Kähler–Ricci flow on blowups along submanifolds

Abstract

In this short note, we study the behavior of Kähler–Ricci flow on Kähler manifolds which contract divisors to smooth submanifolds. We show that the Kähler potentials are Hölder continuous and the flow converges sequentially in Gromov–Hausdorff topology to a compact metric space which is homeomorphic to the base manifold.

1 Introduction

The Ricci flow, introduced by Hamilton [6] in 1982, has been a powerful tool in solving problems in geometry and analysis. It deforms any metric with positive Ricci curvature in real 3-dimensional manifold to a metric with constant curvature [6]. By performing surgery through singular times, Perelman [9] used Ricci flow to solve the geometrization conjecture for 3-dimensional manifolds. On the complex aspect, the Ricci flow preserves the Kähler condition [1] and is reduced to a scalar equation with Monge–Ampère type, which after suitable normalization converges to a solution of the Calabi conjecture [1, 30]. The non-Kähler analogue of Ricci flow also generates much interest recently, among them are the Hermitian curvature flows and the pluriclosed flow [24, 25], the Chern–Ricci flow [27], the Anomaly flow [14] and etc, and we refer to [13] for a survey on the recent development of non-Kähler geometric flows.

The analytic minimal model program, laid out in [19], predicts how the Kähler–Ricci flow behaves on a projective variety. It is conjectured that the Kähler–Ricci flow will either collapse in finite time, or deform any projective variety to its minimal model after finitely many divisorial contractions or flips in the Gromov–Hausdorff (GH) topology. There are various results on the finite time collapsing of Kähler–Ricci flow, see for example [12, 17, 23, 28, 29] and references therein. The behavior of Kähler–Ricci flow on some small contractions is studied in [16, 22] and it is shown that the flow forms a continuous path in GH topology. In [20, 21], Song and Weinkove study the divisorial contractions when the divisor is contracted to discrete points, and it is shown that the flow converges in GH topology to a metric space which is isometric to the metric completion of the base manifold with the smooth limit of the flow outside the divisor, and the flow can be continued on the new space. The main purpose of this note is to generalize their results to divisorial contractions when the divisor is contracted to a higher dimensional subvariety.

Let Y be a Kähler manifold and $N\subset Y$ be a complex submanifold of codimension $k\ge 1$. Let X be the Kähler manifold obtained by blowing up Y along N, $\pi : X\rightarrow Y$ be the blown-down map and $E= \pi ^{-1}(N)$ be the exceptional divisor in X. We consider the (unnormalized) Kähler–Ricci flow on X:

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial \omega }{\partial t} = - {{\,\mathrm{Ric}\,}}(\omega ),\\&\omega (0) = \omega _0, \end{aligned}\right. \end{aligned}$$

(1.1)

for a suitable fixed Kähler metric $\omega _0$ on X. We assume the limit cohomology class satisfies $[\omega _0] + T K_X = [\pi ^* \omega _Y]$ for some Kähler metric $\omega _Y$ on Y, where the maximal existence time (see [26]) of the flow (1.1) is given by

$$\begin{aligned} T = \sup \{t>0: ~ [\omega _0] + t K_X \text { is } \text { K}\ddot{\text {a}}\text {hler}\}<\infty . \end{aligned}$$

We define the reference metrics along the flow

$$\begin{aligned} \hat{\omega }_t = \frac{T-t}{T} \omega _0 + \frac{t}{T}\pi ^*\omega _Y. \end{aligned}$$

In the following for notational simplicity we shall denote $\hat{\omega }_Y = \pi ^*\omega _Y$, which is a nonnegative (1, 1)-form on X.

It is well-known that the flow (1.1) is equivalent to the following parabolic complex Monge–Ampère equation

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial \varphi }{\partial t} = \log \frac{(\hat{\omega }_t + i\partial \bar{\partial }\varphi )^n}{\Omega },\\&\varphi ( 0 ) = 0, \end{aligned}\right. \end{aligned}$$

(1.2)

where $\omega = \hat{\omega }_t + i\partial \bar{\partial }\varphi$ satisfies (1.1) and $\Omega$ is a smooth volume form satisfying $i\partial \bar{\partial }\log \Omega = \frac{1}{T}( \hat{\omega }_Y - \omega _0 )$.

Our main theorem is on the behavior of the metrics $\omega (t)$ as $t\rightarrow T^-$.

Theorem 1.1

Let $\pi : X\rightarrow Y$ and $\omega _t = \omega _0 + i\partial \bar{\partial }\varphi _t$ be as above, then the following hold: there exists a uniform constant $C=C(n,\omega _0,\omega _Y, \pi )>0$

• (1) $\varphi _t$ is uniformly Hölder continuous in $(X,\omega _0)$, i.e. $|\varphi _t(p) - \varphi _t(q)|\le C d_{\omega _0}(p,q)^\delta$, for any $p,q\in X$ and some $\delta \in (0,1)$, and $\varphi _t\xrightarrow {C^\delta (X,\omega _0)} \varphi _T\in PSH(X,\pi ^*\omega _Y)\cap C^\delta (X,\omega _0)$. Moreover, $\varphi _T$ descends to a function $\bar{\varphi }_T\in PSH(Y,\omega _Y)\cap C^{\delta _0}(Y,\omega _Y)$ for some $\delta _0\in (0,1)$.

• (2) $\omega _t$ converge weakly to $\omega _T: = \pi ^*\omega _Y + i\partial \bar{\partial }\varphi _T$ as (1, 1)-currents on X and the convergence is smooth and uniform on any compact subset $K\Subset X\backslash E$.

• (3) diam$(X,\omega _t)\le C$ for any $t\in [0,T)$.

• (4) for any sequence $t_i\rightarrow T^-$, there exists a subsequence $\{t_{i_j}\}$ such that $(X,\omega _{t_{i_j}})$ (as compact metric spaces) converge in Gromov–Hausdorff topology to a compact metric space $(Z,d_Z)$.

• (5) the metric completion of $(Y\backslash N, \omega _T)$ is isometric to $(Y,d_T)$, where the distance function $d_T$ is induced from $\omega _T$ and defined in (3.12). And there exists an open dense subset $Z^\circ \subset Z$ such that $(Y\backslash N, d_T)$ and $(Z^\circ , d_Z)$ are homeomorphic and locally isometric. Furthermore $(Z,d_Z)$ is homeomorphic to $(Y,d_T)$.

The item (2) is known to hold for Kähler–Ricci flow for more general holomorphic maps $\pi :X\rightarrow Y$ with dim$Y = \mathrm {dim} X$ (see e.g. [11, 20, 26]). We include it in the theorem just for completeness. We remark that Theorem 1.1 also holds if the base Y has some mild singularities, for example, if the analytic subvariety N is locally of the form $\mathbb {C}^k\times (\mathbb {C}^{n-k}/\mathbb {Z}_p )$, where $\mathbb {Z}_p$ denotes the $S^1$-action $\{e^{2l\pi i/ p}\}_{l=1}^p$ on $\mathbb {C}^{n-k}$ by

$$\begin{aligned} e^{2l\pi i/p}\cdot (z_{k+1},\ldots , z_n) \rightarrow (e^{2l\pi i/p} z_{k+1},\ldots , e^{2l\pi i/p} z_n). \end{aligned}$$

The proof is by combining the techniques of [21] and this note, so we omit the details.

Lastly we mention that under the same set-up as in Theorem 1.1, the same and even stronger results hold for Kähler metrics along continuity method. More precisely, let $u_t\in PSH(X, \hat{\omega }_Y + t \omega _0)$ be the solution to the complex Monge–Ampère equations

$$\begin{aligned} \omega _t^n = (\hat{\omega }_Y + t \omega _0 + i\partial \bar{\partial }u_t )^n = c_te^{F} \omega _0^n,\quad \sup u_t = 0, \, t\in (0,1], \end{aligned}$$

(1.3)

where F is a given smooth function on X and $c_t$ is a normalizing constant so that the integral of both sides are the same. It has been shown in [3] that $\mathrm {diam}(X, \omega _t)$ is bounded by a constant $C=C(n,\omega _0, \hat{\omega }_Y ,F)>0$ and the Ricci curvature of $\omega _t$ is uniformly bounded below. We can repeat almost identically the proof of Theorem 1.1 to the Eq. (1.3) to get the same conclusions for $u_t$ as the $\varphi _t$ in Theorem 1.1. Furthermore, along the continuity method (1.3), we can improve the Gromov–Hausdorff convergence in Theorem 1.1 in the sense that the full sequence (without the need of passing to a subsequence) $(X,\omega _t)$ converges in GH topology to a compact metric space $(Z,d_Z)$ which is isometric to the metric completion of $(Y\backslash N, \hat{\omega }_0)$, where $\hat{\omega }_0$ is the smooth limit of $\omega _t$ on $X\backslash \pi ^{-1}(N) = Y\backslash N$. The main advantage in this case is that the Ricci curvature has uniform lower bound so we can apply the argument in [2], in particular the Gromov’s lemma to find an almost geodesic connecting any two points away from the singular set $\pi ^{-1}(N)$.

2 Preliminaries

The following estimates are well-known [11, 20, 26, 30], so we just state the results and omit the proofs.

Lemma 2.1

There exists a constant $C>0$ depending only on $(X,\omega _0)$, $(Y,\omega _Y)$ such that

• (i) $\Vert \varphi \Vert _{L^\infty (X)}\le C$ for all $t\in [0,T)$,

• (ii) $\dot{\varphi }: = \frac{\partial \varphi }{\partial t}\le C$ and this is equivalent to $\omega ^n \le C \Omega$ from the Eq. (1.2).

• (iii) as $t\rightarrow T^-$, $\varphi$ converge to a bounded $\hat{\omega }_Y$-PSH function $\varphi _T$ and $\omega$ converge weakly to $\omega _T: = \hat{\omega }_Y + i\partial \bar{\partial }\varphi _T$ as (1, 1)-currents on X.

