6.4. The non-linear problem [02I4]

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6.4. The non-linear problem

We are now ready to deform the triple $\bm{\underline{\omega}}_{\epsilon}$ into a genuine hyperkähler triple by using the following Implicit Function Theorem.

Lemma 6.13.

Let $\Phi\colon\thinspace E\rightarrow F$ be the smooth function between Banach spaces and write $\Phi(x)=\Phi(0)+L(x)+N(x)$ , where $L$ is linear and $N$ contains the non-linearities. Assume that there exists constants $r,C,q$ such that

(i)

$L$ is invertible with $\|L^{-1}\|\leq C$ ;
(ii)

$\|N(x)-N(y)\|_{F}\leq q\|x+y\|_{E}\|x-y\|_{E}$ for all $x,y\in B_{r}(0)\subset E$ ;
(iii)

$\|\Phi(0)\|_{F}<\min\left\{\frac{r}{2C},\frac{1}{4qC^{2}}\right\}$ .

Then there exist a unique $x\in E$ with $\|x\|_{E}\leq 2C\|\Phi(0)\|_{F}$ such that $\Phi(x)=0$ .

In our situation we set

E:=\left(C^{1,\alpha}_{\delta}(T^{\ast}M_{\epsilon})\oplus\mathcal{H}^{+}_{\epsilon}\right)\otimes\mathbb{R}^{3},

where $\mathcal{H}^{+}_{\epsilon}$ denotes the space of self-dual harmonic forms with respect to $g_{\epsilon}$ , i.e. constant linear combinations of $\omega^{1}_{\epsilon},\omega^{2}_{\epsilon},\omega^{3}_{\epsilon}$ . We endow $E$ with the product of the $C^{1,\alpha}_{\delta}$ –norm and the norm on the finite dimensional vector space $\mathcal{H}^{+}_{\epsilon}\otimes\mathbb{R}^{3}\simeq\mathbb{R}^{9}$ induced by the $L^{2}$ –norm. Similarly we set

F:=C^{0,\alpha}_{\delta-1}(\mathbb{R}\oplus\Lambda^{+}T^{\ast}M_{\epsilon})\otimes\mathbb{R}^{3}

endowed with the $C^{0,\alpha}_{\delta-1}$ –norm.

The operator $\Phi$ is the one defined by (6.1). Thus $\Phi(0)=-\mathcal{F}(\text{id}-Q_{\epsilon})$ , $L(\bm{\underline{a}}+\bm{\underline{\zeta}})=D\bm{\underline{a}}+\bm{\underline{\zeta}}$ and the non-linear term is

N(\bm{\underline{a}}+\bm{\underline{\zeta}})=\mathcal{F}\left(\text{id}-Q_{\epsilon}\right)-\mathcal{F}\left(\text{id}-Q_{\epsilon}-d^{-}\bm{\underline{a}}\ast d^{-}\bm{\underline{a}}\right).

We need to check that the hypothesis of Lemma 6.13 are satisfied.

We use Proposition 6.11 to show that $L$ has uniformly bounded inverse for $\delta\in(-\tfrac{1}{2},0)$ .

Lemma 6.14.

For $\delta\in(-\tfrac{1}{2},0)$ and $\epsilon$ sufficiently small there exists a constant $C>0$ independent of $\epsilon$ such that for every triple of self-dual $2$ –forms $\bm{\underline{\xi}}\in C^{0,\alpha}_{\delta-1}$ there exists a unique $(\bm{\underline{a}},\bm{\underline{\zeta}})\in E$ with

\|\bm{\underline{a}}\|_{C^{1,\alpha}_{\delta}}+\|\bm{\underline{\zeta}}\|\leq C\|\bm{\underline{\xi}}\|_{C^{0,\alpha}_{\delta-1}}.

and $L(\bm{\underline{a}},\bm{\underline{\zeta}})=\bm{\underline{\xi}}$ .

Proof.

