9.2. The existence of a hyperkähler triple [03JX]

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9.2. The existence of a hyperkähler triple

Now we are in a position to prove the existence of the hyperkähler triple. For any sufficiently large gluing parameter $\beta\gg 1$ , denote by $\bm{\omega}_{\beta}^{\mathcal{M}}=(\omega_{1},\omega_{2},\omega_{3})$ the approximate definite triple on $\mathcal{M}$ which was constructed in Section 6. To prove the existence of a hyperkähler triple, we will solve the gauge-fixed elliptic system,

(9.90)

\displaystyle d^{+}\bm{\eta}+\bm{\xi}=\mathfrak{F}_{0}\Big(\TF(-Q_{\beta}-S_{d^{-}\bm{\eta}})\Big),\ d^{*}\bm{\eta}=0,

where the renormalized coefficient matrix $Q_{\beta}=(Q_{ij})$ is defined by

(9.91)

\frac{1}{2}\omega_{i}\wedge\omega_{j}=Q_{ij}\dvol_{\bm{\omega}_{\beta}^{\mathcal{M}}},

see Section 1.3 for more details about the setup. A basic tool of solving the elliptic system (9.90) is the following version of the implicit function theorem, see for example [RS05, theorem 4.4.2].

Lemma 9.3.

Let $\mathscr{F}:\mathfrak{A}\to\mathfrak{B}$ be a $C^{1}$ -map between two Banach spaces such that $\mathscr{F}(x)-\mathscr{F}(0)=\mathscr{L}(x)+\mathscr{N}(x)$ , where the operator $\mathscr{L}:\mathfrak{A}\to\mathfrak{B}$ is linear and $\mathscr{N}(0)=0$ . Assume that

(1)

$\mathscr{L}$ is an isomorphism with $\|\mathscr{L}^{-1}\|\leq C_{1}$ ,
(2)
there are constants $r>0$ and $C_{2}>0$ with $r<\frac{1}{3C_{1}C_{2}}$ such that
1. (a)
  
  $\|\mathscr{N}(x)-\mathscr{N}(y)\|_{\mathfrak{B}}\leq C_{2}\cdot(\|x\|_{\mathfrak{A}}+\|y\|_{\mathfrak{A}})\cdot\|x-y\|_{\mathfrak{A}}$ for all $x,y\in B_{r}(0)\subset{\mathfrak{A}}$ ,
2. (b)
  
  $\|\mathscr{F}(0)\|_{\mathfrak{B}}\leq\frac{r}{2C_{1}}$ ,

then there exists a unique solution to $\mathscr{F}(x)=0$ in $\mathfrak{A}$ such that

(9.92)

\|x\|_{\mathfrak{A}}\leq 2C_{1}\cdot\|\mathscr{F}(0)\|_{\mathfrak{B}}.

To apply the above implicit function theorem, we need to verify the above properties in our context. To start with, we define the following Banach spaces,

(9.93)

\mathfrak{A}\equiv\Big(C_{\delta,\nu,\mu}^{1,\alpha}(\mathring{\Omega}^{1}(\mathcal{M}))\oplus\mathcal{H}^{+}(\mathcal{M})\Big)\otimes\mathbb{R}^{3}

and

(9.94)

\mathfrak{B}\equiv\Big(C_{\delta,\nu+1,\mu}^{0,\alpha}(\Lambda^{+}(\mathcal{M}))\Big)\otimes\mathbb{R}^{3},

where $\mathcal{H}^{+}(\mathcal{M})$ is the space of self-dual $2$ -forms on $\mathcal{M}$ , $\Lambda^{+}(\mathcal{M})$ is the space of self-dual $2$ -forms on $\mathcal{M}$ and $\mathring{\Omega}^{1}(\mathcal{M})\equiv\{\eta\in\Omega^{1}(\mathcal{M})|d^{*}\eta=0\}$ . Notice that Proposition 6.6 implies that

(9.95)

\dim(\mathcal{H}^{+}(\mathcal{M}))=b_{2}^{+}(\mathcal{M})=3.