Lemma 2.2

There exists a uniform constant $C>0$ such that

• (i) $\hat{\omega }_Y \le C \omega$ for all $t\in [0,T)$,

• (ii) for any compact subset $K \Subset X\backslash E$, there exists a constant $C_{j,K}>0$ such that $\Vert \varphi \Vert _{C^j(K,\omega _0)}\le C_{j,K}$. Therefore the convergence $\omega _t\rightarrow \omega _T$ and $\varphi \rightarrow \varphi _T$ is smooth on $X\backslash E$, so $\omega _T$ and $\varphi _T$ are both smooth on $X\backslash E$.

In the proof of Lemma 2.2, we need the following Chern–Lu inequality as in the proof the parabolic Schwarz lemma [18]

$$\begin{aligned} \left( \frac{\partial }{\partial t} - \Delta \right) \log {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y \le C {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y, \end{aligned}$$

where $C>0$ depends also on the upper bound of the bisectional curvature of $(Y,\omega _Y)$. In turn this implies the equation below which will be used later.

$$\begin{aligned} \left( \frac{\partial }{\partial t} - \Delta \right) {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y \le - \frac{|\nabla {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y|^2}{{{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y} + C ({{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y)^2 \le - c_0 |\nabla {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y|^2 + C, \end{aligned}$$

(2.1)

where $c_0 = C^{-1}>0$ is the reciprocal of the constant C in (i) Lemma 2.2.

2.1 Kähler metrics from the blown up

We will construct a smooth function $\sigma _Y$ on Y such that $\sigma _Y = 0$ precisely on N. Choose a finite open cover $\{V_\alpha \}_{\alpha = 1}^J$ of N in Y and complex coordinates $\{w_{\alpha ,i}\}_{i=1}^n$ on $V_\alpha$ such that $N\cap V_\alpha = \{w_{\alpha , 1} = \cdots = w_{\alpha , k} = 0\}$. We also denote $V_0 = Y\backslash \cup _\alpha \frac{1}{2}\overline{V_\alpha }$ and we may also assume that $V_0\cap N = \emptyset$. Take a partition of unity $\{\theta _\alpha \}_{\alpha = 0}^J$ subordinate to the open cover $\{V_\alpha \}_{\alpha = 0}^J$, and we define a smooth function

$$\begin{aligned} \sigma _Y = \theta _0\cdot 1 \;+\; \sum _{\alpha = 1}^J \theta _\alpha \cdot \sum _{j=1}^k |w_{\alpha , j}|^2\in C^\infty (Y), \end{aligned}$$

and it is straightforward to see from the construction that $\sigma _Y$ vanishes precisely along N. Since $\{w_{\alpha ,i}\}_{i=1}^k$ are defining functions of N, it follows that if $V_\alpha \cap V_\beta \ne \emptyset$, then the function

$$\begin{aligned} f_{\alpha \beta } := \frac{\sum _{j=1}^k |w_{\alpha , j}|^2}{\sum _{j=1}^k |w_{\beta ,j}|^2},\quad \text {on }V_\alpha \cap V_\beta \end{aligned}$$

is never-vanishing and bounded from above. Since the cover is finite we have

$$\begin{aligned} 0<c\le \inf _{\alpha ,\beta } \inf _{y\in V_\alpha \cap V_\beta \ne \emptyset } f_{\alpha \beta }(y)\le \sup _{\alpha ,\beta } \sup _{y\in V_\alpha \cap V_\beta \ne \emptyset } f_{\alpha \beta }(y) \le C<\infty . \end{aligned}$$

(2.2)

We denote $\sigma _X = \pi ^* \sigma _Y$ to be the pull-back of $\sigma _Y$ to X.

Lemma 2.3

(see also [10]) There exists an $\varepsilon _0>0$ such that for all $\varepsilon \in (0, \varepsilon _0]$ the (1, 1)-form

$$\begin{aligned} \omega _\varepsilon : = \pi ^*\omega _Y + \varepsilon i\partial \bar{\partial }\log \sigma _X \end{aligned}$$

is positive definite on $X\backslash E$ and extends to a smooth Kähler metric on X.

Proof

We only need to prove the positivity of $\omega _\varepsilon$ near E, which is in fact local. So we may assume the map $\pi$ is defined from an open set $U\subset X$ to $V_\alpha$ given by

$$\begin{aligned} w_{\alpha , 1} = z_1,\, w_{\alpha ,2} = z_1 z_2,\ldots , w_{\alpha , k} = z_1 z_k,\, w_{\alpha , k+1} = z_{k+1},\ldots , w_{\alpha , n} = z_n, \end{aligned}$$

where $\{z_i\}$ are the complex coordinates on U such that $E\cap U = \{z_1 = 0\}$. It loses no loss of generality to assume $\omega _Y$ on $V_\alpha$ is just the Euclidean metric $\omega _{\mathbb {C}^n} = \sum _j i dw_{\alpha , j}\wedge d\bar{w}_{\alpha , j}$.

We note that on $V_\alpha$

$$\begin{aligned} \sigma _Y =\left( \sum _{\beta = 1, V_\beta \cap V_\alpha \ne \emptyset }^J \theta _\beta f_{\beta \alpha } \right) \cdot \sum _{j=1}^k |w_{\alpha , j}|^2 = : \phi _\alpha \cdot \sum _{j=1}^k |w_{\alpha ,j}|^2. \end{aligned}$$

From (2.2), we know that $\phi _\alpha$ is a smooth function with a strict positive lower bound on $V_\alpha$. In particular $\omega _Y + \varepsilon i\partial \bar{\partial }\log \phi _\alpha >0$ on $V_\alpha$ for any $0< \varepsilon \le \varepsilon _0<<1$.

We calculate

$$\begin{aligned} \pi ^* \omega _Y&= \left( 1+\sum _{j=2}^k |z_j|^2\right) dz_1\wedge d\bar{z}_1 + \sum _{j=2}^k (z_1 \bar{z}_j dz_j \wedge d\bar{z}_1 + \bar{z}_1 z_j dz_1 \wedge d\bar{z}_j)\nonumber \\&\quad +\, |z_1|^2 \sum _{j=2}^k dz_j \wedge d\bar{z}_j +\sum _{j=k+1}^n dz_j\wedge d\bar{z}_j, \end{aligned}$$

(2.3)

and note that on U

$$\begin{aligned} \log \sigma _X = \log \phi _\alpha + \log |z_1|^2 + \log \left( 1+ \sum _{j=2}^k |z_j|^2 \right) , \end{aligned}$$

so on $U\backslash E$ we have

$$\begin{aligned} i\partial \bar{\partial }\log \sigma _X = i\partial \bar{\partial }\log \phi _\alpha + \frac{\sum _{i, j=2}^k ( (1+|z'|^2)\delta _{ij} - \bar{z}_i z_j) \sqrt{-1}dz_i \wedge d\bar{z}_j}{(1+|z'|^2)^2}, \end{aligned}$$

(2.4)

where $z' = (z_2,\ldots , z_k)$ and the second term on RHS is nonnegative in $z'$-directions, which is just the Fubini-Study metric in the coordinates $z'$. By straightforward calculations, we see that if $\varepsilon$ is small enough the (1, 1)-form $\pi ^* \omega _Y + \varepsilon i\partial \bar{\partial }\log \sigma _X$ is positive on $X\backslash E$ and extends to a Kähler metric on X. $\square$

Remark 2.1

Globally from the above calculations we see that

$$\begin{aligned} \omega _\varepsilon = \pi ^*\omega _Y + \varepsilon i\partial \bar{\partial }\log \sigma _X - \varepsilon [E], \end{aligned}$$

where [E] denotes the current of integration along E.

We will denote $\omega _X = \pi ^* \omega _Y + \varepsilon _0i\partial \bar{\partial }\log \sigma _X - \varepsilon _0[E]$ to be a fixed Kähler metric obtained from the Lemma 2.3. The following inequality follows from the local expression of $\pi ^* \omega _Y$ as in the proof of Lemma 2.3.

Lemma 2.4

There exists a uniform constant $C>1$ such that

$$\begin{aligned} C^{-1} \hat{\omega }_Y \le \omega _X \le \frac{C}{\sigma _X}\hat{\omega }_Y, \end{aligned}$$

(2.5)

where the second inequality is understood on $X\backslash E$.

3 The proof of the main theorem

Now we are ready to derive the crucial estimates on $\omega$ along the Kähler–Ricci flow (1.1).

Lemma 3.1

There exists uniform constants $C>0$ and $\delta \in (0,1)$ such that along the flow (1.1) we have

$$\begin{aligned} \omega \le C \frac{\omega _0}{\sigma _X^{1-\delta }},\quad \text {on }X\backslash E \times [0, T). \end{aligned}$$

(3.1)

The proof is almost the same as that of Lemma 2.5 in [20], with minor modification using Lemma 2.3. For completeness, we provide a sketched proof.

