4. Collapsing of Ricci-flat metrics [00WD]

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4. Collapsing of Ricci-flat metrics

In this section we use the estimates from section 3 to prove that the Ricci-flat metrics collapse to the base of the fibration.

We first explain the meaning of the Weil-Petersson metric, following the discussion in [ST2]. Recall that the Ricci-flat Kähler metric on $X$ cohomologous to $\omega_{1}=\omega_{0}+\omega_{X}$ is denoted by $\tilde{\omega}_{1}$ . We will call $\Omega=\tilde{\omega}_{1}^{n}$ its volume form. The generic fiber $X_{y}$ of $f$ is an $(n-m)$ -dimensional Calabi-Yau manifold, and it is equipped with the Kähler form $\omega_{y}=\omega_{X}|_{X_{y}}$ .

Recall that the volume of $X_{y}$ is a homological constant independent of $y$ , and that we assume that it is equal to $1$ . Since $c_{1}(X_{y})=0$ , there is a smooth function $F_{y}$ such that $\mathrm{Ric}(\omega_{y})=\sqrt{-1}\partial\overline{\partial}F_{y}$ and $\int_{X_{y}}(e^{F_{y}}-1)\omega_{y}^{n-m}=0$ . The functions $F_{y}$ vary smoothly in $y$ , since so do the Kähler forms $\omega_{y}$ . By Yau’s theorem there is a unique Ricci-flat Kähler metric $\omega_{SF,y}$ on $X_{y}$ cohomologous to $\omega_{y}$ , given by the solution of

(4.1)

\omega_{SF,y}^{n-m}=e^{F_{y}}\omega_{y}^{n-m}.

If we write $\omega_{SF,y}=\omega_{y}+\sqrt{-1}\partial\overline{\partial}\zeta_{y}$ , the functions $\zeta_{y}$ vary smoothly in $y$ and so they define a smooth function $\zeta$ on $X\backslash S$ . We then define a real closed $(1,1)$ -form $\omega_{SF}$ on $X\backslash S$ by $\omega_{SF}=\omega_{X}+\sqrt{-1}\partial\overline{\partial}\zeta$ , and call it the semi-flat form. Notice that $\omega_{SF}$ is not necessarily nonnegative (it is Kähler only in the fiber directions), but on $X\backslash S$ the $(n,n)$ -form $\omega_{SF}^{n-m}\wedge\omega_{0}^{m}$ is strictly positive, and so we can define a smooth positive function $F$ on $X\backslash S$ by

(4.2)

F=\frac{\Omega}{\omega_{SF}^{n-m}\wedge\omega_{0}^{m}}.

We claim that $F$ is actually constant on each fiber $X_{y}$ , and so it is the pullback of a function on $Y\backslash f(S)$ . To see this, fix a point $y\in Y\backslash f(S)$ and choose local coordinates $z^{1},\dots,z^{n-m}$ on the fiber $X_{y}$ , which extend locally to coordinates in a ball in $X$ . Then take local coordinates $w^{n-m+1},\dots,w^{n}$ near $y\in Y\backslash f(S)$ , so that $z^{1},\dots,z^{n-m},z^{n-m+1}=f^{*}(w^{n-m+1}),\dots,z^{n}=f^{*}(w^{n})$ give local holomorphic coordinates on $X$ . In these coordinates write

\omega_{0}=\sqrt{-1}\sum_{i,j=n-m+1}^{n}g^{0}_{i\overline{j}}dz^{i}\wedge d\overline{z}^{j},

\omega_{SF,y}=\sqrt{-1}\sum_{i,j=1}^{n-m}g^{SF}_{i\overline{j}}dz^{i}\wedge d\overline{z}^{j},

\Omega=G(\sqrt{-1})^{n}dz^{1}\wedge\dots\wedge d\overline{z}^{n}.