Now we give a basis of $\mathcal{H}^{+}(\mathcal{M})$ . Let $\bm{\omega}_{\beta}^{\mathcal{M}}\equiv(\omega_{1},\omega_{2},\omega_{3})$ be the gluing definite triple on $\mathcal{M}$ constructed in Section 6 which induces a Riemannian metric $g$ such that the triple $\bm{\omega}^{\mathcal{M}}$ is self-dual with respect to $g$ . Immediately, $d^{*}\omega_{k}=d\omega_{k}=0$ and hence for every $1\leq k\leq 3$ , $\omega_{k}$ is a self-dual harmonic $2$ -form. Then by Corollary 6.5, $\{\omega_{1},\omega_{2},\omega_{3}\}$ is actually a basis of $\mathcal{H}^{+}(\mathcal{M})$ .

Let $\mathfrak{A}$ and $\mathfrak{B}$ equipped with the following weighted Hölder norms: Let $(\bm{\eta},\bm{\bar{\xi}}^{+})\in\mathfrak{A}$ and $\bm{\xi}^{+}\in\mathfrak{B}$ , then

(9.96)

\|(\bm{\eta},\bm{\bar{\xi}}^{+})\|_{\mathfrak{A}}\equiv\|\bm{\eta}\|_{C_{\delta,\nu,\mu}^{1,\alpha}(\mathcal{M})}+\|\bm{\bar{\xi}}^{+}\|_{L^{2}}

and

(9.97)

\|\bm{\xi}^{+}\|_{\mathfrak{B}}\equiv\|\bm{\xi}^{+}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})},

where the above $L^{2}$ norm is defined with respect to a fixed basis $\{\omega_{1},\omega_{2},\omega_{3}\}\subset\mathcal{H}^{+}(\mathcal{M})$ . The operator $\mathscr{F}:\mathfrak{A}\to\mathfrak{B}$ is defined by

(9.98)

\mathscr{F}(\bm{\eta},\bar{\bm{\xi}}^{+})\equiv d^{+}\bm{\eta}+\bar{\bm{\xi}}^{+}-\mathfrak{F}_{0}\Big(\TF(-Q_{\beta}-S_{d^{-}\bm{\eta}})\Big),

which is given by the system (9.90). The corresponding linearization is

(9.99)

\mathscr{L}\equiv(d^{+}\oplus\Id)\otimes\mathbb{R}^{3}:\mathfrak{A}\longrightarrow\mathfrak{B}.

So the nonlinear part is given by

(9.100)

\mathscr{N}(\bm{\eta},\bar{\bm{\xi}}^{+})\equiv\mathfrak{F}_{0}\Big(\TF(-Q_{\beta})\Big)-\mathfrak{F}_{0}\Big(\TF(-Q_{\beta}-S_{d^{-}\bm{\eta}})\Big).

First, we will check Property (1) in Lemma 9.3 and we will prove that the linearized operator $\mathscr{L}_{g}$ is an isomorphism from $\mathfrak{A}$ to $\mathfrak{B}$ .

Proposition 9.4.

For $(\mathcal{M},g_{\beta})$ with sufficiently large gluing parameter $\beta\gg 1$ , then there exists some constant $C>0$ , independent of $\beta$ , such that for every triple

(9.101)

\bm{\xi}^{+}\equiv(\xi_{1}^{+},\xi_{2}^{+},\xi_{3}^{+})\in\mathfrak{B},

there exists a unique pair

(9.102)

(\bm{\eta},\bar{\bm{\xi}}^{+})\equiv\Big((\eta_{1},\eta_{2},\eta_{3}),(\bar{\xi}_{1}^{+},\bar{\xi}_{2}^{+},\bar{\xi}_{3}^{+})\Big)\in\mathfrak{A}

which satisfies

(9.103)

\mathscr{L}_{g}(\bm{\eta},\bar{\bm{\xi}}^{+})=\bm{\xi}^{+}

and

(9.104)

\|\bm{\eta}\|_{C_{\delta,\nu,\mu}^{1,\alpha}(\mathcal{M})}+\|\bar{\bm{\xi}}^{+}\|_{L^{2}}\leq Ce^{10\delta\cdot\beta}\cdot\|\bm{\xi}^{+}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})},

where $\delta$ , $\nu$ and $\mu$ are the constants in Proposition 9.2.