Proof

Fix an $\epsilon \in (0,1)$ and define

$$\begin{aligned} Q_\epsilon = \log {{\,\mathrm{tr}\,}}_{\omega _0} \omega + A \log \sigma _X^{1+\epsilon } {{\,\mathrm{tr}\,}}_{\hat{\omega }_Y }\omega - A^2 \varphi , \end{aligned}$$

where $A>0$ is a constant to be determined later. First of all, $Q_\epsilon |_{t= 0}\le C$ for a constant C independent of $\epsilon \in (0,1)$, which can be seen from (2.5). Observe that for each time $t_0\in (0,T)$, $\max _X Q_\epsilon$ can only be achieved on $X\backslash E$, since $Q_\epsilon (x)\rightarrow -\infty$ as $x\rightarrow E$. Thus we assume the maximum of $Q_\epsilon$ is obtained at $(x_0,t_0)$ for some $x_0\in X\backslash E$. From the Chern–Lu inequality (e.g. Eq. (2.1)) the following holds on $X\backslash E$

$$\begin{aligned} \left( \frac{\partial }{\partial t} - \Delta \right) Q_\epsilon \le C {{\,\mathrm{tr}\,}}_\omega \omega _0 - A {{\,\mathrm{tr}\,}}_\omega ( A\hat{\omega }_t + (1+\epsilon )i\partial \bar{\partial }\log \sigma _X)+ A^2 \log \frac{\Omega }{\omega ^n} + C, \end{aligned}$$

where the constant C depends on the lower bound of the bisectional curvature of $(X,\omega _0)$ and the upper bound of bisectional curvature of $(Y,\omega _Y)$. Since $\hat{\omega }_t\ge c_1 \hat{\omega }_Y$ for a uniform $c_1>0$ and any $t\in [0,T)$, by Lemma 2.3 for $A>0$ large enough $A \hat{\omega }_t + (1+ \epsilon )i\partial \bar{\partial }\log \sigma _X \ge c_2 \omega _0$ on $X\backslash E$ for some $c_2>0$. If $A>0$ is taken even larger then at $(x_0,t_0)$, we have

$$\begin{aligned} 0\le \left( \frac{\partial }{\partial t} - \Delta \right) Q_\epsilon \le - 2 {{\,\mathrm{tr}\,}}_\omega \omega _0 + A^2 \log \frac{\Omega }{\omega ^n} + C\le - {{\,\mathrm{tr}\,}}_\omega \omega _0 + C, \end{aligned}$$

where in the last inequality we use

$$\begin{aligned} -{{\,\mathrm{tr}\,}}_\omega \omega _0 + A^2 \log \frac{\Omega }{\omega ^n}\le -{{\,\mathrm{tr}\,}}_\omega \omega _0 + n A^2 \log {{\,\mathrm{tr}\,}}_\omega \omega _0 + C \le C, \end{aligned}$$

as $\log x \le \varepsilon x + C(\varepsilon )$ for any $x\in (0,\infty )$. So we have ${{\,\mathrm{tr}\,}}_\omega \omega _0(x_0,t_0)\le C$. Then

$$\begin{aligned} {{\,\mathrm{tr}\,}}_{\omega _0}\omega |_{(x_0,t_0)}\le \frac{\omega ^n}{\omega _0^n } ({{\,\mathrm{tr}\,}}_\omega \omega _0)^{n-1}|_{(x_0,t_0)}\le C. \end{aligned}$$

Observing that from (2.5), $\sigma _X {{\,\mathrm{tr}\,}}_{\hat{\omega }_Y} \omega \le C {{\,\mathrm{tr}\,}}_{\omega _0} \omega$ on $X\backslash E$, thus $\sup _X Q_\epsilon \le C$ for some uniform constant $C>0$. Letting $\epsilon \rightarrow 0$, we get

$$\begin{aligned} \log {{\,\mathrm{tr}\,}}_{\omega _0} \omega + A \log \sigma _X {{\,\mathrm{tr}\,}}_{\hat{\omega }_Y } \omega \le C,\quad \text {on }X\backslash E \times [0,T). \end{aligned}$$

Finally from $C {{\,\mathrm{tr}\,}}_{\hat{\omega }_Y}\omega \ge {{\,\mathrm{tr}\,}}_{\omega _0}\omega$ we see from the above that

$$\begin{aligned} \log {{\,\mathrm{tr}\,}}_{\omega _0} \omega + \log \sigma _X^A ({{\,\mathrm{tr}\,}}_{\omega _0} \omega )^A \le C, \end{aligned}$$

so ${{\,\mathrm{tr}\,}}_{\omega _0}\omega \le C {\sigma _X^{- A/(1+A)}}$ on $X\backslash E$, and we can then take $\delta = \frac{1}{1+A}\in (0,1)$. $\square$

Next we will show the distance function defined by $\omega _t$ is Hölder-continuous with respect to the fixed metric $(X,\omega _0)$.

Lemma 3.2

There exists a uniform constant $C>0$ such that for any $p,q\in X$, it holds that

$$\begin{aligned} d_{\omega _t}(p,q)\le C d_{\omega _0}(p,q)^\delta ,\quad \forall ~ t\in [0,T), \end{aligned}$$

where $\delta \in (0,1)$ is the constant determined in Lemma 3.1.

Proof

It suffices to prove the estimate near E, say on T(E), a tubular neighborhood of E, since $\omega _t$ is uniformly equivalent to $\omega _0$ outside T(E). Choose coordinates charts $\{U_\alpha \}$ covering T(E) and local coordinates $\{z_{\alpha , i}\}_{i=1}^n$ such that $U_\alpha \cap E = \{z_{\alpha ,1} = 0\}$. We may assume that the cover is fine enough such that any $p,q\in T(E)$ with $d_{\omega _0}(p,q)\le \frac{1}{2}$ must lie in the same $U_\alpha$. Since we have only finitely many such $U_\alpha$, we will work on one of them only and omit the subscript $\alpha$ for simplicity. Furthermore the fixed Kähler metric $\omega _0$ is uniformly equivalent to the Euclidean metric $\omega _{\mathbb {C}^n}$ on U, so without loss of generality we assume $\omega _0 = \omega _{\mathbb {C}^n}$ on U. Recall that Lemma 3.1 implies that on $U\backslash E$ it holds that

$$\begin{aligned} \omega _t \le C \frac{\omega _{\mathbb {C}^n}}{|z_1|^{2(1-\delta )}},\quad \forall t\in [0,T), \end{aligned}$$

(3.2)

since $\sigma _X \sim |z_1|^2$ on U.

Take any two points $p,q\in U$ with $d_{\omega _0}(p,q) = d< \frac{1}{4}$. We will consider different cases depending on the positions of p, q.

$\bullet$Case 1$p,q\in E$. Rotating the coordinates if necessary we may assume $p = 0$ and $q = (0,d,0,\ldots ,0)$. We pick two points $\tilde{p} = (d, 0 ,\ldots , 0)$ and $\tilde{q} = (d,d,0,\ldots , 0)$ as shown the picture below. From (3.2), we have

$$\begin{aligned} d_{\omega _t} (p,\tilde{p}) \le L_{\omega _t}(\overline{p \tilde{p}})\le C \int _0^d \frac{1}{r^{1-\delta }}dr \le C d^\delta , \end{aligned}$$

where $\overline{p\tilde{p}}$ denotes the (Euclidean) line segment connecting p and $\tilde{p}$. Similarly $d_{\omega _t}(q,\tilde{q})\le C d^\delta$. On the other hand,

$$\begin{aligned} d_{\omega _t}(\tilde{p},\tilde{q}) \le L_{\omega _t}(\overline{\tilde{p}\tilde{q}}) \le \frac{C}{d^{1-\delta }} L_{\omega _{\mathbb {C}^n}}(\overline{\tilde{p}\tilde{q}}) = C d^\delta . \end{aligned}$$

If we denote $\gamma = \overline{p\tilde{p}} + \overline{\tilde{p}\tilde{q}} + \overline{q \tilde{q}}$ to be the piecewise line segment connecting p and q, then we have

$$\begin{aligned} d_{\omega _t}(p,q)\le L_{\omega _t}(\gamma )\le C d^\delta = C d_{\omega _0}(p,q)^{\delta }. \end{aligned}$$

We remark that $\gamma \subset X\backslash E$, except the two end points p, q.

$\bullet$Case 2$\min (d_{\omega _0}(p,E), d_{\omega _0}(q,E) )\le d$. The (Euclidean) projections of p, q to E, denoted by $p', q'$, respectively, must satisfy $d_{\omega _0}(p',q')\le d$. From the assumption it follows that $d_{\omega _0}(p,p')\le 2d$ and $d_{\omega _0}(q,q')\le 2d$. By similar arguments as above using (3.2) we have

$$\begin{aligned} d_{\omega _t}(p,p')\le C d^\delta ,\quad d_{\omega _t}(q,q')\le C d^\delta , \end{aligned}$$

and by Case 1 $d_{\omega _t}(p',q')\le C d_{\omega _0}(p',q')^\delta \le C d^\delta$. By triangle inequality we get the desired estimate $d_{\omega _t}(p,q)\le C d^\delta$.

$\bullet$Case 3$\min (d_{\omega _0}(p,E), d_{\omega _0}(q,E) )\ge d$. Every point in the (Euclidean) line segment $\overline{pq}$ has norm of $z_1$-coordinates no less than d, therefore

$$\begin{aligned} d_{\omega _t} (p,q)\le L_{\omega _t} (\overline{pq})\le C d^{-(1-\delta )} L_{\omega _{\mathbb {C}^n}} (\overline{p,q}) = C d^\delta . \end{aligned}$$

Combining the all the cases above, we finish the proof of the lemma. $\square$

Next we will prove the Hölder continuity of $\varphi _t$ with respect to $(X,\omega _0)$. To begin with, we first prove the gradient estimate of $\Phi := (T-t)\dot{\varphi }+ \varphi$ with respect to the evolving metrics $(X,\omega _t)$ (c.f. [3]).

