarrow 0$ as $r_{0}\rightarrow 0$. In order to work transversally to the diffeomorphisms, $\xi$ will be restricted to $\Omega^{2,\phi}_{14}$ since the tangent space to the diffeomorphism orbit of $\phi$ is

[TABLE]

See next section for the decomposition of forms on a manifold with G2 holonomy.

There are two major steps to solve the equation:

(1) Find out the infinitisimal deformation space and show that the linearized operator is invertible in the perpendicular space of the infinitisimal deformation space of links of G2 “cones” in the sense of Adams-Simon. Note that Lemma 2 of [1] can not be used directly because of the difference in the boundary conditions in order to solve the second problem.

(2) Study the obstruction term in the deformation of links of G2 “cones” in the sense of Adams-Simon. If the obstruction term is non-zero, then it is possible to construct a solution $\xi$ with slow convergence rate to 0. Note the Foscolo’s result can not be used directly because of the first problem.

The first step will be done in Section 3 and the second step will be done in Section 4.

In Section 5, the following main theorem will be proved:

Theorem 1.1**.**

There exists $\xi\in\Omega^{2,\phi}_{14}(B_{1})$ such that

[TABLE]

[TABLE]

and $(-\ln r)(r^{-3}F_{r}^{*}(\phi+d\xi)-\phi)$ converges to a non-zero limit in $C^{k,\alpha}_{B_{1}\setminus B_{1/2}}$ when $r\rightarrow 0$, where $F_{a}(r,x)=(ar,x):CM\rightarrow CM$. It induces a metric on $B_{1}\subset CM$ with G2 holonomy whose rate of convergence to the cone metric is $(-\ln r)^{-1}$.

⁢

Acknowledgement: The author is grateful to the insightful and helpful discussions with Xiuxiong Chen, Lorenzo Foscolo, Song Sun and Yuanqi Wang.

2 Notations and definitions

Even though Foscolo’s result can not be applied directly, his notations and several facts in his paper [9] can still be used.

$M$ will be the flag manifold $\mathrm{SU}(3)/T^{2}$. The Lie algebra $\mathfrak{u}_{3}$ is spanned by the following matrices:

[TABLE]

[TABLE]

[TABLE]

where $E_{ij}$ is the $3\times 3$ matrix with 1 in position $ij$ and all other entries zero. Compared to [9] and [19], the sign of $e_{3}$ has been changed in order to simplify the calculation. There exists a metric $h$ on $\mathfrak{u}_{3}$ given by making the frames $\{e_{i},\sqrt{2}h_{j}\}$ orthonormal. It induces to a metric on $M$. Let $\{e^{i},h^{j}\}$ be the dual basis and $e^{i_{1},...i_{m}}=e^{i_{1}}\wedge...\wedge e^{i_{m}}$. They can be extended to left-invariant forms on $\mathrm{U}(3)$. The left-invariant 2-form

[TABLE]

on $\mathrm{SU}(3)$ is projectable to $M$. It induces an almost complex structure $J$ on $M$. The (3,0) form $\Omega$ on $M$ is induced from the left-invariant 3-form

[TABLE]

on $\mathrm{SU}(3)$. It is easy to verify that [19]

[TABLE]

in other words, $(M,h,\Omega,\omega)$ is a nearly-Kähler manifold. It provides orthogonal decompositions on forms:

[TABLE]

where $\Lambda^{2}_{1}=\mathbb{R}\omega,\Lambda^{2}_{6}=\{X\lrcorner\mathrm{Re}\Omega\}$ and $\Lambda^{2}_{8}$ is the space of primitive (1,1)-forms. Following the notation of [9] and [19], 1-forms will be identified with vector fields using the the metric $h$. For example, $Je^{1}=Je_{1}=e_{2}=e^{2}.$ On the contrary, the dual of $J$ acts on 1-form by $J^{*}=-J$.

Given any $\zeta\in\mathfrak{su}(3)$, define functions $x_{i}$ and $h_{i}$ on $\mathrm{SU}(3)$ by

[TABLE]

The functions $v_{i}$ are projectable to $M$ but $x_{i}$ are not. However, some functions of $x_{i}$ may be projectable to $M$. Let

[TABLE]

then

[TABLE]

In [19], Moroianu and Semmelmann proved that the space of all co-closed primitive (1,1)-form on $M$ satisfying $\triangle\eta=12\eta$ is exactly the space of $\eta$ for all $\zeta\in\mathfrak{su}(3)$.