First of all, note that the $2$ –forms $\omega_{\epsilon}^{i}$ have uniformly bounded $C^{0,\alpha}_{\delta-1}$ –norm. Indeed, outside the gluing regions $\bm{\underline{\omega}}_{\epsilon}$ is a hyperkähler triple and thus $\omega_{\epsilon}^{i}$ is parallel and bounded. On the gluing regions, $\bm{\underline{\omega}}_{\epsilon}$ differs from the hyperkähler triple $\bm{\underline{\omega}}_{q_{j},\epsilon}$ or $\bm{\underline{\omega}}_{p_{i},\epsilon}$ by terms of order $O(\epsilon\rho^{2}+\epsilon^{3}\rho^{-3})$ (with similar estimates on their derivatives). Finally, $\rho_{\epsilon}^{-\delta+1}$ is bounded above since $\delta<0$ .

Now, let $\widetilde{\bm{\underline{\omega}}}$ be an $L^{2}$ –orthonormal triple of harmonic self-dual forms with respect to $g_{\epsilon}$ . Since

\int_{M_{\epsilon}}{\omega_{\epsilon}^{i}\wedge\omega_{\epsilon}^{j}}=2\int_{M_{\epsilon}}{(Q_{\epsilon})_{ij}\,\operatorname{dv}_{g_{\epsilon}}},

$Q_{\epsilon}$ is close to the identity and $\operatorname{Vol}_{g_{\epsilon}}(M_{\epsilon})=O(\epsilon)$ we can assume that

\|\widetilde{\bm{\underline{\omega}}}\|_{C^{0,\alpha}_{\delta-1}}\leq C\epsilon^{-\frac{1}{2}}\|\bm{\underline{\omega}}_{\epsilon}\|_{C^{0,\alpha}_{\delta-1}}\leq C\epsilon^{-\frac{1}{2}}.

Finally, observe that for every $u\in C^{0,\alpha}_{\delta-1}$ we have

\|u\|_{L^{2}}\leq\|\rho_{\epsilon}^{\delta-1}\|_{L^{2}}\|u\|_{C^{0,\alpha}_{\delta-1}}\leq C(\epsilon^{\frac{1}{2}}+\epsilon^{\delta+1})\|u\|_{C^{0,\alpha}_{\delta-1}}.

Indeed, using the definition (6.7) of $\rho_{\epsilon}$ and the construction of $\bm{\underline{\omega}}_{\epsilon}$ it is not difficult to estimate $\|\rho_{\epsilon}^{\delta-1}\|_{L^{2}}\leq C(\epsilon^{\frac{1}{2}}+\epsilon^{\delta+1})$ .

Now let $\pi\colon\thinspace C^{0,\alpha}_{\delta-1}(\Lambda^{+}T^{\ast}M_{\epsilon})\rightarrow\mathcal{H}^{+}_{\epsilon}$ be the $L^{2}$ –orthogonal projection

\pi(\xi)=\sum_{i=1}^{3}{\lambda_{i}\,\widetilde{\omega}_{i}},\qquad\lambda_{i}=\int{\xi\wedge\widetilde{\omega}_{i}},

and regard $\text{id}-\pi$ as a map $C^{0,\alpha}_{\delta-1}(\Lambda^{+}T^{\ast}M_{\epsilon})\rightarrow C^{0,\alpha}_{\delta-1}(\Lambda^{+}T^{\ast}M_{\epsilon})$ . By the remarks above we have

|\lambda_{i}|\leq C(1+\epsilon^{\delta+1})\|\xi\|_{C^{0,\alpha}_{\delta-1}},\qquad\|\pi(\xi)\|_{C^{0,\alpha}_{\delta-1}}\leq C(1+\epsilon^{\delta+\frac{1}{2}})\|\xi\|_{C^{0,\alpha}_{\delta-1}}.