Lemma 3.3

There exists a uniform constant $C>0$ such that

$$\begin{aligned} \sup _X |\nabla _{\omega _t} \Phi |_{\omega _t}\le C,\quad \forall ~ t\in [0,T). \end{aligned}$$

Proof

Taking $\frac{\partial }{\partial t}$ on both sides of (1.2), we get

$$\begin{aligned} \frac{\partial }{\partial t} \dot{\varphi }= \Delta \dot{\varphi }+ \frac{1}{T} {{\,\mathrm{tr}\,}}_\omega (\hat{\omega }_Y - \omega _0 ) = \Delta \dot{\varphi }+ \frac{1}{T-t}{{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y + \frac{1}{T-t}\Delta \varphi , \end{aligned}$$

where we used the equation $- \frac{1}{T} {{\,\mathrm{tr}\,}}_\omega \omega _0 = - \frac{n}{T-t} + \frac{t}{T(T-t)} {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y + \frac{1}{T-t} \Delta \varphi$. Then we have the equation

$$\begin{aligned} \left( \frac{\partial }{\partial t} - \Delta \right) \Phi = {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y - n\ge -n. \end{aligned}$$

(3.3)

By maximum principle, it follows that $\inf _X \Phi \ge - C$ for some constant depending also on T. Recall $\Phi$ is also bounded above by Lemma 2.1. And combining (3.3) with Bochner formula the following equation holds:

$$\begin{aligned} (\frac{\partial }{\partial t} - \Delta )|\nabla \Phi |^2_{\omega } = - |\nabla \nabla \Phi |^2 - |\nabla \bar{\nabla } \Phi | ^2 + 2 Re \left\langle \nabla \Phi , \bar{\nabla } {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y\right\rangle . \end{aligned}$$

Fix a constant $B: = \sup _{X\times [0,T)}|\Phi | + 2$. By direct calculations the following equation holds

$$\begin{aligned} \left( \frac{\partial }{\partial t} - \Delta \right) \frac{|\nabla \Phi |^2}{ B - \Phi }&= \frac{ \left( \frac{\partial }{\partial t}- \Delta \right) |\nabla \Phi |^2}{B - \Phi } + \frac{|\nabla \Phi |^2 \left( \frac{\partial }{\partial t}- \Delta \right) \Phi }{(B-\Phi )^2} + 2 Re \left\langle \nabla \log (B-\Phi ), \bar{\nabla } \frac{|\nabla \Phi |^2}{B-\Phi }\right\rangle \nonumber \\&= \frac{ - |\nabla \nabla \Phi |^2 - |\nabla \bar{\nabla } \Phi |^2 + 2 Re \left\langle \nabla \Phi , \bar{\nabla } {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y \right\rangle }{B-\Phi } + \frac{|\nabla \Phi |^2 ( {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y - n )}{(B-\Phi )^2}\nonumber \\&\quad + 2 Re \left\langle \nabla \log (B-\Phi ), \bar{\nabla } \frac{|\nabla \Phi |^2}{B- \Phi } \right\rangle . \end{aligned}$$

(3.4)

From the Eq. (2.1), we have

$$\begin{aligned} \begin{aligned}&\left( \frac{\partial }{\partial t}- \Delta \right) \frac{{{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y}{ B - \Phi }\\&\quad \le ~ \frac{-c_0 |\nabla {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y|^2 + C}{ B-\Phi } + \frac{{{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y ({{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y - n)}{(B-\Phi )^2} + 2 Re \left\langle \nabla \log (B-\Phi ), \bar{\nabla } \frac{{{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y}{ B - \Phi } \right\rangle . \end{aligned} \end{aligned}$$

(3.5)

Denote

$$\begin{aligned} G = \frac{|\nabla \Phi |^2}{ B - \Phi } + A \frac{{{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y}{ B - \Phi },\quad \text {where } A = 10 c_0^{-1}. \end{aligned}$$

By (3.4), (3.5) and Cauchy–Schwarz inequality we have

$$\begin{aligned}&\left( \frac{\partial }{\partial t}- \Delta \right) G\\&\quad \le \frac{ - |\nabla \nabla \Phi |^2 - |\nabla \bar{\nabla } \Phi |^2 - 9 |\nabla {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y |^2 }{B-\Phi } + C G + C + 2 Re\left\langle \nabla \log (B-\Phi ),\bar{\nabla } G\right\rangle . \end{aligned}$$

Assuming the maximum of G is attained at $(x_0,t_0)$, we may assume at this point $|\nabla \Phi |\ge A$, otherwise we are done yet. Then at this point $\left( \frac{\partial }{\partial t}- \Delta \right) G \ge 0$ and $\nabla G = 0$, hence we have $2 |\nabla \Phi |\cdot \nabla |\nabla \Phi | = - G \nabla \Phi - A \nabla {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y$. Taking norm on both side we get at $(x_0,t_0)$

$$\begin{aligned} \frac{|G \nabla \Phi + A \nabla {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y |^2}{2 |\nabla \Phi |^2} = 2 |\nabla |\nabla \Phi ||^2\le |\nabla \nabla \Phi |^2 + |\nabla \bar{\nabla } \Phi |^2, \end{aligned}$$

(3.6)

where we used the Kato’s inequality in the last inequality. Therefore at $(x_0,t_0)$, we have

$$\begin{aligned} 0&\le (B-\Phi )^{-1} \Big ( - \frac{1}{2} G^2 + A G \frac{|\nabla {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y|}{|\nabla \Phi |} + \frac{A^2}{2|\nabla \Phi |^2} |\nabla {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y|^2 - 9 |\nabla {{\,\mathrm{tr}\,}}_\omega \hat{\omega }_Y|^2 \Big ) + C G + C\\&\le -\frac{G^2}{4 (B-\Phi )} + C G + C, \end{aligned}$$

so at $(x_0,t_0)$, $G \le C$. From this we get the desired bound on $|\nabla \Phi |$. $\square$

An immediate consequence of the gradient bound is the uniform Hölder continuity of $\varphi _t$ on $(X,\omega _0)$.

Corollary 3.1

There exists a uniform constant $C>0$ such that

$$\begin{aligned} | \varphi _t(p) - \varphi _t(q) |\le C d_{\omega _0}(p,q)^\delta ,\quad \forall p,q\in X, \text { and }\forall t\in [0,T), \end{aligned}$$

where $\delta >0$ is the same constant as in Lemma 3.1.

Proof

Recall the definition of $\Phi$, that

$$\begin{aligned} \Phi _t = (T-t) \dot{\varphi } + \varphi = (T- t)^2 \frac{\partial }{\partial t} \Big ( \frac{\varphi _t}{T-t} \Big ). \end{aligned}$$

By the gradient bound in Lemma 3.3 and distance estimate in Lemma 3.2, for any fixed points $p, q\in X$, we have

$$\begin{aligned} | \Phi _t(p) - \Phi _t (q) |\le C d_{\omega _t}(p,q)\le C d_{\omega _0}(p,q)^\delta . \end{aligned}$$

So

$$\begin{aligned} \Big | \frac{\partial }{\partial t}\Big ( \frac{\varphi _t}{T-t} \Big )(p) - \frac{\partial }{\partial t}\Big ( \frac{\varphi _t}{T-t} \Big )(q) \Big | \le \frac{C d_{\omega _0}(p,q)^\delta }{(T-t)^2}, \end{aligned}$$

(3.7)

integrating (3.7) over $t\in [0,t_1)$ we get by noting that $\varphi _0 \equiv 0$ that

$$\begin{aligned} \big |\frac{\varphi _{t_1}(p)}{T - t_1} - \frac{\varphi _{t_1}(q)}{T- t_1} \big |\le C d_{\omega _0}(p,q)^\delta \int _0^{t_1} \frac{1}{(T-t)^2}dt= C d_{\omega _0}(p,q)^\delta \frac{t_1}{T(T-t_1)}, \end{aligned}$$

cancelling the common factor $\frac{1}{T-t_1}$ on both sides we get the desired estimate since $t_1\in (0,T)$ is arbitrarily chosen. $\square$

Remark 3.1

By an argument in [8], the Hölder continuity of $\varphi _t$ implies that the distance functions satisfy the estimate in Lemma 3.2.

Recall that the exceptional divisor E is a $\mathbb {CP}^{k-1}$-bundle over N and we identify N with the zero section of this bundle. Denote the bundle map by $\hat{\pi }: E \rightarrow N$ which is the restriction of $\pi : X\rightarrow Y$ to E. From Corollary  3.1, we see that the limit $\varphi _T\in PSH(X,\hat{\omega }_Y)$ is also Hölder continuous in $(X,\omega _0)$. Since $\hat{\omega }_Y |_{\hat{\pi }^{-1}(y)} = 0$ for any $y\in N$, we know that $\varphi _T|_{\hat{\pi }^{-1}(y)} = \text {const}$ for each $y\in N$ since $\hat{\pi }^{-1}(y)$ is connected. Thus $\varphi _T$ descends to a bounded function in $PSH(Y,\omega _Y)$, which we will still denote by $\varphi _T$. We shall show $\varphi _T$ is also Hölder continuous in $(Y,\omega _Y)$ with a possible different Hölder component.

Lemma 3.4

There exists a uniform constant $C>0$ such that

$$\begin{aligned} |\varphi _T(p) - \varphi _T(q) |\le C d_{\omega _Y}(p,q)^{\delta _Y},\quad \forall ~ p,q\in Y, \end{aligned}$$

(3.8)

where $\delta _Y = \min \{\delta (1-\delta ),\delta ^2\}\in (0,1)$.

Proof

We denote the zero section of the $\mathbb {CP}^{k-1}$-bundle $\hat{\pi }: E\rightarrow N$ by $\hat{N}$, and it is well-known that $\hat{N} \cong N$. It suffices to show (3.8) for p, q in a fixed tubular neighborhood T(N) of N, since on $Y\backslash T(N)$ the metric $\pi ^*\omega _Y = \hat{\omega }_Y$ is equivalent to $\omega _0$, and the estimate follows from Corollary 3.1.

Choose coordinates charts $(V_\alpha , w_{\alpha , j})$ covering T(N) such that $V_\alpha \cap N = \{ w_{\alpha ,1} = \cdots = w_{\alpha , k} = 0 \}$. We also assume that any $p,q\in T(N)$ with $d_{\omega _Y}(p,q)\le 1$ lie in the same $V_\alpha$, if the charts are chosen sufficiently fine. We will work in a fixed chart $(V, w_i)$, and omit the subscript $\alpha$. On this open set $\omega _Y$ is equivalent to the Euclidean metric $\omega _{\mathbb {C}^n}$ in $(\mathbb {C}^n, w_i)$, so without loss of generality, we may assume $\omega _Y = \omega _{\mathbb {C}^n}$ on V. The map $\pi : U\rightarrow V$ can be locally expressed as

$$\begin{aligned} w_1 = z_1,\, w_2 = z_1 z_2,\ldots , w_k = z_1 z_k,\, w_{k+1} = z_{k+1},\ldots , w_n = z_n, \end{aligned}$$

(3.9)

where $(U,z_i)$ is an open chart on X. The zero section $\hat{N}$ can be locally expressed as $\hat{N}\cap U = \{z_1=\cdots = z_k = 0\}$.