According to [9], the space $\mathcal{K}$ of Killing vector fields is given by $K$ satisfying

[TABLE]

where $\alpha$ is the operator dual to $X\rightarrow X\lrcorner\mathrm{Re}\Omega$. It is easy to prove the following propositions on the flag manifold $M$:

Proposition 2.1**.**

If $K$ is a 1-form on the $M$ satisfying $\triangle(JK)=18JK$, then $K$ is a Killing vector field.

Proof.

First of all, $\triangle d^{*}(JK)=18d^{*}(JK)$. In [19], Moroianu and Semmelmann proved that all possible eigenvalues for the Laplacian operator on functions on $M$ are $2(k(k+2)+l(l+2))=0,6,12,22,30,...$ So $d^{*}(JK)$ must vanish.

By [9], $dJK$ can be decomposed to:

[TABLE]

So

[TABLE]

The $\Omega^{3}_{6}$ component of the previous equation is [9]

[TABLE]

So

[TABLE]

Therefore,

[TABLE]

using the formula

[TABLE]

which can be derived from Proposition 3.6.(v) of [9]. Since 14 is not the eigenvalue for the Laplacian operator on functions on $M$, $d^{*}K$ must vanish, too. So

[TABLE]

In other words, $K$ can be written as $K=K_{3}+K_{-6}$, where $\alpha dJK_{3}=3K_{3}$ and $\alpha dJK_{-6}=-6K_{-6}.$ So

[TABLE]

and

[TABLE]

Therefore $d^{*}K_{3}=d^{*}JK_{3}=0$. In particular $\triangle JK_{3}=18JK_{3}$. Let $\bar{\triangle}$ be the Hermitian Laplace operator defined in [19], then

[TABLE]

Possible $\bar{\triangle}$ eigenvalues are also $2(k(k+2)+l(l+2))=0,6,12,22,30,...$ [19]. So $K_{3}=0$. It follows that $K=K_{-6}$ is a Killing vector field. ∎

Proposition 2.2**.**

If $A$ is a 1-form on $M$ satisfying $d^{*}(JA)=0$, then $JA=d^{*}\eta$ has a solution $\eta\in\Omega^{2}_{8}$ if and only if $A$ is perpendicular to the space of Killing vector fields.

Proof.

If $d^{*}(JA)=0$ and $JA$ is perpendicular to the image of $d^{*}$ from $\Omega^{2}_{8}$ to $\Omega^{1}$, then $0=(JA,d^{*}\eta)=(dJA,\eta)=(\pi_{8}(dJA),\eta)$ for all $\eta\in\Omega^{2}_{8}$. So $\pi_{8}(dJA)=0$. Therefore,

[TABLE]

So

[TABLE]

using the formula

[TABLE]

However, since $M$ is not isometric to 6-sphere, 6 is not an eigenvalue of the Laplacian operator on functions. Therefore, $d^{*}A=0$. So

[TABLE]

In other words, $A$ can be written as $A=A_{0}+A_{-6}$, where $\alpha dJA_{0}=0$ and $\alpha dJA_{-6}=-6A_{-6}.$ As before,

[TABLE]

Since $\alpha dJA_{0}=0$,

[TABLE]

It also equals to

[TABLE]

because $dJA_{0}\in\Omega^{2}_{8}.$ So both $dJA_{0}$ and $d^{*}JA_{0}$ vanish. In other words, $JA_{0}$ is harmonic. By Bochner technique, $A_{0}=0$. So $A=A_{-6}$ satisfies $d^{*}A=0$, $d^{*}(JA)=0$ and $\alpha dJA=-6A.$ In other words, $A$ is a Killing vector field.

Conversely, by Proposition 3.19 of [9], any Killing vector field $K$ satisfies $\pi_{8}(dJK)=0$. ∎

In general, any 3-form $\phi$ on a 7-manifold determines a bilinear form

[TABLE]

When it is positive definite, it determines a metric $g$ by

[TABLE]

It is natural to use notations like $*_{\phi}$ instead of $*_{g}$ because $g$ is determined by $\phi$. By the result of Fernandez and Gray [8], a manifold has G2 holonomy if and only if $d\phi=0$ and $d*_{\phi}\phi=0$.