Thus if $\delta\geq-\tfrac{1}{2}$ the projections $\pi$ and $\text{id}-\pi$ are uniformly bounded. Proposition 6.11 and the surjectivity of $L$ then yield the result. ∎

Next, we consider the non-linear term $N$ . Note that this does not involve the harmonic part $\bm{\underline{\zeta}}$ . The function $\mathcal{F}$ is pointwise smooth with uniformly controlled norm for $\epsilon$ sufficiently small. Using the Taylor expansion of $\mathcal{F}$ at $\text{id}-Q_{\epsilon}$ and Lemma 6.9 to control products we can therefore find $r>0$ and $C$ independent of $\epsilon$ such that assumption (ii) in Lemma 6.13 is satisfied with $r$ and $q=C\epsilon^{\delta-1}$ for some $\epsilon$ –independent constant $C$ .

Finally,

\|\mathcal{F}(\text{id}-Q_{\epsilon})\|_{C^{0,\alpha}_{\delta-1}}\leq C\|\text{id}-Q_{\epsilon}\|_{C^{0,\alpha}_{\delta-1}}\leq C\epsilon^{2+\frac{1}{5}-\frac{2}{5}\delta}.

Indeed, setting $\rho=\rho_{j}$ for $j=1,\dots,8$ or $\rho=\rho_{i}$ for some $i=1,\dots,n$ , in the region $\epsilon^{\frac{2}{5}}\leq\rho\leq 2\epsilon^{\frac{2}{5}}$ we have $|\text{id}-Q_{\epsilon}|=O(\epsilon\rho^{2}+\epsilon^{3}\rho^{-3})$ by (5.8) and $\rho_{\epsilon}=\rho$ by (6.7).

Thus assumption (iii) in Lemma 6.13 is therefore satisfied as soon as $\epsilon^{2+\frac{1}{5}-\frac{2}{5}\delta}\ll\epsilon^{1-\delta}$ , i.e.

\epsilon^{\frac{3}{5}(\delta+2)}\ll 1.

If $\delta>-2$ this condition is satisfied for $\epsilon>0$ sufficiently small.

Theorem 6.15.

Let $(\mathbb{T},g_{\mathbb{T}})$ be a flat $3$ –torus with standard involution $\tau\colon\thinspace\mathbb{T}\rightarrow\mathbb{T}$ . Let $q_{1},\dots,q_{8}$ be the fixed points of $\tau$ and let $p_{1},\tau(p_{1}),\dots,p_{n},\tau(p_{n})$ be further $2n$ distinct points. Denote by $\mathbb{T}^{\ast}$ the punctured torus $\mathbb{T}\setminus\{q_{1},\dots,q_{8},p_{1},\dots,\tau(p_{n})\}$ .

Let $m_{1},\dots,m_{8}\in\mathbb{Z}_{\geq 0}$ and $k_{1},\dots,k_{n}\in\mathbb{Z}_{\geq 1}$ satisfy

\sum_{j=1}^{8}{m_{j}}+\sum_{i=1}^{n}{k_{i}}=16.

For each $j=1,\dots,8$ fix a $D_{m_{j}}$ ALF space $M_{j}$ and for each $i=1,\dots,n$ an $A_{k_{i}-1}$ ALF space $N_{i}$ .

Then there exists a $1$ –parameter family of hyperkähler metrics $\{g_{\epsilon}\}_{\epsilon\in(0,\epsilon_{0})}$ on the K3 surface with the following properties. We can decompose the K3 surface into the union of open sets $K^{\epsilon}\cup\bigcup_{j=1}^{8}{M_{j}^{\epsilon}}\cup\bigcup_{i=1}^{n}{N_{i}^{\epsilon}}$ such that

(i)

$(K^{\epsilon},g_{\epsilon})$ collapses to the flat orbifold $\mathbb{T}^{\ast}/\mathbb{Z}_{2}$ with bounded curvature away from the punctures;
(ii)

for each $j=1,\dots,8$ and $k\geq 0$ , $(M_{j}^{\epsilon},\epsilon^{-2}g_{\epsilon})$ converges in $C^{k,\alpha}_{loc}$ to the $D_{m_{j}}$ ALF space $M_{j}$ ;
(iii)

for each $i=1,\dots,n$ and $k\geq 0$ , $(N_{j}^{\epsilon},\epsilon^{-2}g_{\epsilon})$ converges in $C^{k,\alpha}_{loc}$ to the $A_{k_{i}-1}$ ALF space $N_{i}$ .