Proof.

First, we prove the surjectivity of the linear operator $\mathscr{L}_{g}$ . By standard Hodge theory, it holds that

(9.105)		$\displaystyle\Omega^{2}_{+}(\mathcal{M})$	$\displaystyle=\mathcal{H}^{+}(\mathcal{M})\oplus d^{+}(\Omega^{1}(\mathcal{M}))$
(9.106)		$\displaystyle\Omega^{1}(\mathcal{M})$	$\displaystyle=d(\Omega^{0}(\mathcal{M}))\oplus\mathring{\Omega}^{1}(\mathcal{M}),$

where $\mathring{\Omega}^{1}(\mathcal{M})$ denotes the space of divergence-free $1$ -forms on $\mathcal{M}$ , therefore

(9.107)

\Omega^{2}_{+}(\mathcal{M})=\mathcal{H}^{+}(\mathcal{M})\oplus d^{+}(\mathring{\Omega}^{1}(\mathcal{M})).

This clearly implies that

(9.108)

\mathscr{L}_{g}=(d^{+}\oplus\Id)\otimes\mathbb{R}^{3}:\mathfrak{A}\longrightarrow\mathfrak{B}.

is surjective.

The remainder of the proof is a contradiction argument. We will argue on the level of forms, and this will imply the result for triples. If (9.104) does not hold for a uniform constant, then there exists a sequence of gluing parameters $\beta_{j}\rightarrow\infty$ and $\eta_{j}$ , $\bar{\xi}^{+}_{j}$ with

(9.109)		$\displaystyle e^{10\delta\cdot\beta_{j}}\\|d^{+}\eta_{j}+\bar{\xi}^{+}_{j}\\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}$	$\displaystyle\rightarrow 0,$
(9.110)		$\displaystyle\\|\eta_{j}\\|_{C_{\delta,\nu,\mu}^{1,\alpha}(\mathcal{M})}+\\|\bar{{\xi}}^{+}_{j}\\|_{L^{2}(\mathcal{M})}$	$\displaystyle=1,$

as $j\to\infty$ . Pairing $d^{+}\eta_{j}+\bar{\xi}^{+}_{j}$ with $\bar{\xi}^{+}_{j}$ and integrating, and using (9.109), we obtain that

(9.111)

\displaystyle\begin{split}\|\bar{\xi}^{+}_{j}\|_{L^{2}(\mathcal{M})}^{2}&\leq\epsilon_{j}e^{-10\delta\cdot\beta_{j}}\int_{\mathcal{M}}|\bar{\xi}_{j}^{+}|(\rho_{\delta,\nu+1,\mu}^{(0+\alpha)})^{-1}\dvol_{g_{\beta_{j}}}\\ &\leq\epsilon_{j}e^{-10\delta\cdot\beta_{j}}\|\bar{\xi}^{+}_{j}\|_{L^{2}(\mathcal{M})}\Big\{\int_{\mathcal{M}}(\rho_{\delta,\nu+1,\mu}^{(0+\alpha)})^{-2}\dvol_{g_{\beta_{j}}}\Big\}^{\frac{1}{2}},\end{split}

where $\epsilon_{j}\to 0$ as $j\to\infty$ . It is easy to check that

(9.112)

\displaystyle\int_{\mathcal{M}}(\rho_{\delta,\nu+1,\mu}^{(0+\alpha)})^{-2}\dvol_{g_{\beta_{j}}}<C,

where $C$ is independent of $\beta$ , so this implies that

(9.113)

e^{10\delta\cdot\beta_{j}}\|\bar{\xi}^{+}_{j}\|_{L^{2}(\mathcal{M})}\to 0

as $j\to\infty$ .