We consider different cases depending on the positions of p, q in V. Denote $0<d= d_{\omega _Y}(p,q) \le 1/4$.

$\bullet$Case 1 we assume $p,q\in N$. Take the unique pre-images under $\hat{p}$ of p, q in $\hat{N}$, $\hat{p}, \hat{q}$, respectively. We know that $\varphi _T(p) = \varphi _T(\hat{p})$ and $\varphi _T(q) = \varphi _T(\hat{q})$. The line segment $\overline{pq}$ is contained in N and similarly $\overline{\hat{p} \hat{q}}$ is contained in $\hat{N}$ as well. From the local expressions (2.3) and (2.4) of $\omega _X :=\pi ^*\omega _Y + \varepsilon _0 i\partial \bar{\partial }\log \sigma _X$, we conclude that $d_{\omega _Y}(p,q) = L_{\omega _Y}(\overline{pq})$ is comparable to $L_{\omega _X} (\overline{\hat{p}\hat{q}})$, which is no less than $c_1 d_{\omega _0}(\hat{p},\hat{q})$, for some uniform $c_1>0$. Therefore

$$\begin{aligned} | \varphi _T(p) - \varphi _T(q) | =&|\varphi _T(\hat{p}) - \varphi _T(\hat{q})| \le C d_{\omega _0}(\hat{p},\hat{q})^\delta \le C d_{\omega _Y}(p,q)^\delta , \end{aligned}$$

as desired.

$\bullet$Case 2 if $0<\min \{d_{\omega _Y}(p,N), d_{\omega _Y} (q,N) \}\le 2 d^{1-\delta }$. Take the orthogonal projections of p and q to N, $p', q'$ respectively. In other words, $p'$ ($q'$ resp.) has the same $(w_{k+1},\ldots , w_n)$-coordinates as p (q resp.) but the first k-coordinates are zero. From the assumption we know that $d_{\omega _Y}(p, p') = L_{\omega _Y}(\overline{pp'})\le 3d^{1-\delta }$ and $d_{\omega _Y}(q, q') = L_{\omega _Y}(\overline{q q'})\le 3d^{1-\delta }$. The pull-back of the line segment $\overline{pp'}$ under $\pi$ is also a line segment $\overline{\pi ^{-1}({p}) \hat{p}'}$ in $(U,z_i)$ connecting $\pi ^{-1}(p)$ and a unique point $\hat{p}'\in \hat{\pi } ^{-1}(p')\subset E$, and $\hat{p}' = (0,\frac{w_2}{w_1},\ldots , \frac{w_k}{w_1},w_{k+1},\ldots , w_n)$, where $w_j$ denotes the $w_j$-coordinate at p. It holds that $\varphi _T(p') = \varphi _T(\hat{p}')$ since $\hat{p}'$ lies at the fiber over $p'$. Again from the local expressions (2.3) and (2.4) of $\omega _X$, it follows that $L_{\omega _X} (\overline{\pi ^{-1}(p) \hat{p}'})$ is comparable to the length of $\overline{p p'}$ under $\omega _Y$, therefore

$$\begin{aligned} d_{\omega _0}(\pi ^{-1}(q), \hat{p}')\le C L_{\omega _X} ( \overline{ \pi ^{-1}(p) \hat{p}' } )\le C L_{\omega _Y}(\overline{p p'})\le C d^{1-\delta }, \end{aligned}$$

from which we derive that

$$\begin{aligned} |\varphi _T(p) - \varphi _T(p') | = |\varphi _T(\pi ^{-1}(p)) - \varphi _T(\hat{p} ') |\le C d_{\omega _0}(\pi ^{-1}(p), \hat{p} ' )^\delta \le C d^{\delta _Y}. \end{aligned}$$

Similar estimate also holds for $|\varphi _T(\pi ^{-1}(q)) - \varphi _T(q') |$. Since $p',q'\in E$ and $d_{\omega _Y}(p',q')\le d$, by Case 1 we also have $|\varphi _T(p') - \varphi _T(q')|\le C d^\delta$. The desired estimate (3.8) in this case then follows from triangle inequality.

$\bullet$Case 3$\min \{d_{\omega _Y} (p,N), d_{\omega _Y}(q, N)\}> 2d^{1-\delta }$. The line segment $\gamma (s) = \overline{pq}$ is strictly away from N, in fact, $\sigma _Y (\gamma (s) ) \ge d^{2(1-\delta )}$. Therefore the pull-back $\hat{\gamma }(s) = \pi ^{-1}(\gamma (s))$ joins $\pi ^{-1}(p)$ to $\pi ^{-1}(q)$ and $\sigma _X ( \hat{\gamma }(s) ) \ge d^{2(1-\delta )}$. From (2.5) that $\omega _X \le C \frac{\pi ^*\omega _Y}{ \sigma _X}$ on $X\backslash E$ we have

$$\begin{aligned} d_{\omega _0}(\pi ^{-1}(p), \pi ^{-1}(q)) \le C L_{\omega _X} (\hat{\gamma }) \le \frac{C}{d^{1-\delta }} L_{\omega _Y}(\gamma ) \le C d ^{\delta }. \end{aligned}$$

Therefore

$$\begin{aligned} | \varphi _T(p) - \varphi _T(q) | = | \varphi _T(\pi ^{-1}(p)) - \varphi _T(\pi ^{-1}(q)) |\le C d_{\omega _0}\big ( \pi ^{-1}(p), \pi ^{-1}(q) \big )^\delta \le C d^{\delta ^2}\le C d^{\delta _Y}, \end{aligned}$$

as desired.

Combining the cases discussed above, we finish the proof of the lemma. $\square$

The positive (1, 1)-form $\omega _T = \omega _Y + i\partial \bar{\partial }\varphi _T$ defines a Kähler metric $g_T$ on $Y\backslash N$, with the associated function $\tilde{d}_T: Y\backslash N \times Y\backslash N \rightarrow [0,\infty )$ defined by

$$\begin{aligned} \tilde{d}_T(p,q): = \inf \Big \{ \int _{\gamma \backslash N} \sqrt{g_T( \gamma ',\gamma ' )} |~ \gamma \subset Y \text { and }\gamma \text { joins }p\text { to }q \Big \} \end{aligned}$$

for any $p,q\in Y\backslash N$ and $\gamma$ is taken over all piecewise smooth curves in Y with only finitely many intersections with N. With this distance function, $(Y\backslash N, \tilde{d}_T)$ becomes a metric space, which may not be complete. We want to extend the distance function to the whole Y. To begin with, we need a trick from [8].

Lemma 3.5

There exists a uniform constant $C>0$ such that for any $p\in Y\backslash N$ and $r_p = d_{\omega _Y}(p, N)>0$

$$\begin{aligned} \tilde{d}_T(p,q)\le C d_{\omega _Y}(p,q)^{\delta _Y/2},\quad \forall q\in B_{\omega _Y}(p,r_p/2). \end{aligned}$$

Proof

The ball $B:=B_{\omega _Y}(p, r_p/2)$ is strictly away from N so $\omega _T$ is smooth on B. The function $d_p(x) = \tilde{d}_T(p, x)$ is Lipschtiz continuous and satisfies $|\nabla d_p|_{\omega _T}\le 1$ a.e.. For any $r\le \frac{r_p}{2}$, we have

$$\begin{aligned} \int _{B_{\omega _Y}(p,r)} |\nabla d_p|_{\omega _Y}^2 \omega _Y^n&\le \int _{B_{\omega _Y}(p,r)} |\nabla d_p|_{\omega _T}^2 ({{\,\mathrm{tr}\,}}_{\omega _Y} \omega _T )\omega _Y^n \\&\le \int _{B_{\omega _Y}(p,r)} ( n + \Delta _{\omega _Y} \varphi _T ) \omega _Y^n\\&\le C r^{2n} + \int _{B_{\omega _Y}(p, 1.5 r)} |\varphi _T(x) - \varphi _T(p)| |\Delta _{\omega _Y} \eta | \omega _Y^n\\&\le C r^{2n} + C r^{\delta _Y + 2n - 2} \le C r^{2n - 2 + \delta _Y}, \end{aligned}$$

where $\eta$ is a standard cut-off function supported in $B_{\omega _Y}(p, 1.5 r)$ and identically equal to 1 on $B_{\omega _Y}(p,r)$, and it satisfies $|\Delta _{\omega _Y} \eta |\le C r^{-2}$. Then by Poincare inequality and Campanato’s lemma (see Theorem 3.1 in [7]) we get

$$\begin{aligned} \tilde{d}_T(p,q) = d_p(q) = | d_p(q) - d_p(p) |\le C d_{\omega _Y}(p,q)^{\delta _Y/2}, \end{aligned}$$

for any $q\in B_{\omega _Y}(p, r_p/2)$. $\square$

Lemma 3.6

There exist constants $C>0$ and $\delta _0\in (0,1)$ such that

$$\begin{aligned} \tilde{d}_{T}(p,q)\le C d_{\omega _Y}(p,q)^{\delta _0},\quad \forall p, q\in Y\backslash N. \end{aligned}$$

(3.10)

Proof

We use the same notation as in the proof of Lemma 3.4. It suffices to show (3.10) for $p,q\in V$ where V is a fixed coordinate chart in Y and recall locally the map $\pi : U\rightarrow V$ is given by (3.9). Let $d = d_{\omega _Y}(p,q)<1/4$.