The 7-dimensional cone $CM$ is given by $CM=(M\times(0,1])\cup\{o\}$, where $o$ is the tip point. Let $r$ be the coordinate of the $(0,1]$ factor. There is a 3-form $\phi$ on $CM$ given by

[TABLE]

It determines a metric

[TABLE]

and a 4-form

[TABLE]

From now on, define $d_{\phi},d^{*}_{\phi}$ as the exterior differential operator on $CM$ and its $g$ adjoint. Let $d$ and $d^{*}$ be the exterior differential operator on $M$ and its $h$ adjoint. Then $d_{\phi}=dr\wedge\frac{\partial}{\partial r}+d$. It follows that $d_{\phi}\phi=d_{\phi}*_{\phi}\phi=0$. In other words, the cone metric $g$ on $CM$ has G2 holonomy.

The G2 structure provides a $g$-orthogonal decomposition of forms on $CM$.

[TABLE]

[TABLE]

where $\Lambda^{2}_{7}=\{X\lrcorner\phi\},\Lambda^{3}_{1}=\mathbb{R}\phi,\Lambda^{3}_{7}=\{X\lrcorner*_{\phi}\phi\}$ and the orthogonal complements are $\Lambda^{2}_{14}$ and $\Lambda^{3}_{27}$. Let $X$ be a tangent vector on $M$, then

[TABLE]

$\Omega^{2}_{7}(CM)$ consists of the linear combination of them with $\frac{\partial}{\partial r}\lrcorner\phi=r^{2}\omega$. So any form in $\Omega^{2}_{8}(M)$ is perpendicular to $\Omega^{2}_{7}(CM)$. By direct calculation,

[TABLE]

So

[TABLE]

Finally, define $t=-\ln r$. Choose large enough $T$ and define

[TABLE]

[TABLE]

3 Estimate for the linearized equation

The first step to apply Adams and Simon’s result [1] is the computation of the linearization equation. Let $\xi\in\Omega^{2}_{14,\phi}(CM)$. For any compactly supported vector field $X$,

[TABLE]

So

[TABLE]

For any compactly supported function $f$,

[TABLE]

So

[TABLE]

According to [12], the linearization of $*_{\phi+d_{\phi}\xi}(\phi+d_{\phi}\xi)$ is

[TABLE]

So the linearization of $\pi_{14}^{\phi}((*_{\phi}d_{\phi}*_{\phi+d_{\phi}\xi}(\phi+d_{\phi}\xi)+\frac{3}{2}d_{\phi}d^{*}_{\phi}\xi)$ is

[TABLE]

Let $\mathcal{D}$ be the eigenspace of eigenvalue 12 of the Laplacian operator acting on co-closed forms in $\Omega^{2}_{8}(M)$. As an analogy of Lemma 2 of [1], the proof of the following lemma will be the main goal of this section:

Lemma 3.1**.**

Suppose $q>0$, $||r^{-3}f||_{C^{k,\alpha}_{q}}<\infty$ and $f(t)\in\mathcal{D}^{\perp}\subset\Omega^{2}_{14,\phi},\forall t\geq 0$. Then there exists a solution $\xi(t)\in\mathcal{D}^{\perp}\subset\Omega^{2}_{14,\phi},t\geq 0$ to

[TABLE]

Moreover, $\xi$ satisfies

[TABLE]

and boundary condition

[TABLE]

for some constant $C$ independent of $T$. As a corollary,

[TABLE]

Lemma 2 of [1] can not be applied directly because the boundary condition is in a different form. However, their method of proof can be combined with a long calculation in this section to produce a solution satisfying the boundary conditions.