Next, since the triple $\bm{\omega}^{\mathcal{M}}$ is harmonic and spans $\mathcal{H}_{+}(\mathcal{M})$ at every point, we can write

(9.114)

\bar{\xi}^{+}=\lambda_{1}\omega_{1}+\lambda_{2}\omega_{2}+\lambda_{3}\omega_{3}.

Recall by the definition of the triple $\bm{\omega}_{\beta}^{\mathcal{M}}$ , for every $1\leq p,q\leq 3$ ,

(9.115)

\frac{1}{2}\int_{\mathcal{M}}\omega_{p}\wedge\omega_{q}=\int_{\mathcal{M}}Q_{pq}\dvol_{\bm{\omega}_{\beta}^{\mathcal{M}}},

and so for any self-dual harmonic form $\bar{\xi}^{+}\in\mathcal{H}_{+}(\mathcal{M})$ ,

(9.116)

\displaystyle\|\bar{\xi}^{+}\|_{L^{2}(\mathcal{M})}^{2}

\displaystyle=2\sum_{p,q=1}^{3}\lambda_{p}\lambda_{q}\int_{\mathcal{M}}Q_{pq}\dvol_{\bm{\omega}_{\beta}^{\mathcal{M}}},

so applying the volume estimate

(9.117)

C^{-1}\beta^{2}\leq\Vol_{g}(\mathcal{M})\leq C\beta^{2},

and Proposition 6.4, we have the estimate

(9.118)

\displaystyle C^{-1}\beta_{j}^{2}(\lambda_{1,j}^{2}+\lambda_{2,j}^{2}+\lambda_{3,j}^{2})\leq\|\bar{\xi}_{j}^{+}\|_{L^{2}(\mathcal{M})}^{2}.

The above and (9.113) imply that $\beta_{j}\lambda_{k,j}e^{10\delta\cdot\beta_{j}}\to 0$ as $j\to\infty$ for $k=1,2,3$ . We then have

(9.119)

\displaystyle\begin{split}\|\bar{\xi}^{+}_{j}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}&=\|\lambda_{1,j}\omega_{1}+\lambda_{2,j}\omega_{2}+\lambda_{3,j}\omega_{3}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}\\ &\leq\lambda_{1,j}\|\omega_{1}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}+\lambda_{2,j}\|\omega_{2}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}+\lambda_{3,j}\|\omega_{3}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}.\end{split}

Since

(9.120)

\displaystyle\|\omega_{k}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}\leq Ce^{5\delta\cdot\beta_{j}},

for $1\leq k\leq 3$ , the above implies that

(9.121)

\displaystyle\|\bar{\xi}^{+}_{j}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}\leq C\epsilon_{j}\beta_{j}^{-1}e^{-5\delta\cdot\beta_{j}},

for some sequence $\epsilon_{j}\to 0$ as $j\to\infty$ , so we have proved that

(9.122)

\displaystyle\|\bar{\xi}^{+}_{j}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}\rightarrow 0,

as $j\to\infty$ . Consequently, our sequence satisfies

(9.123)		$\displaystyle\\|d^{+}\eta_{j}\\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}$	$\displaystyle\rightarrow 0,$
(9.124)		$\displaystyle\\|\eta_{j}\\|_{C_{\delta,\nu,\mu}^{1,\alpha}(\mathcal{M})}$	$\displaystyle\to 1,$

as $j\to\infty$ , which contradicts Proposition 9.2. ∎

In the following proposition, we will prove the nonlinear error estimate which corresponds to Property (2) in Lemma 9.3.