In case $\min \{d_{\omega _Y}(p,N), d_{\omega _Y}(q,N)\}> 2d$, then $q\in B_{\omega _Y}(p, \frac{1}{2} d_{\omega _Y}(p,N) )$. By Lemma 3.5, it follows that $\tilde{d}_T(p,q)\le C d^{\delta _Y/2}$. So it only remains to consider the case when the minimum above is $\le 2d$. Let $p',q'\in N\cap V$ be the orthogonal projection (assuming $\omega _Y = \omega _{\mathbb {C}^n}$) of p, q to N, respectively. Then $\max \{d_{\omega _Y}(p, p '), d_{\omega _Y}(q,q')\} \le 3d$ and $d_{\omega _Y}(p',q')\le d$. Choose the unique pre-images $\hat{p},\hat{q}\in \hat{N}\subset E$ in the zero section $\hat{N}$ of the bundle $\hat{\pi }: E\rightarrow N$, of $p', q'$, i.e. $\hat{\pi }(\hat{p}) = p'$ and $\hat{\pi }(\hat{q}) = q'$. From the local expressions (2.3) and (2.4) of $\omega _X = \pi ^*\omega _Y + \varepsilon _0i\partial \bar{\partial }\log \sigma _X$, it can be shown that $d_{\omega _X}(\hat{p},\hat{q}) \le C d_{\omega _Y} (p',q')\le C d$. As in the proof of Case 1 in Lemma 3.2 with p, q in that lemma replaced by $\hat{p}, \hat{q}$ here. Recall that the piecewise line segment $\gamma$ which connects $\hat{p}$ and $\hat{q}$ lies outside E, except the two end points. Furthermore $\gamma$ is chosen independent of $t\in [0,T)$ and we have

$$\begin{aligned} \int _\gamma \sqrt{ g_t( \gamma ',\gamma ' ) } = L_{\omega _t}(\gamma ) \le C d_{\omega _X}(\hat{p},\hat{q})^\delta \le C d^\delta , \end{aligned}$$

(3.11)

since $g_t \rightarrow g_T$ (locally) smoothly on $\gamma \backslash \{\hat{p}, \hat{q}\}$, letting $t\rightarrow T^-$ and applying Fatou’s lemma to (3.11), we get

$$\begin{aligned} \int _\gamma \sqrt{ g_T( \gamma ',\gamma ' ) } \le C d^{\delta }. \end{aligned}$$

Denote the image curve $\gamma _0 = \pi (\gamma )\subset Y$ which joins $p'$ to $q'$ and is contained in $Y\backslash N$ except the end points. It follows then that $L_{\omega _T}(\gamma _0)\le C d^\delta$. The line segment $\gamma _1(s) = \overline{p p'}$ is given by

$$\begin{aligned} \gamma _1(s) = ( s w_1(p), \ldots , s w_k(p), w_{k+1} (p),\ldots , w_n(p) ) ,\quad s\in [0,1] \end{aligned}$$

and its pull-back to X, $\hat{\gamma }_1(s) = \pi ^{-1}(\gamma (s))$ is locally given by

$$\begin{aligned} \hat{\gamma }_1(s) = ( s w_1(p), \frac{w_2(p)}{w_1(p)}, \ldots ,\frac{w_{k}(p)}{w_1(p)}, w_{k+1}(p),\ldots , w_{n}(p) ),\quad s\in [0,1]. \end{aligned}$$

By the estimate in Lemma 3.1, it follows that

$$\begin{aligned} \int _{\hat{\gamma }_1}\sqrt{ g_t( \hat{\gamma }_1',\hat{\gamma }_1 ' ) } \le C \int _{\hat{\gamma }_1} \sqrt{ \frac{g_X(\hat{\gamma }_1',\hat{\gamma }_1')}{s^{2(1-\delta )} |\mathbf {w}(p)|^{2(1-\delta )} } } \le C \int _{\hat{\gamma }_1} \frac{\sqrt{\pi ^*\omega _Y( \hat{\gamma }_1',\hat{\gamma }_1 ' )}}{ s^{1-\delta } |\mathbf {w}(p)|^{1-\delta } } \le C |\mathbf {w}(p)|^\delta \le C d^\delta , \end{aligned}$$

where $\mathbf {w}(p) = ( w_1(p),\ldots , w_k(p))$. By Fatou’s lemma and letting $t\rightarrow T^-$, we get

$$\begin{aligned} L_{\omega _T}(\gamma _1) = \int _{\hat{\gamma }_1} \sqrt{ g_T( \hat{\gamma }_1' , \hat{\gamma }_1' ) } \le C d^\delta . \end{aligned}$$

Similarly the line segment $\gamma _2 = \overline{q q'}$ also have $L_{\omega _T}(\gamma _2)\le C d^\delta$. Now we define a piecewise smooth curve

$$\begin{aligned} \bar{\gamma } = \gamma _1 + \gamma _0 + \gamma _2, \end{aligned}$$

which joins p to q and lies entirely outside N, except the two points $p'$ and $q'$. And combining the estimates above we get

$$\begin{aligned} L_{\omega _T}(\bar{\gamma }) = L_{\omega _T}(\gamma _1) + L_{\omega _T}(\gamma _0) + L_{\omega _T}(\gamma _2)\le C d^\delta . \end{aligned}$$

Then by definition

$$\begin{aligned} \tilde{d}_T (p,q)\le L_{\omega _T} (\bar{\gamma }) \le C d^\delta = C d_{\omega _Y}(p,q)^\delta . \end{aligned}$$

From the discussions above, (3.10) follows for $\delta _0 = \min (\delta _Y/2, \delta )$. $\square$

We now extend the distance function $\tilde{d}_T$ to Y, for any $p\in Y\backslash N$ and $q\in N$, we define the distance

$$\begin{aligned} d_T(p,q):= \lim _{i\rightarrow \infty } \tilde{d}_T(p, q_i), \end{aligned}$$

(3.12)

where $\{q_i\}\subset Y\backslash N$ is a sequence of points such that $d_{\omega _Y} (q,q_i)\rightarrow 0$. We need to justify $d_T$ is well-defined, i.e. the limit exists and is independent of the choice of the sequence $\{q_i\}$.

Lemma 3.7

The limit in (3.12) exists and for any other sequence $\{q_i'\}\subset Y\backslash N$ converging to q in $(Y,\omega _Y)$, the following holds

$$\begin{aligned} \lim _{i\rightarrow \infty } \tilde{d}_T(p,q_i) = \lim _{i\rightarrow \infty } \tilde{d}_T(p, q_i'). \end{aligned}$$

Proof

This is in fact an immediate consequence of Lemma 3.6. Observe that

$$\begin{aligned} | \tilde{d}_T(p,q_i) - \tilde{d}_T(p,q_j) | \le \tilde{d}_T(q_i, q_j) \le C d_{\omega _Y}(q_i,q_j)^{\delta _0}\rightarrow 0,\quad \text {as }i,j\rightarrow \infty . \end{aligned}$$

Thus $\{\tilde{d}_T(p,q_i)\}_{i=1}^\infty$ is a Cauchy sequence hence it converges. On the other hand, similarly we have

$$\begin{aligned} |\tilde{d}_T(p,q_i) - \tilde{d}_T(p,q_i') |\le C d_{\omega _Y}(q_i,q_i')^\delta \rightarrow 0,\quad \text {as }i\rightarrow \infty , \end{aligned}$$

and it then follows that the limit is independent of the choice of $\{q_i\}$ converging to q. $\square$

We then define the distance between points in N as follows: for any $p,q\in N$

$$\begin{aligned} d_T(p,q): = \lim _{i\rightarrow \infty } \tilde{d}_T ( p_i, q_i ), \end{aligned}$$

for two sequences $Y\backslash N\supset \{p_i\}\rightarrow p$ and $Y\backslash N\supset \{q_i\}\rightarrow q$ under $d_{\omega _Y}$. It can be checked similar as Lemma 3.7 that the limit exists and is independent of the choice of sequences converging to p or q. Thus $(Y, d_T)$ defines a compact metric space, since $d_T(p,q)\le C d_{\omega _Y}(p,q)^{\delta _0}$ for any $p,q\in Y$, which follows from Lemma 3.6.

We now turn to the Gromov–Hausdorff (GH) convergence of the flow. The proof is motivated by [15] (see also [3, 5, 29]).

Lemma 3.8

For any $t_i\rightarrow T^-$, there exists a subsequence which we still denote by $\{t_i\}$ such that as compact metric spaces

$$\begin{aligned} (X,\omega _{t_i})\xrightarrow {d_{GH}} (Z,d_Z) \end{aligned}$$

for some compact metric space $(Z,d_Z)$.

Proof

For any $\epsilon >0$, we choose an $\epsilon$-net $\{x_j\}_{j=1}^{N_{i,\epsilon }}\subset (X,\omega _{t_i})$, in the sense that $d_{\omega _{t_i}}(x_j, x_{j'})>\epsilon$ and the open balls $\{B_{\omega _{t_i}}(x_j, 2\epsilon )\}_{j}$ cover $(X,\omega _{t_i})$. From Lemma 3.2, we have

$$\begin{aligned} \epsilon < d_{\omega _{t_o}} (x_j, x_{j'}) \le C d_{\omega _0} (x_j, x_{j'})^\delta , \end{aligned}$$

thus under the fixed metric $d_{\omega _0}$, each pair of points $(x_j, x_{j'})$ from the $\epsilon$-net has distance at least $C^{-1/\delta } \epsilon ^{1/\delta }$, thus the balls $\{B_{\omega _0}(x_j, C^{-1/\delta } \epsilon ^{1/\delta }/2)\}_j$ are disjoint, so for some $c>0$

$$\begin{aligned} N_{i,\epsilon } c \epsilon ^{1/\delta ^{2n}} = \sum _{j=1}^{N_{i,\epsilon }} c \epsilon ^{1/\delta ^{2n}} \le \int _{\cup _j B_{\omega _0}(x_j, C^{-1/\delta } \epsilon ^{1/\delta }/2)} \omega _0^n \le \int _X \omega _0^n, \end{aligned}$$

from which we derive an upper bound of $N_{i,\epsilon }\le N_\epsilon$, which is independent of i. Then by Gromov’s precompactness theorem [4], there exists a compact metric space $(Z,d_Z)$, such that up to a subsequence $(X,\omega _{t_i})\xrightarrow {d_{GH}} (Z,d_Z)$. $\square$

Lemma 3.9

There exists an open and dense subset $Z^\circ \subset Z$ such that $(Z^\circ , d_Z)$ and $(Y\backslash N, d_T)$ are homeomorphic and locally isometric.