Then locally

F=\frac{G}{\det(g^{0}_{i\overline{j}})\det(g^{SF}_{i\overline{j}})},

and so on the fiber $X_{y}$ we have

\sqrt{-1}\partial\overline{\partial}\log F=-\mathrm{Ric}(\tilde{\omega}_{1})+\mathrm{Ric}(\omega_{SF,y})=0,

because $\omega_{0}$ is the pullback of a metric from $Y$ , and so $F$ is indeed constant on $X_{y}$ . Moreover, it is easy to check [ST2, Lemma 3.3] that on $Y\backslash f(S)$ we have

F=\frac{f_{*}\Omega}{\omega_{Y}^{m}},

and so

\int_{Y}F\omega_{Y}^{m}=\int_{X}\Omega=\int_{X}\omega_{1}^{n}

is finite. In fact there is a positive $\varepsilon$ so that $\int_{Y}F^{1+\varepsilon}\omega_{Y}^{m}$ [ST2, Proposition 3.2]. Then we apply [ST2, Theorem 3.2], which relies on the seminal work of Kołodziej [K] and further generalizations [EGZ1, Z], to solve (uniquely) the complex Monge-Ampère equation

(4.3)

(\omega_{Y}+\sqrt{-1}\partial\overline{\partial}\psi)^{m}=\frac{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}{\int_{X}\omega_{1}^{n}}F\omega_{Y}^{m},

with $\psi\in L^{\infty}(Y)$ and moreover $\psi$ is smooth on $Y\backslash f(S)$ (the proof of this follows the arguments of Yau in [Y1]). We will call $\omega=\omega_{Y}+\sqrt{-1}\partial\overline{\partial}\psi$ the Kähler metric on $Y\backslash f(S)$ that we’ve just constructed. Its Ricci curvature is the Weil-Petersson metric that we are about to define. Recall that the fibers $X_{y}$ have torsion canonical bundle, so that there is a number $k$ such that $K_{X_{y}}^{\otimes k}$ is trivial for all $y\in Y\backslash f(S)$ . The Weil-Petersson metric is a smooth nonnegative $(1,1)$ -form on $Y\backslash f(S)$ defined as the curvature form of a pseudonorm on the relative canonical line bundle $f_{*}(\Omega^{n-m}_{X/Y})^{\otimes k}$ : if $\Psi_{y}$ is a local nonzero holomorphic section of $f_{*}(\Omega^{n-m}_{X/Y})^{\otimes k}$ , which means that $\Psi_{y}$ is a nonzero holomorphic $k$ -pluricanonical form on $X_{y}$ that varies holomorphically in $y$ , then we let its length be

|\Psi_{y}|^{2}_{h_{WP}}=\int_{X_{y}}(\Psi_{y}\wedge\overline{\Psi_{y}})^{\frac{1}{k}}.

For $k>1$ this is not a Hermitian metric, but just a pseudonorm. The Weil-Petersson metric $\omega_{WP}$ on $Y\backslash f(S)$ is just formally the curvature of $h_{WP}$ , that is locally we set

\omega_{WP}=-\sqrt{-1}\partial\overline{\partial}\log|\Psi_{y}|^{2}_{h_{WP}},

and this is well-defined because the bundle $K_{X_{y}}^{\otimes k}$ is trivial. It is a classical fact (see [FS]) that $\omega_{WP}$ is pointwise nonnegative. As an aside, we note here that one can realize $\omega_{WP}$ as the actual curvature form of an honest Hermitian metric on a relative canonical bundle if one takes a finite unramified $k$ -sheeted cyclic cover $\tilde{X}\to X$ so that the smooth fibers of $\tilde{X}\to Y$ now have trivial canonical bundle.

Proposition 4.1 (cfr. [ST2]).

On $Y\backslash f(S)$ we have

(4.4)

\mathrm{Ric}(\omega)=\omega_{WP}.

Proof.

Differentiating (4.3) we see that

\mathrm{Ric}(\omega)=\mathrm{Ric}(\omega_{Y})-\sqrt{-1}\partial\overline{\partial}\log F.

If we fix $y\in Y\backslash f(S)$ and choose $\Psi$ a local never vanishing holomorphic section of $f_{*}(\Omega^{n-m}_{X/Y})^{\otimes k}$ , then we can define a local function $u=\frac{(\Psi\wedge\overline{\Psi})^{1/k}}{\omega_{SF}^{n-m}}$ on $X\backslash S$ , which is constant on each fiber $X_{y}$ . Since $\int_{X_{y}}\omega_{SF}^{n-m}=1$ , we see that

-\sqrt{-1}\partial\overline{\partial}\log u=\omega_{WP}.

Then

(4.5)

\mathrm{Ric}(\omega)=\mathrm{Ric}(\omega_{Y})-\sqrt{-1}\partial\overline{\partial}\log\frac{u\Omega}{(\Psi\wedge\overline{\Psi})^{\frac{1}{k}}\wedge\omega_{0}^{m}}.