Lemma 9.5 (Nonlinear Errors).

Consider $(\mathcal{M},g_{\beta})$ with sufficiently large gluing parameter $\beta\gg 1$ . Let $\delta$ , $\nu$ and $\mu$ be the constants in Proposition 9.2, then there are constants $r_{0}>0$ and $C>0$ which are independent $\beta$ , such that for every $\bm{v}_{1}\equiv(\bm{\eta}_{1},\bar{\bm{\xi}}_{1}^{+})\in B_{r}(0)\subset\mathfrak{A}$ and $\bm{v}_{2}\equiv(\bm{\eta}_{2},\bar{\bm{\xi}}_{2}^{+})\in B_{r}(0)\subset\mathfrak{A}$ , where $r<r_{0}$ , we have

(9.125)

\|\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})\|_{\mathfrak{B}}\leq C(\|\bm{v}_{1}\|_{\mathfrak{A}}+\|\bm{v}_{2}\|_{\mathfrak{A}})\cdot\|\bm{v}_{1}-\bm{v}_{2}\|_{\mathfrak{A}}.

Proof.

By definition, for any $\bm{v}\equiv(\bm{\omega},\bar{\bm{\xi}}^{+})$ ,

(9.126)

\mathscr{N}(\bm{v})\equiv\mathfrak{F}_{0}\Big(\TF(-Q_{\beta})\Big)-\mathfrak{F}_{0}\Big(\TF(-Q_{\beta}-S_{d^{-}\bm{\eta}})\Big).

and hence

(9.127)

\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})=\mathfrak{F}_{0}\Big(\TF(-Q_{\beta}-S_{d^{-}\bm{\eta}_{2}})\Big)-\mathfrak{F}_{0}\Big(\TF(-Q_{\beta}-S_{d^{-}\bm{\eta}_{1}})\Big).

Since $\mathfrak{F}_{0}:\mathscr{S}_{0}(\mathbb{R}^{3})\to\mathscr{S}_{0}(\mathbb{R}^{3})$ is a smooth map on the space of trace-free symmetric $(3\times 3)$ -matrices, there is some universal constant $C>0$ such that

(9.128)

\displaystyle\begin{split}|\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})|&\leq C|d^{-}\bm{\eta}_{1}*d^{-}\bm{\eta}_{1}-d^{-}\bm{\eta}_{2}*d^{-}\bm{\eta}_{2}|\\ &\leq C(|d^{-}\bm{\eta}_{1}|+|d^{-}\bm{\eta}_{2}|)\cdot|d^{-}(\bm{\eta}_{1}-\bm{\eta}_{2})|.\end{split}

Multiplying by the weight function,

(9.129)

\displaystyle\begin{split}\rho_{\delta,\nu+1,\mu}^{(0)}(x)\cdot|\mathscr{N}(v_{1})&-\mathscr{N}(v_{2})|\leq C\cdot\rho_{\delta,\nu+1,\mu}^{(0)}(x)\cdot(|d^{-}\bm{\eta}_{1}|+|d^{-}\bm{\eta}_{2}|)\cdot|d^{-}(\bm{\eta}_{1}-\bm{\eta}_{2})|\\ &\leq C\Big(\rho_{\delta,\nu,\mu}^{(1)}(x)\cdot(|d^{-}\bm{\eta}_{1}|+|d^{-}\bm{\eta}_{2}|)\Big)\cdot\Big(\rho_{\delta,\nu,\mu}^{(1)}(x)\cdot|d^{-}(\bm{\eta}_{1}-\bm{\eta}_{2})|\Big).\end{split}

Taking sup norms,

(9.130)

\displaystyle\|\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})\|_{C^{0}_{\delta,\nu+1,\mu}(\mathcal{M})}\leq C\Big(\|\bm{v}_{1}\|_{C^{1}_{\delta,\nu,\mu}(\mathcal{M})}+\|\bm{v}_{2}\|_{C^{1}_{\delta,\nu,\mu}(\mathcal{M})}\Big)\cdot\Big(\|\bm{v}_{1}-\bm{v}_{2}\|_{C^{1}_{\delta,\nu,\mu}(\mathcal{M})}\Big).