The first step of the proof is writing the Laplacian operator on $CM$ in terms of operators in $M$. Recall that $d_{\phi},d^{*}_{\phi}$ were defined as the exterior differential operator on $CM$ and its $g$ adjoint, $d$, $d^{*}$ were defined as the exterior differential operator on $M$ and its $h$ adjoint. So $d_{\phi}=dr\wedge\frac{\partial}{\partial r}+d$.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Let

[TABLE]

then the main goal of this section is to solve the equations

[TABLE]

such that

[TABLE]

[TABLE]

After changing coordinate $r=e^{-t}$, the equations are reduced to

[TABLE]

There are several steps to achieve it

Step 1: Solve the equation

[TABLE]

By Lemma 2 of [1], since $\int_{M}d^{*}(JA)(t)=0$ for all $t$, it is possible to get a solution $f_{1}(t)$ satisfying $\int_{M}f_{1}(t)=0$ for all $t$. Moreover,

[TABLE]

Step 2: Since $\int_{M}f_{1}(t)=0$, it is possible to solve $d^{*}(JX_{1})(t)=f_{1}(t)$ so that $||X_{1}||_{C^{k+1,\alpha}_{q}}\leq C||f_{1}||_{C^{k+1,\alpha}_{q}}$. Write $X=X_{1}+X_{2}$, then the equations become

[TABLE]

Moreover, $d^{*}(JA_{2})=0$, and $||A_{2}||_{C^{k-1,\alpha}_{q}}+||B_{2}||_{C^{k-1,\alpha}_{q}}\leq C(||A||_{C^{k,\alpha}_{q}}+||B||_{C^{k,\alpha}_{q}}).$ The boundary condition is replaced by

[TABLE]

where $||G(0)||_{C^{k,\alpha}(M)}\leq C(||A||_{C^{k,\alpha}_{q}}+||B||_{C^{k,\alpha}_{q}}).$

Step 3: Suppose $A_{3}=\pi_{\mathcal{K}}A_{2}(t,x)=\sum_{i}A_{i}(t)K_{i}(x)$ for the basis of Killing vector fields $K_{i}(x)$ on $M$ and some scalar functions $A_{i}(t)$. This step deals with equations

[TABLE]

such that

[TABLE]

Since

[TABLE]

The equations are reduced to

[TABLE]

with

[TABLE]

They can be solved by adjusting the coefficients $\alpha_{i}$ in

[TABLE]

The solution $X_{3}=\sum_{i}X_{i}(t)K_{i}(x)$ satisfies

[TABLE]

Step 4: Let $X_{2}=X_{3}+X_{4}\in\mathcal{K}\oplus\mathcal{K}^{\perp}$, $A_{2}=A_{3}+A_{4}\in\mathcal{K}\oplus\mathcal{K}^{\perp}$. This step deals with the equation

[TABLE]

with

[TABLE]

By Proposition2.2, $d^{*}B_{2}\in\mathrm{Ker}d^{*}\cap J(\mathcal{K}^{\perp})$. Let $\phi_{j}$ be orthonormal eigenvectors of $\triangle$ on $\mathrm{Ker}d^{*}\cap J(\mathcal{K}^{\perp})$ with eigenvalues $\lambda_{j}$. Let $JX_{4}=\sum_{j}w_{j}(t)\phi_{j}(x)$, and

[TABLE]

The equation is reduced to

[TABLE]

with restriction on

[TABLE]

Let

[TABLE]

then the equations can be solved by adjusting the coefficients $\beta_{j}$ in

[TABLE]

[TABLE]

if $\gamma_{j}^{-}>0$,

[TABLE]

if $\gamma_{j}^{-}<0$.

Note that $5-\sqrt{4\lambda_{j}+9}<0$ because $\lambda_{j}\geq 5$ by Bochner technique. By [1], it is possible to get a solution satisfying

[TABLE]

as long as $\gamma_{j}^{-}\not=0$ on $\mathrm{Ker}d^{*}\cap J(\mathcal{K}^{\perp})$, or equivalently $\lambda_{j}\not=18$. It is true by Proposition 2.1.

Step 5: Solve $\eta_{4}\in\Omega^{2}_{8}\cap(\mathrm{Ker}d^{*})^{\perp}$ satisfying

[TABLE]

The right hand side is co-coclosed and is perpendicular to $J\mathcal{K}$. So the equation can be solved by proposition 2.2. Moreover,

[TABLE]

Let $\eta=\eta_{4}+\eta_{5}$, then the equation is reduced to

[TABLE]

with restriction $d^{*}\eta_{5}=0$ for some given $B_{5}\in\Omega^{2}_{8}$ satisfying $d^{*}B_{5}=0$. Moreover

[TABLE]

Let $\phi_{j}$ be orthonormal eigenvectors of $d^{*}d+dd^{*}$ on $\Omega^{2}_{8}\cap\mathrm{Ker}d^{*}$ with eigenvalues $\lambda_{j}$. The equation is reduced to