Picking local coordinates $z^{i}$ as above, and writing

\Psi=K[(\sqrt{-1})^{n-m}dz^{1}\wedge\dots\wedge dz^{n-m}]^{\otimes k},

\omega_{0}=\sqrt{-1}\sum_{i,j=n-m+1}^{n}g^{0}_{i\overline{j}}dz^{i}\wedge d\overline{z}^{j},

\Omega=G(\sqrt{-1})^{n}dz^{1}\wedge\dots\wedge d\overline{z}^{n},

we see that

\frac{u\Omega}{(\Psi\wedge\overline{\Psi})^{\frac{1}{k}}\wedge\omega_{0}^{m}}=\frac{uG}{|K|^{\frac{2}{k}}\det(g^{0}_{i\overline{j}})},

and since $K$ is holomorphic and $\Omega$ is Ricci-flat we see that

-\sqrt{-1}\partial\overline{\partial}\log\frac{uG}{|K|^{\frac{2}{k}}\det(g^{0}_{i\overline{j}})}=\omega_{WP}-\mathrm{Ric}(\omega_{Y}),

which together with (4.5) gives (4.4). ∎

With these preparations, we can now show Theorem 2.4, which can be recast as follows

Theorem 4.1.

Consider the Ricci-flat metrics $\tilde{\omega}_{t}$ on $X$ , which can be written as $\tilde{\omega}_{t}=\omega_{0}+t\omega_{X}+\sqrt{-1}\partial\overline{\partial}\varphi_{t}$ . As $t\to 0$ we have that $\varphi_{t}\to\psi$ in the $C^{1,\beta}_{loc}$ topology on $X\backslash S$ , for any $0<\beta<1$ , and so $\tilde{\omega}_{t}$ converges in this topology to $\omega$ , which satisfies (4.4). Moreover $\tilde{\omega}_{t}$ also converge to $\omega$ weakly as currents on $X$ .

Proof.

We first prove that $\tilde{\omega}_{t}$ converges to $\omega$ in the weak topology of currents. Since the cohomology class of $\tilde{\omega}_{t}$ is bounded, weak compactness of currents implies that from any sequence $t_{i}\to 0$ we can extract a subsequence so that $\tilde{\omega}_{t_{i}}$ converges weakly to a limit closed positive $(1,1)$ -current $\hat{\omega}$ , which a priori depends on the sequence. If we write $\hat{\omega}=\omega_{0}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi}$ , it follows that $\varphi_{t_{i}}\to\hat{\varphi}$ in $L^{1}$ , and from the bound (2.8) we infer that $\hat{\varphi}$ is in $L^{\infty}$ . Moreover restricting $\hat{\omega}$ to any smooth fiber $X_{y}$ we see that

\sqrt{-1}\partial\overline{\partial}\hat{\varphi}|_{X_{y}}\geq 0,

and the maximum principle implies that $\hat{\varphi}$ is constant on each fiber, and so descends to a bounded function $\hat{\varphi}$ on $Y\backslash f(S)$ . We will show that $\hat{\varphi}$ satisfies the same equation (4.3) as $\psi$ , and so by uniqueness $\hat{\varphi}=\psi$ . To this end we fix an arbitrary compact set $K\subset Y\backslash f(S)$ , and we wish to show that $\hat{\varphi}$ satisfies (4.3) on $K$ .

We then fix $\eta$ a smooth function with support contained in $K$ , and we will also denote by $\eta$ its pullback to $X$ via $f$ . Recall that we have called $\tilde{\omega}_{1}$ the Ricci-flat metric in the class $[\omega_{1}]$ , and $\Omega=\tilde{\omega}_{1}^{n}$ . Then from the Monge-Ampère equation (2.5) we have

(4.6)

\int_{X}\eta\Omega=\frac{1}{a_{t}}\int_{X}\eta(\omega_{0}+t\omega_{X}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n},

where the constants $a_{t}$ are equal to

\frac{\int_{X}\omega_{t}^{n}}{\int_{X}\omega_{1}^{n}},

and behave like (2.6). We can also write

(4.7)

\int_{X}\eta\Omega=\int_{X}\eta F\omega_{SF}^{n-m}\wedge\omega_{0}^{m}.