Proof.

Given data as in the statement we constructed a $4$ –manifold $M_{\epsilon}$ and a $1$ –parameter family of closed definite triples $\bm{\underline{\omega}}_{\epsilon}$ which are approximately hyperkähler. For $\epsilon$ sufficiently small we can apply Lemma 6.13 to find unique $\bm{\underline{a}}_{\epsilon}\in C^{1,\alpha}_{\delta}(T^{\ast}M_{\epsilon})$ for $\delta\in(-\tfrac{1}{2},0)$ and $\bm{\underline{\zeta}}_{\epsilon}\in\mathcal{H}^{+}_{\epsilon}$ such that $\|\bm{\underline{a}}_{\epsilon}\|_{C^{1,\alpha}_{\delta}}+\|\bm{\underline{\zeta}}_{\epsilon}\|\leq C\epsilon^{\frac{11-2\delta}{5}}$ and $\bm{\underline{\omega}}_{\epsilon}+d\bm{\underline{a}}_{\epsilon}+\bm{\underline{\zeta}}_{\epsilon}$ is a hyperkähler structure on $M_{\epsilon}$ . In particular, since $b_{1}(M_{\epsilon})=0$ by Proposition 5.1, $M_{\epsilon}$ must be diffeomorphic to the K3 surface.

Away from the gluing regions $\bm{\underline{a}}_{\epsilon}$ solves the elliptic PDE $d^{+}\bm{\underline{a}}_{\epsilon}=\mathcal{F}(d^{-}\bm{\underline{a}}_{\epsilon}\ast d^{-}\bm{\underline{a}}_{\epsilon})-\bm{\underline{\zeta}}_{\epsilon}$ , $d^{\ast}\bm{\underline{a}}_{\epsilon}=0$ . By elliptic regularity, for any $k\geq 2$ the $C^{k,\alpha}$ –norm of $\bm{\underline{a}}_{\epsilon}$ on compact sets of $M^{\textup{gh}}_{\epsilon}$ and (after rescaling) on compact sets of the gravitational instantons $M_{j}$ and $N_{i}$ is controlled in terms of $\|\bm{\underline{a}}_{\epsilon}\|_{C^{1,\alpha}_{\delta}}+\|\bm{\underline{\zeta}}_{\epsilon}\|$ . In particular, on compact sets of $M^{\textup{gh}}_{\epsilon}$ the hyperkähler metric induced by $\bm{\underline{\omega}}_{\epsilon}+d\bm{\underline{a}}_{\epsilon}+\bm{\underline{\zeta}}_{\epsilon}$ is $C^{k,\alpha}$ –close to $g_{\mathbb{T}}+\epsilon^{2}\theta^{2}$ . The statements (i), (ii) and (iii) about the limit $\epsilon\rightarrow 0$ now follow. ∎

By varying all parameters involved in the construction we can in fact realise a whole open set in the moduli space of hyperkähler metrics on the K3 surface. Indeed,

(i)

the moduli space of flat tori is $6$ –dimensional;
(ii)

the choice of punctures $p_{1},\dots,p_{n}$ yields additional $3n$ parameters;
(iii)

once the punctured torus and weights are fixed, the moduli space of abelian Dirac monopoles with prescribed singularities is $4$ dimensional (one has to choose $\epsilon$ and the $3$ –moduli of a flat connection);
(iv)

each $D_{m_{j}}$ ALF space contributes $3m_{j}$ parameters and every $A_{k_{i}-1}$ ALF space contributes $3(k_{i}-1)$ parameters.

Hence the total number of parameters in the construction is

6+3n+4+3\sum_{j=1}^{8}{m_{j}}+3\sum_{i=1}^{n}{k_{i}}-3n=10+3\times 16=58,

which is exactly the dimension of the moduli space of Ricci-flat metrics on the K3 surface (without any normalisation on volume).