[TABLE]

where

[TABLE]

Let

[TABLE]

then the equation can be solved by

[TABLE]

if $\gamma_{j}>0$,

[TABLE]

if $\gamma_{j}<0$. By [1], as long as $\gamma_{j}\not=0$ on $\mathcal{D}^{\perp}$, or equivalently $\lambda_{j}\not=12$, there exists a solution $\eta_{5}=\sum_{j}w_{j}\phi_{j}$ satisfying

[TABLE]

However, this is true by definition of $\mathcal{D}$. In conclusion, it is possible to solve $\xi\in\Omega^{2}_{14,\phi}$ on $t\geq 0$ such that

[TABLE]

[TABLE]

and

[TABLE]

By standard elliptic estimate,

[TABLE]

4 Calculation of obstruction term

Another essential step to apply [1] is the computation of the quadratic term $Q$ in $s$ in

[TABLE]

The proof of the following lemma is the main goal of this section:

Lemma 4.1**.**

There exists $v\not=0\in\mathcal{D}$ such that $Q(v,v)=v$. Moreover, $Q(v,.)$ is a symmetric map from $\mathcal{D}$ to $\mathcal{D}$.

$Q$ is a linear map from $\mathrm{Sym}^{2}\mathcal{D}$ to $\mathcal{D}$. The deformation space $\mathcal{D}=\mathfrak{su}(3)$. As in [9], $Q$ belongs to the 1-dimensional space $\mathrm{Hom_{SU(3)}}(\mathrm{Sym}^{2}\mathfrak{su}(3),\mathfrak{su}(3))$, where $\mathrm{SU(3)}$ acts by $\mathrm{Ad}$. So it must be a multiple of the element $Q_{0}$ defined by

[TABLE]

using the following identification of $\mathfrak{su}(3)$ with $\Lambda^{2}_{8}(\mathbb{R}^{6})$:

[TABLE]

[TABLE]

[TABLE]

By direct calculation

[TABLE]

[TABLE]

[TABLE]

[TABLE]

So

[TABLE]

Moreover, the map $Q_{0}(e_{2}+e_{4}+e_{6},.)$ is symmetric.

Therefore, it suffices to show that $Q$ is a non-zero multiple of $Q_{0}$.

The term $\frac{3}{2}d_{\phi}d^{*}_{\phi}(sr^{3}\eta)$ is linear in $s$, so $Q$ is also the quadratic term in $s$ in

[TABLE]

Therefore, it suffices to show that the quadratic term in $s$ in the integral

[TABLE]

is non-zero. Let

[TABLE]

Then

[TABLE]

So it is necessary to compute the metric $g_{ij}$ induced by $\phi+sd(r^{3}\eta)$. Let $e_{0}=r\frac{\partial}{\partial r}$ and $\tilde{e}_{i}=r^{-1}e_{i}$, then $e^{0}=r^{-1}dr$ and $\tilde{e}^{i}=re^{i}$. Since

[TABLE]

[TABLE]

satisfy

[TABLE]

Moreover,

[TABLE]

So

[TABLE]

The term

[TABLE]

So

[TABLE]

because $\frac{\omega^{2}}{2}\wedge\eta=(v_{1}+v_{2}+v_{3})e^{123456}=0.$

[TABLE]

for $i=1,2,...6$ because by direct calculation

[TABLE]

[TABLE]

for $i,j=1,2...6$ because

[TABLE]

and

[TABLE]

So

[TABLE]

[TABLE]

[TABLE]

[TABLE]

After cyclic permutation

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The following table can be obtained by the symmetry of $B_{ij}^{(1)}$:

[TABLE]

Therefore

[TABLE]

So

[TABLE]

[TABLE]

It is possible to compute $B_{ij}^{(2)}$ for $i,j=0,1,...6$. However, it is enough to compute $B_{ij}^{(2)}$ for $i,j=1,...6$ because they are the only terms in further calculation.