We are now going to estimate $\frac{1}{a_{t}}\int_{X}\eta(\omega_{0}+t\omega_{X}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n}.$ We have

\begin{split}&\frac{1}{a_{t}}\int_{X}\eta(\omega_{0}+t\omega_{X}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n}\\ &=\frac{1}{a_{t}}\int_{X}\eta\left((\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})+(t\omega_{X}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}})\right)^{n}\\ &=\frac{1}{a_{t}}\int_{X}\eta\sum_{k=0}^{n}\binom{n}{k}(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge(t\omega_{X}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{n-k}\end{split}

First of all observe that the form $\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}}$ is the pullback of a form on $Y$ , and it can be wedged with itself at most $m$ times, so all terms in the sum with $k>m$ are zero. Next, we claim that all the terms with $k<m$ go to zero as $t\to 0$ . To see this, start by observing that on the compact set $K$ the estimate (3.29) gives a constant $C$ (that depends on $K$ ) such that

(4.8)

-C\omega_{X}\leq\sqrt{-1}\partial\overline{\partial}\varphi_{t}\leq C\omega_{X}.

Moreover from the equation

\partial\overline{\partial}\underline{\varphi_{t}}=f_{*}(\partial\overline{\partial}\varphi_{t}\wedge\omega_{X}^{n-m})

together with (4.8), (3.12), we see that on $f(K)$ we have

(4.9)

-C\omega_{Y}\leq\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}}\leq C\omega_{Y}.

We also need to use (3.9) which on $K$ gives

(4.10)

\sup_{K}|\varphi_{t}-\underline{\varphi_{t}}|\leq Ct.

Then any term with $k<m$ is equal to

\frac{\binom{n}{k}}{a_{t}}\int_{X}\eta(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge(t\omega_{X}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{n-k},

and it can be expanded into

\frac{\binom{n}{k}}{a_{t}}\sum_{i=0}^{n-k}\binom{n-k}{i}\int_{X}\eta(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge(t\omega_{X})^{n-k-i}\wedge(\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{i}.

On $K$ the $(1,1)$ -form $\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}}$ is bounded by (4.9). Since $a_{t}=O(t^{n-m})$ from (2.6), we see that the term in this sum with $i=0$ goes to zero. Any term with $i>0$ is comparable to

(4.11)

\frac{1}{t^{n-m}}\int_{X}(\varphi_{t}-\underline{\varphi_{t}})\sqrt{-1}\partial\overline{\partial}\eta\wedge(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge(t\omega_{X})^{n-k-i}\wedge(\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{i-1}.

Notice that all the $(1,1)$ -forms appearing inside the integral are bounded by (4.8), (4.9), and that the function $\varphi_{t}-\underline{\varphi_{t}}$ is $O(t)$ by (4.10). On $K$ the estimate (2.10) gives

(4.12)

-Ct\omega_{y}\leq(\sqrt{-1}\partial\overline{\partial}\varphi_{t})|_{X_{y}}=(\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))|_{X_{y}}\leq Ct\omega_{y}.

The form $\sqrt{-1}\partial\overline{\partial}\eta\wedge(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}$ is the pullback of a form from $Y$ , and so we can use (4.12) to estimate

\left|\frac{\sqrt{-1}\partial\overline{\partial}\eta\wedge(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge(t\omega_{X})^{n-k-i}\wedge(\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{i-1}}{\omega_{X}^{n}}\right|\leq Ct^{n-m},

and so the term (4.11) goes to zero. This proves our claim.

We are then left with only the term with $k=m$ , which is

\frac{1}{a_{t}}\int_{X}\eta\binom{n}{m}(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{m}\wedge(t\omega_{X}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{n-m},

and if we expand the term $(t\omega_{X}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}-\underline{\varphi_{t}}))^{n-m}$ , we get

\begin{split}\frac{1}{a_{t}}&\int_{X}\eta\binom{n}{m}(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{m}\wedge(t\omega_{X})^{n-m}\\ &+\frac{1}{a_{t}}\int_{X}\sqrt{-1}\partial\overline{\partial}\eta\wedge(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{m}\wedge\dots,\end{split}

and the second term is zero because $\partial\overline{\partial}\eta$ is the pullback of a form from the base. We are then left with the term

(4.13)

\frac{1}{a_{t}}\int_{X}\eta\binom{n}{m}(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{m}\wedge(t\omega_{X})^{n-m},

which we need to further estimate. Using (4.8) we see that, up to taking a further subsequence, the functions $\varphi_{t_{i}}$ converge to $\hat{\varphi}$ in the $C^{1,\beta}(K)$ topology, and (4.10) implies that the functions $\underline{\varphi_{t_{i}}}$ also converge to $\hat{\varphi}$ uniformly. We can then rewrite (4.13) as

\frac{t^{n-m}\binom{n}{m}}{a_{t}}\int_{X}\eta(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{m}\wedge\omega_{X}^{n-m}.