By similar computations, we also have the estimate for the Hölder seminorm

(9.131)

\displaystyle\Big[\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})\Big]_{C^{0,\alpha}_{\delta,\nu+1,\mu}(\mathcal{M})}\leq C\Big(\|\bm{v}_{1}\|_{C^{1,\alpha}_{\delta,\nu,\mu}(\mathcal{M})}+\|\bm{v}_{2}\|_{C^{1,\alpha}_{\delta,\nu,\mu}(\mathcal{M})}\Big)\cdot\Big(\|\bm{v}_{1}-\bm{v}_{2}\|_{C^{1,\alpha}_{\delta,\nu,\mu}(\mathcal{M})}\Big).

So we obtain the effective estimate (9.125) for the nonlinear errors. ∎

Proposition 9.6.

Consider $(\mathcal{M},g_{\beta})$ with sufficiently large gluing parameter $\beta\gg 1$ . Let $\delta$ , $\nu$ and $\mu$ be the constants in Proposition 9.2, then there exists some constant $C>0$ which is independent of $\beta$ such that

(9.132)

\|\mathscr{F}(0)\|_{\mathfrak{B}}\leq Ce^{-\frac{\delta_{q}\beta}{2}},

where $\delta_{q}>0$ is the constant in Corollary 6.5.

Proof.

In our context, it holds that

(9.133)

\mathscr{F}(0)=-\mathfrak{F}_{0}\Big(\TF(-Q_{\beta})\Big).

(9.134)

\|\mathscr{F}(0)\|_{\mathfrak{B}}\leq C\|\TF(Q_{\beta})\|_{\mathfrak{B}}.

The proof immediately follows from the estimate in Corollary 6.5. ∎

Now we are ready to prove the existence of a hyperkähler triple on $\mathcal{M}$ which implies that $\mathcal{M}$ is diffeomorphic to the $\K 3$ surface.

Theorem 9.7.

Consider $(\mathcal{M},g_{\beta})$ with sufficiently large gluing parameter $\beta\gg 1$ . Denote by $\bm{\omega}_{\beta}^{\mathcal{M}}$ the gluing definite triple which is constructed by Proposition 6.4. Let $\delta$ , $\nu$ and $\mu$ be the constants in Proposition 9.2, then there exists a hyperkähler triple $\bm{\omega}_{\beta}^{\HK}$ with the effective estimate

(9.135)

\|\bm{\omega}_{\beta}^{\mathcal{M}}-\bm{\omega}_{\beta}^{\HK}\|_{C_{\delta,\nu+1,\mu}^{0,\alpha}(\mathcal{M})}\leq Ce^{-\delta_{0}\beta}

for some constants $C>0$ and $\delta_{0}>0$ independent of $\beta$ . In particular, $\mathcal{M}$ is diffeomorphic to the $\K 3$ surface.

Proof.

It suffices to verify the conditions in Lemma 9.3. In fact, Proposition 9.4, Lemma 9.5 and Proposition 9.6 verify Property (1), Property (2a) and Property (2b) in Lemma 9.3 respectively. So applying the implicit function theorem given by Lemma 9.3, the existence of the hyperkähler triple $\bm{\omega}_{\beta}^{\HK}$ just follows. The Hölder type error estimate (9.135) follows directly from the implicit function theorem and the definition of the weight functions.

Since the hyperkähler triple $\bm{\omega}_{\beta}^{\HK}$ determines a hyperkähler metric on $\mathcal{M}$ . By Proposition 6.6, $\chi(\mathcal{M})=24$ and hence $\mathcal{M}$ is diffeomorphic to the $\K 3$ surface.

∎