[TABLE]

So

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The following table can be obtained by cyclic permutation:

[TABLE]

[TABLE]

Let

[TABLE]

[TABLE]

then

[TABLE]

[TABLE]

[TABLE]

By cyclic permutation,

[TABLE]

[TABLE]

The rest terms are

[TABLE]

In conclusion,

[TABLE]

[TABLE]

because

[TABLE]

The term $\frac{r^{-3}}{2}\phi_{ijk}(4e^{0}\wedge\eta-d\eta)_{lmk}B_{il}^{(1)}B_{jm}^{(1)}$ will be computed first:

[TABLE]

By cyclic permutation,

[TABLE]

[TABLE]

[TABLE]

By cyclic permutation,

[TABLE]

[TABLE]

So

[TABLE]

Similarly

[TABLE]

[TABLE]

[TABLE]

Adding all seven terms together,

[TABLE]

Then, the second term $\frac{r^{-3}}{2}\phi_{ijk}(4e^{0}\wedge\eta-d\eta)_{ljk}(B_{ip}^{(1)}B_{pl}^{(1)}-B_{il}^{(2)})$ will be computed:

[TABLE]

[TABLE]

[TABLE]

So

[TABLE]

Similarly,

[TABLE]

[TABLE]

[TABLE]

Adding everything together

[TABLE]

In conclusion

[TABLE]

As in [9], the integration is non-zero. In other words, Lemma 4.1 has been proved.

5 Proof of Main Theorem

The main goal of this section is to find out $w(t)$ such that

[TABLE]

satisfies the equation

[TABLE]

with boundary conditions $d^{*}_{\phi}\xi|_{r=1}=0$. The method is essentially same as Adams-Simon. However, a detailed proof using contraction mapping theorem will be included.

Write $w(t)=w^{T}(t)+w^{\perp}(t)\in\mathcal{D}\oplus\mathcal{D}^{\perp}$. The constant term is 0 because $\phi$ is closed and co-closed. The linear term is

[TABLE]

The rest terms are

[TABLE]

where for $T$ large enough and $||w||_{C^{k+2,\alpha}},||w_{1}||_{C^{k+2,\alpha}},||w_{2}||_{C^{k+2,\alpha}}$ small enough,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here, the $C^{k,\alpha}$ norm means $C^{k,\alpha}_{dt^{2}+h}(t\in[t_{0},t_{0}+1])$, and $C$ means a constant independent of $T$ and $t_{0}$.

The equations are reduced to

[TABLE]

By Lemma 3.1, the first two equations can be solved if $w^{\perp}=L_{1}(R^{\perp}_{4}(w))$ for some linear operator $L_{1}$ satisfying

[TABLE]

By Lemma 4 of [1], for all except a finite number of $q>0$, the last equation can be solved if

[TABLE]

for some linear operator $L_{2}$ satisfying

[TABLE]

The combination of the two equations can be rewritten as $w=F(w)$, where

[TABLE]

For all except a finite number of $1<q<3/2$, large enough $T$ and $w$ with small enough

[TABLE]

it is easy to see that

[TABLE]

and

[TABLE]

So $w=F(w)$ has solution with small $||w||$ by contraction mapping theorem.

After the solution of the equation, integration by parts using the boundary conditions provides

[TABLE]

and

[TABLE]

So

[TABLE]

It is well known that $d_{\phi}*_{\phi+d_{\phi}\xi}(\phi+d_{\phi}\xi)\in\Omega^{5,\phi+d_{\phi}\xi}_{14}$ [3]. So if $\xi$ is small enough, then $\pi_{14}^{\phi}(*_{\phi}d_{\phi}*_{\phi+d_{\phi}\xi}(\phi+d_{\phi}\xi))$ is much larger than $\pi_{7}^{\phi}(*_{\phi}d_{\phi}*_{\phi+d_{\phi}\xi}(\phi+d_{\phi}\xi))$. It means that $\pi_{14}^{\phi}d_{\phi}d^{*}_{\phi}\xi$ and therefore $d_{\phi}d^{*}_{\phi}\xi$ vanish. So $\pi_{14}^{\phi}(*_{\phi}d_{\phi}*_{\phi+d_{\phi}\xi}(\phi+d_{\phi}\xi))$ and therefore $*_{\phi}d_{\phi}*_{\phi+d_{\phi}\xi}(\phi+d_{\phi}\xi)$ also vanish. So $\phi+d_{\phi}\xi$ is both closed and co-closed. It induces a metric on $B_{1}\subset CM$ with G2 holonomy whose rate of convergence to the cone metric is $(-\ln r)^{-1}$. Moreover, after integration by parts,

[TABLE]