Bounded Diameter Under Mean Curvature Flow

Suppose towards a contradiction, there exists $\varepsilon>0$, such that for all $\delta_{i}=\frac{1}{i}$, we can find $X_{i}=(x_{i},t_{i})$ in $B(x_{0},\delta)\times(T-\delta^{2})$ that is neither $\varepsilon$-close to a one $\mathbb{R}$-factor cylinder nor to a bowl soliton.

Let $r_{i}=R(X_{i})$ be the regularity scale of $X_{i}$ (see Definition 2.8). Because $X_{0}$ is singular, we have that $r_{i}$ converges to $0$. Let $\mathcal{M}^{i}=\mathcal{D}_{1/r_{i}}(\mathcal{M}-{X_{i}})$ be the flow which is obtained from $\mathcal{M}$ by shifting $X_{i}$ to the space-time origin and parabolically rescaling by $1/r_{i}$. By the uniform boundedness of the entropy of $\mathcal{M}^{i}$ and by Brakke’s compactness theorem in [Ilm94, Thm 7.1], we can pass to a subsequential limit $\mathcal{M}^{\infty}$, which is an integral Brakke flow.

Claim. $\mathcal{M}^{\infty}$ is an ancient low entropy flow.

Let $l\in\mathbb{N}_{+}$ and $b>0$ be fixed. For ease of notation, we suppose that $\mathcal{M}$ is defined on $[-1,0)$ with the only conical singularity at $(0,0)$. Now, we consider the renormalized flow $\widehat{\mathcal{M}}=\{\widehat{M}_{\tau}\}_{\tau\in[\tau^{\prime},+\infty)}$. By Theorem 4.1 (uniqueness of conical tangent flow), $\widehat{M}_{\tau}$ converges to conical shrinker $\Sigma$ smoothly. Hence, for the given $l\in\mathbb{N}_{+}$, we can find monotone functions $\rho(\tau)\rightarrow+\infty$, and $\sigma(\tau)\rightarrow 0$ as $\tau\rightarrow+\infty$, such that $\widehat{M}_{\tau}$ is a graph of some function $\hat{u}(\tau)$ over $\Sigma\cap B_{\rho(\tau)}$ with

$$\|\hat{u}(\tau)\|_{C^{l+3}(B_{\rho(\tau)})}\leq\sigma(\tau),$$

(4.2)

provided that $\tau$ is large enough.

Rescaling this back to the original flow, we see that $M_{t}\cap B_{\sqrt{|t|}\rho(\tau)}(0)$ is a graph of some function $u(t)$ over $\sqrt{-t}\Sigma$ for $|t|$ small enough. Moreover, for the given $b>0$ and $l\in\mathbb{N}_{+}$, we can find some $t_{1}=-e^{-\tau_{1}}$, where $\tau_{1}=\tau_{1}(b,l)$ is large enough, such that

$$\|u(t)\|_{C^{l+3}(B_{\sqrt{|t|}\rho(\tau_{1})})}\leq\|u(t)\|_{C^{l+3}(B_{\sqrt{|t|}\rho(\tau)})}\leq\sqrt{|t|}\sigma(\tau)\leq\sqrt{|t_{1}|}\sigma(\tau_{1})<b$$

(4.3)

holds for all $t=-e^{-\tau}\in(t_{1},0)$. Therefore, we have obtained the desired estimates (4.1) in the parabolic region $P=\{(x,t):|x|^{2}\leq\rho(\tau_{1})|t|,t\in[t_{1},0)\}$.

Next, we need to extend the estimates (4.1) from $P$ to some parabolic ball with center at $(0,0)$. Fo the given $b$ and $t_{1}$ from above, let $\delta_{*}>0$ be a small constant to be fixed later. Then, $\gamma_{*}=\gamma_{*}(\frac{\delta_{*}}{2})<\frac{\delta_{*}}{2}$ and $\rho_{*}=\rho_{*}(\frac{\delta_{*}}{2})$ will be fixed according to Corollary 4.4 (pseudolality for renormalized flow). Because $\Sigma$ is an asymptotically conical shrinker, we can find $R_{1}=R_{1}(\Sigma,\gamma_{*},\rho_{*})<\infty$ such that for $x\in\Sigma\cap B^{c}_{R_{1}}$, $\Sigma\cap B_{4\rho_{*}}(x)$ can be written as graph over $T_{x}\Sigma$ with $C^{1}$ norm less than $\frac{\gamma_{*}}{4}$. Since $\widehat{M}_{\tau}$ converges to $\Sigma$ smoothly, for any $\tau_{2}=-\log(-t_{2})\geq\tau_{1}$, $\widehat{M}_{\tau_{2}}\cap B_{\rho(\tau_{2})}(0)\cap B^{c}_{R_{1}}(0)$ can be written as graphs over $2\rho_{*}$-size balls on tangent planes of $\Sigma$ with $C^{1}$ norm less than $\frac{\gamma_{*}}{2}$, provided that $\tau_{1}$ is large enough.