Proof

For notational convenience we denote $Y^\circ = Y\backslash N$. The maps $\pi _i = \pi : (X,\omega _{t_i}) \rightarrow (Y, \omega _Y)$ are Lipschitz by the estimate $\pi ^*\omega _Y \le C \omega _{t_i}$ as in (ii) of Lemma 2.2. The target space $(Y,\omega _Y)$ is compact, so by Arzela–Ascoli theorem up to a subsequence of $\{t_i\}$, along the GH convergence $(X,\omega _{t_i})\xrightarrow {d_{GH}}(Z,d_Z)$, the maps $\pi _i \xrightarrow {{GH}} \pi _Z$, for some map $\pi _Z : (Z,d_Z)\rightarrow (Y,\omega _Y)$, in the sense that for any $(X,\omega _{t_i})\ni x_i\xrightarrow {d_{GH}} z\in Z$, $\pi _i(x_i)\xrightarrow {d_{\omega _Y}} \pi _Z(z)$ in Y. $\pi _Z$ is also Lipschitz from $(Z,d_Z)$ to $(Y,\omega _Y)$, i.e. $d_{\omega _Y}(\pi _Z(z_1),\pi _Z(z_2))\le C d_Z(z_1, z_2)$ for any $z_1,z_2\in Z$. We denote $Z^\circ = \pi _Z^{-1}(Y^\circ )$, and we will show that $\pi _Z|_{Z^\circ }: (Z^\circ , d_Z) \rightarrow (Y^\circ , d_T)$ is homeomorphic and locally isometric, and $Z^\circ \subset Z$ is open and dense. The openness of $Z^\circ \subset Z$ follows from the continuity of the map $\pi _Z : (Z,d_Z)\rightarrow (Y,d_{\omega _Y})$ and the fact that $Y^\circ \subset Y$ is open.

$\bullet$$\pi _Z|_{Z^\circ }$is injective: suppose $z_1, z_2\in Z^\circ = \pi _Z^{-1}(Y^\circ )$ are mapped to the same point $y\in Y^\circ$, $\pi _Z(z_1) = \pi _Z(z_2) = y$. Since $(Y^\circ , \omega _T)$ is an incomplete smooth Riemannian manifold and locally in $Y^\circ$, $d_T$ is induced from the Riemannian metric, we can find a small $r = r_y>0$ such that the metric ball $(B_{\omega _T}(y, 2r), \omega _T)$ is geodesically convex. Choose two sequence of points $z_{1,i},z_{2,i}\in (X,\omega _{t_i})$ converging in GH sense to $z_1, z_2\in Z$, respectively. From the convergence of $\pi _i\xrightarrow {GH} \pi _Z$, we obtain $d_{\omega _Y} ( \pi _i(z_{1,i}), \pi _Z(z_1) )\xrightarrow {i\rightarrow \infty } 0$ and $d_{\omega _Y} ( \pi _i(z_{2,i}), \pi _Z(z_2) )\xrightarrow {i\rightarrow \infty } 0$. By Lemma 3.6, the same limits hold with $d_{\omega _Y}$ replaced by $d_T$. In particular this implies that $d_T(\pi _i(z_{1,i}), \pi _{i}(z_{2,i}) )\xrightarrow {i\rightarrow \infty } 0$ and both $\pi _{i}(z_{1,i})$ and $\pi _i(z_{2,i})$ lie inside $B_{\omega _T}(y, r/2)$ when i is large enough. We can find $\omega _T$-geodesics $\gamma _i\subset B_{\omega _T} (y, r)$ connecting $\pi _i(z_{1,i})$ and $\pi _i(z_{2,i})$, and by the uniform and smooth convergence of $\omega _{t_i}\rightarrow \omega _T$ on $\overline{\pi ^{-1}(B_{\omega _T} (y, 2r) ) }$, it follows that

$$\begin{aligned} 0\le d_{\omega _{t_i}}(z_{1,i}, z_{2,i}) \le L_{\omega _{t_i}} (\hat{\gamma }_i) \le L_{\omega _T}(\gamma _i) + \epsilon _i \!=\! d_{T}( \pi (z_{1,i}),\pi _i( z_{2,i}) ) + \epsilon _i \xrightarrow {i\rightarrow \infty } 0, \end{aligned}$$

where $\hat{\gamma }_i = \pi ^{-1}(\gamma _i)$ is a curve joining $z_{1,i}$ to $z_{2,i}$ and $\{\epsilon _i\}$ is a sequence tending to zero. From the definition of GH convergence we see that

$$\begin{aligned} d_Z(z_1, z_2) = \lim _{i\rightarrow \infty }d_{\omega _{t_i}} (z_{1,i}, z_{2,i}) = 0. \end{aligned}$$

Hence $z_1 = z_2$ and $\pi _{Z}|_{Z^\circ }$ is injective.

$\bullet$$\pi _{Z} |_{Z^\circ }: (Z^\circ , d_Z) \rightarrow (Y^\circ , d_T)$is a local isometry. We first explain what the local isometry means. It says that for any $z\in Z^\circ$ and $y = \pi _Z(z)\in Y^\circ$, we can find open sets $z\in U\subset Z^\circ$ and $y\in V\subset Y^\circ$ such that $\pi _Z|_U: (U,d_Z) \rightarrow (V,d_T)$ is an isometry.

There exists a small $r = r_y>0$ such that the metric ball $(B_{\omega _T}(y, 3r),\omega _T) \subset Y^\circ$ and is geodesically convex. Take $U = (\pi _Z|_{Z^\circ })^{-1} (B_{\omega _T}(y, r) )$. Since $B_{\omega _T}(y, r)$ is also open in $(Y, \omega _Y)$, it can be seen that U is open in $Z^\circ$ and is a neighborhood of $z\in Z^\circ$. We will show $\pi _Z|_{U}: (U,d_Z) \rightarrow ( B_{\omega _T}(y,r),\omega _T )$ is an isometry, i.e. for any $z_1,z_2\in U$, and $y_1 = \pi _Z(z_1)$, $y_2 = \pi _Z(z_2)$, we have $d_Z(z_1, z_2) = d_T(y_1, y_2)$.

We choose sequences of points $z_{1,i},z_{2,i}\in (X,\omega _{t_i})$ converging in GH sense to $z_1, z_2$, respectively, as before. It then follows from $\pi _i \xrightarrow {GH} \pi _Z$ and Lemma 3.6 that $d_{T}( \pi _i(z_{a,i}), y_a ) \rightarrow 0$ as $i\rightarrow \infty$, for each $a= 1,2$. In particular when i is large enough, $\pi _i(z_{a,i})\in B_{\omega _T}(y, 1.1 r)$. Choose a minimal $\omega _{t_i}$-geodesic $\hat{\gamma }_i$ joining $z_{1,i}$ to $z_{2,i}$, and we have

$$\begin{aligned} d_{\omega _{t_i}}(z_{1,i}, z_{2,i}) = L_{\omega _{t_i}}(\hat{\gamma }_i) \xrightarrow {i\rightarrow \infty } d_Z(z_1, z_2). \end{aligned}$$

Denote the image $\gamma _i = \pi _i(\hat{\gamma }_i)$ which is a continuous curve joining $\pi _i(z_{1,i})$ to $\pi _i(z_{2,i})$. If $\gamma _i\subset B_{\omega _T}(y, 3r)$ (for a subsequence of i), since $\omega _{t_i}$ converge smoothly and uniformly to $\omega _T$ on the compact subset $\overline{\pi ^{-1} ( B_{\omega _T} (y, 3r) ) }$, it follows

$$\begin{aligned} d_{T}( \pi _i(z_{1,i}), \pi _i (z_{2,i}) ) \le L_{\omega _T} (\gamma _i) \le L_{\omega _{t_i}} (\hat{\gamma }_i) + \epsilon _i \xrightarrow {i\rightarrow \infty } d_Z(z_1,z_2). \end{aligned}$$

In case $\gamma _i\not \subset B_{\omega _T}(y, 3 r)$ for i large enough, we have

$$\begin{aligned} d_T( \pi _i(z_{1,i}), \pi _i (z_{2,i}) )\le & {} 2.5 r \le L_{\omega _T}( \gamma _i\cap B_{\omega _T} (y, 3r) ) \\\le & {} L_{\omega _{t_i}} (\gamma _i) + \epsilon _i \xrightarrow {i\rightarrow \infty } d_Z(z_1, z_2). \end{aligned}$$

Observe that $d_T( \pi _i(z_{1,i}), \pi _i (z_{2,i}) )\xrightarrow {i\rightarrow \infty } d_{T} (\pi _Z(z_1) , \pi _Z(z_2) ) = d_T(y_1, y_2)$. So by the discussion in both cases, it follows that $d_T(y_1, y_2)\le d_Z(z_1, z_2)$. To see the reverse inequality, by the geodesic convexity of $(B_{\omega _T}(y, 3r), \omega _T )$, we can find minimal $\omega _T$-geodesics $\sigma _i\subset B_{\omega _T}(y, 3r)$ connecting $\pi _i(z_{1,i})$ and $\pi _i(z_{2,i})$ for i large enough. The pull-back $\hat{\sigma }_i = \pi ^{-1}(\sigma _i)\subset \overline{B_{\omega _T}(y, 3r) }$ joins $z_{1,i}$ to $z_{2,i}$, again by the local smooth convergence of $\omega _{t_i}$ to $\omega _T$, we have

$$\begin{aligned} d_{\omega _{t_i}}(z_{1,i}, z_{2,i})\le & {} L_{\omega _{t_i}} ( \hat{\sigma }_i )\le L_{\omega _T}( \sigma _i ) + \epsilon _i \\= & {} d_{T}(\pi _i(z_{1,i}), \pi _i(z_{2,i}) ) + \epsilon _i\xrightarrow {i\rightarrow \infty } d_{T} (y_1,y_2), \end{aligned}$$

letting $i\rightarrow \infty$ we get $d_Z(z_1, z_2)\le d_T(y_1, y_2)$. Thus we show that $d_{Z}(z_1,z_2) = d_T(y_1, y_2)$, as desired.