Using (2.6) we see that as $t$ goes to zero the coefficient $\frac{t^{n-m}\binom{n}{m}}{a_{t}}$ converges to

\frac{\int_{X}\omega_{1}^{n}}{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}.

On the other hand we have

\int_{X}\eta(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{m}\wedge\omega_{X}^{n-m}=\sum_{k=0}^{m}\binom{m}{k}\int_{X}\eta\omega_{0}^{m-k}\wedge(\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k}\wedge\omega_{X}^{n-m}.

The term with $k=0$ is independent of $t$ , while any term with $k>0$ can be written as

(4.14)

\int_{X}\underline{\varphi_{t}}\sqrt{-1}\partial\overline{\partial}\eta\wedge\omega_{0}^{m-k}\wedge(\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k-1}\wedge\omega_{X}^{n-m}.

The $(n,n)$ -form $\sqrt{-1}\partial\overline{\partial}\eta\wedge\omega_{0}^{m-k}\wedge(\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k-1}\wedge\omega_{X}^{n-m}$ is supported in $K$ and is uniformly bounded by (4.9), and the functions $\underline{\varphi_{t_{i}}}$ converge uniformly to $\hat{\varphi}$ , and so along the sequence $t_{i}$ the term (4.14) has the same limit as

\int_{X}\hat{\varphi}\sqrt{-1}\partial\overline{\partial}\eta\wedge\omega_{0}^{m-k}\wedge(\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k-1}\wedge\omega_{X}^{n-m}.

But this is equal to

\int_{X}\underline{\varphi_{t}}\sqrt{-1}\partial\overline{\partial}\eta\wedge\omega_{0}^{m-k}\wedge(\sqrt{-1}\partial\overline{\partial}\underline{\varphi_{t}})^{k-2}\wedge\sqrt{-1}\partial\overline{\partial}\hat{\varphi}\wedge\omega_{X}^{n-m},

and repeating the same argument $k-1$ times we see that along the sequence $t_{i}$ the term (4.14) converges to

\int_{X}\eta\omega_{0}^{m-k}\wedge(\sqrt{-1}\partial\overline{\partial}\hat{\varphi})^{k}\wedge\omega_{X}^{n-m}.

It follows that along the sequence $t_{i}$ the term (4.13) converges to

\frac{\int_{X}\omega_{1}^{n}}{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}\int_{X}\eta(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi})^{m}\wedge\omega_{X}^{n-m},

and using (4.6), (4.7) we get

\int_{X}\eta F\omega_{SF}^{n-m}\wedge\omega_{0}^{m}=\frac{\int_{X}\omega_{1}^{n}}{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}\int_{X}\eta(\omega_{0}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi})^{m}\wedge\omega_{X}^{n-m}.

We then integrate first along the fibers and get

\begin{split}\int_{Y}&\eta F\omega_{Y}^{m}\left(\int_{X_{y}}\omega_{SF,y}^{n-m}\right)\\ &=\frac{\int_{X}\omega_{1}^{n}}{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}\int_{Y}\eta(\omega_{Y}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi})^{m}\left(\int_{X_{y}}\omega_{y}^{n-m}\right),\end{split}

and since $\omega_{y}$ is cohomologous to $\omega_{SF,y}$ , we get

\int_{Y}\eta F\omega_{Y}^{m}=\frac{\int_{X}\omega_{1}^{n}}{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}\int_{Y}\eta(\omega_{Y}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi})^{m},

which is just the weak form of (4.3). This shows that any weak limit $\hat{\omega}$ of $\tilde{\omega}_{t}$ as $t\to 0$ satisfies (4.3) weakly, and we have already remarked that we can write $\hat{\omega}=\omega_{Y}+\sqrt{-1}\partial\overline{\partial}\hat{\varphi}$ with $\hat{\varphi}$ in $L^{\infty}$ . By Kołodziej’s uniqueness of $L^{\infty}$ weak solutions of (4.3) (see [ST2, Theorem 3.2] and [EGZ1, Z]), we must have $\hat{\varphi}=\psi$ , and so the whole sequence $\tilde{\omega}_{t}$ converges weakly to $\omega$ as $t\to 0$ . Then the bound (2.9) implies that $\varphi_{t}$ actually converges to $\psi$ in the $C^{1,\beta}_{loc}$ topology on $X\backslash S$ . ∎