$\bullet$$\pi _Z|_{Z^\circ }$is surjective. This follows from the definition. Indeed, for any $y\in Y^\circ$, take $z = z_i = \pi ^{-1}(y)\in (X,\omega _{t_i})$, up to a subsequence $z_i\xrightarrow {d_{GH}} z_0\in Z$. Since $\pi _i \xrightarrow {GH} \pi _Z$, we get $d_{\omega _Y} (y, \pi _Z(z_0)) = d_{\omega _Y} ( \pi _i(z_i), \pi _Z(z_0) ) \rightarrow 0$ as $i\rightarrow \infty$. So $\pi _Z(z_0) = y$ and $z_0\in Z^\circ$ is the pre-image of y under $\pi _Z|_{Z^\circ }$.

Combining the discussions above, we see that $\pi _Z|_{Z^\circ }: (Z^\circ , d_Z) \rightarrow (Y^\circ , d_T)$ is a bijection and thus a homeomorphism (noting that the continuity of the maps $\pi _Z|_{Z^\circ }$ and $(\pi _Z|_{Z^\circ })^{-1}$ follow from the local isometry property).

It only remains to show $Z^\circ \subset Z$ is dense. Suppose not, there exists a point $z_0\in Z$ such that $B_{d_Z} (z_0,\bar{\varepsilon }) \subset Z\backslash Z^\circ$ for some $\bar{\varepsilon }>0$. Choose a sequence of points $x_{i}\in (X,\omega _{t_i})$ such that $x_{i}\xrightarrow {d_{GH}} z_0$. We claim that $d_{\omega _{t_i}}(x_{i}, E) \rightarrow 0$ as $i\rightarrow \infty$, where E is the exceptional divisor of the blown-down map $\pi : X\rightarrow Y$. If not, then $d_{\omega _{t_i}}(x_{i}, E)\ge a_0>0$ for a sequence of large i’s, by Lemma 3.2, under the fixed metric $w_0$, $d_{\omega _0} (x_{i}, E)\ge C^{-1/\delta } a_0^{1/\delta }>0$, thus $\{x_{i}\} \subset K$, for some compact subset $K\Subset X\backslash E$. It then follows that $\pi _i(x_{i})\in \pi (K)\Subset Y^\circ$, and this contradicts the fact that $d_{\omega _Y}( \pi _i(x_{i}), \pi _Z(z_0) )\rightarrow 0$ and $\pi _Z(z_0)\not \in Y^\circ$. Therefore, we may assume without loss of generality that $x_{i}\in E$ for all i. Moreover, from Lemma 3.10 below, we may replace $x_{i}\in E$ by the point in the same fiber as $x_{i}$ of the $\mathbb {CP}^{k-1}$-bundle $\hat{\pi }: E\rightarrow N$ and the zero section $\hat{N}$. So we can assume in addition that $x_{i}\in \hat{N}$. Denote the points $y_{i} = \pi _i(x_i)\in N$ and $y_0 = \pi _Z(z_0)\in N$. From $\pi _i\xrightarrow {GH} \pi _Z$ and $x_i\xrightarrow {d_{GH}} z_0$, we have $d_{\omega _Y}(y_i, y_0)\rightarrow 0$ as $i\rightarrow \infty$.

We may choose a coordinates chart $(V, w_j)$ as before, which is centered at $y_0$ and contains all but finitely many $y_i$, and $N\cap V = \{w_1 = \cdots = w_k = 0\}$. We take an open set $(U,z_j)$ over $(V,w_j)$, such that the map $\pi : U\rightarrow V$ is expressed as in (3.9). We fix a point $p\in V\backslash N$ whose w-coordinate is $w(p) = (r, 0,\ldots , 0)$ for some $r>0$ to be determined. Take $\hat{p} = \pi ^{-1}(p)$ and its z-coordinate is $z(\hat{p}) = (r,0,\ldots , 0)$. The point(s) $\hat{p}_i = \hat{p}\in (X,\omega _{t_i})$ converge (up to a subsequence) in GH sense to some point $p_Z\in Z$, and as above, we have $d_{\omega _Y}(p, \pi _Z(p_Z)) = d_{\omega _Y}(\pi _i(\hat{p}_i), \pi _Z(p_Z)) \rightarrow 0$ as $i\rightarrow \infty$, so $p = \pi (p_Z)\in Y^\circ$ and $p_Z\in Z^\circ$. From the assumption we have $d_Z(z_0, p_Z)\ge \bar{\varepsilon }>0$. On the other hand, by the local expressions (2.3) and (2.4) of $\omega _X = \pi ^*\omega _Y + \varepsilon _0 i\partial \bar{\partial }\log \sigma _X$, we find that line segments $\overline{\hat{p} \hat{z}_0} + \overline{\hat{z}_0 x_i}$ in $(U,z_j)$ have $\omega _X$-length $\le C r + \epsilon _i$ for some sequence $\epsilon _i\rightarrow 0$, where we denote $\hat{z}_0 = \pi ^{-1}(p_0)\cap \hat{N}$, i.e. $\hat{z}_0$ is the origin in $(U,z_j)$. So $d_{\omega _0}(\hat{p}_i, x_i)\le C (r + \epsilon _i)$ and by Lemma 3.2, $d_{\omega _{t_i}}(\hat{p}_i, x_i)\le C(r + \epsilon _i )^\delta$. Letting $i\rightarrow \infty$ we get $d_Z(p_Z, z_0)\le C r^\delta$. If we choose r small such that $C r^{\delta } = \bar{\varepsilon } /2$, we would get a contradiction. Therefore $Z^\circ \subset Z$ is dense. $\square$

By exactly the same proof of Lemma 3.2 in [20], we have

Lemma 3.10

There is a uniform constant $C>0$ such that

$$\begin{aligned} \mathrm {diam}( \hat{\pi }^{-1}(y), \omega _t )\le C (T-t)^{1/3},\quad \forall t\in [0,T), \text { and }\forall y\in N. \end{aligned}$$

That is to say, the diameters of the fibers of $\hat{\pi }: E\rightarrow N$ degenerate at a uniform rate as $O((T-t)^{1/3})$.

Lemma 3.11

The map $\pi _Z : (Z,d_Z) \rightarrow (Y,d_T)$ is a homeomorphism.

Note that the target space is equipped with the metric $d_T$, not the metric $d_{\omega _Y}$.

Proof

From Lemma 3.6, we get for any $z_1, z_2\in Z$

$$\begin{aligned} d_T(\pi _Z(z_1), \pi _Z(z_2) )\le C d_{\omega _Y} ( \pi _Z(z_1),\pi _Z(z_2) )^{\delta _0}\le C d_Z(z_1,z_2)^{\delta _0}, \end{aligned}$$

so the map $\pi _Z: (Z,d_Z) \rightarrow (Y,d_T)$ is continuous.

$\bullet$$\pi _Z$is injective. Suppose $z_1,z_2\in Z$ satisfies $\pi _Z(z_1) = \pi _Z(z_2) = y\in Y$. If $y\in Y^\circ$, then $z_1,z_2\in Z^\circ$, $z_1 = z_2$ by the injectivity of $\pi _Z|_{Z^\circ }$. So we only need to consider the case $y\in Y\backslash Y^\circ = N$ and thus $z_1, z_2 \in Z\backslash Z^\circ$. Pick sequences of points $x_{1,i}, x_{2,i}\in (X,\omega _{t_i})$ converging in GH sense to $z_1, z_2$, respectively. By similar arguments as in the proof of Lemma 2.3, without loss of generality we can assume $x_{1,i}, x_{2,i}\in \hat{N}\subset E$. Denote $y_{1,i} = \pi (x_{1,i})$ and $y_{2,i} = \pi (x_{2,i})$. We then have

$$\begin{aligned} d_{\omega _Y} ( y_{1,i}, y ) = d_{\omega _Y}( \pi _i(x_{1,i}), \pi _Z(z_1) )\xrightarrow {i\rightarrow \infty } 0, \end{aligned}$$

and similarly $d_{\omega _Y} (y_{2,i}, y )\rightarrow 0$ as well, and this implies that $d_{\omega _Y}(y_{1,i}, y_{2,i})\rightarrow 0$. Since $x_{1,i}$ and $x_{2,i}$ are both in the zero section $\hat{N}$, from the local expressions (2.3) and (2.4) of the metric $\omega _X = \pi ^*\omega _Y + \varepsilon _0i\partial \bar{\partial }\log \sigma _X$, we see that $d_{\omega _X} (x_{1,i}, x_{2,i})\rightarrow 0$ as $i\rightarrow \infty$. Then by Lemma 3.2 again, we get $d_{\omega _{t_i}}(x_{1,i}, x_{2,i})\le C d_{\omega _X} (x_{1,i}, x_{2,i})^\delta \rightarrow 0$. Letting $i\rightarrow \infty$ we get $d_Z(z_1, z_2) = 0$, thus $z_1 = z_2$. This proves the injectivity of $\pi _Z$.

$\bullet$$\pi _Z$is surjective. This follows from the definition. In fact, we only need to show any $p \in Y\backslash Y^\circ = N$ lies in the image of $\pi _Z$. We fix the point $\hat{p}\in \hat{N}$ with $\hat{\pi }(\hat{p}) = p$. $\hat{p}_i = \hat{p}\in (X,\omega _{t_i})$ converge up to subsequence in GH sense to a point $p_Z\in Z$. Then $d_{\omega _Y} (p, \pi _Z(p_Z)) = d_{\omega _Y} ( \pi _i (\hat{p}_i), \pi _Z(p_Z) ) \rightarrow 0$ by definition of $\pi _i\xrightarrow {GH} \pi _Z$. It then follows that $\pi _Z(p_Z) = p$.

Thus, $\pi _Z: (Z,d_Z) \rightarrow (Y, d_T)$ is bijective and continuous. It is also a homeomorphism since $(Z,d_Z)$ is compact. $\square$