By Corollary 4.4 (pseudolality for renormalized flow), we see that for all $\tau\in[\tau_{2},+\infty)$, $\widehat{M}_{\tau}\cap B_{\rho(\tau_{2})}(0)\cap B^{c}_{R_{1}}(0)$ can be written as pieces of $\frac{\delta_{*}}{2}$ Lipschitz graphs over $e^{\frac{\tau-\tau_{2}}{2}}\frac{\delta_{*}}{2}$ size balls on tangent planes of $\Sigma$. By Ecker-Huisken’s curvature estimates for graphical flow in [EH91, Thm 3.1, Thm 3.4] (renormalized version), we see that $\widehat{M}_{\tau}\cap B_{\rho(\tau_{2})}(0)\cap B^{c}_{R_{1}}(0)$ satisfies

$$|\nabla^{l}A_{\widehat{M}_{\tau}}|\leq C\delta^{2}_{*}|\delta^{2}_{*}e^{\tau-\tau_{2}}|^{-\frac{l+1}{2}}\leq C\delta^{2}_{*},$$

(4.4)

for all $\hat{x}\in\widehat{M}_{\tau}\cap B_{\rho(\tau_{2})}(0)\cap B^{c}_{R_{1}}(0)$ and $\tau>\tau_{2}-\log(|t_{1}|-\omega)-2\log\delta_{*}$. Here, $0<\omega\ll|t_{1}|$ is a small constant, and $C<\infty$ is a constant depending on $\omega$ and the datum at time $\tau_{1}$. This implies that $\widehat{M}_{\tau}\cap B_{\rho(\tau_{2})}(0)\cap B^{c}_{R_{1}}(0)$ is a graph of some function $\hat{u}(\tau)$ over $\Sigma$ with

$$\|\hat{u}(\tau)\|_{C^{l}(B_{\rho(\tau_{2})}\cap B^{c}_{R_{1}})}\leq C\delta^{2}_{*},$$

(4.5)

for $\tau>\tau_{2}-\log(|t_{1}|-\omega)-2\log\delta_{*}$.

Now, we rescale this back to the original flow. Then, $M_{t}\cap B_{\sqrt{|t_{2}|}\rho(\tau_{2}))}\cap B^{c}_{\sqrt{|t_{2}|}R_{1}}$ can be written as graph of some function $u(t)$ over $\sqrt{-t}\Sigma$. Note that $u(t)=\sqrt{|t|}\hat{u}(\tau)$ and $y=\sqrt{|t|}y^{\prime}\in\sqrt{|t|}\Sigma$, we see that

$$\|u(t)\|_{C^{l}(B_{\sqrt{|t_{2}|}\rho(\tau_{2}))}\cap B^{c}_{\sqrt{|t_{2}|}R_{1}})}\leq C\delta^{2}_{*}$$

(4.6)

holds for $t\in(\delta^{2}_{*}(|t_{1}|-\omega)t_{2},0)$.

Now, we choose $\delta_{*}>0$ small enough, such that

$$C\delta^{2}_{*}<b.$$

(4.7)

Let $\varepsilon=\delta_{*}(|t_{1}|-\omega|)$. Noticing that $t_{2}$ is arbitrary and combing this with (4.6), (4.7) and (4.3), we obtain the estimation (4.1) in $B_{\varepsilon}(0)\times(-\varepsilon^{2},0)$. This completes the proof of the theorem.

∎

By the local regularity theorem (or Theorem 2.9), we know that the set of all regular space-time points is open. This implies that the singular set $S_{T}(\mathcal{M})$ at first singular time $T$ is closed and bounded, hence it is compact.

By the isolated property of conical singularities from Theorem 4.1, we know there are only finitely many conical singularities $x_{1},\dots,x_{m}$ at time $T$. For each $i\in\{1,\dots,m\}$, we denote by $\{\sqrt{-t}\Sigma_{i}\}$ the tangent flow at the conical singularity $(x_{i},T)$. Now, according to Theorem 4.1, we can find $\delta^{\prime}_{i}>0$, such that $B_{\delta^{\prime}_{i}}(x_{i})$ are disjoint, and for $t\in(T-\delta_{i}^{\prime 2},T)$, $M_{t}\cap B_{\delta^{\prime}_{i}}(x_{i})$ are smooth graphs over $\left(\sqrt{T-t}\Sigma_{i}+x_{i}\right)\cap B_{\delta^{\prime}_{i}}(x_{i})$ with $C^{l+1}$ norm less than $\varepsilon$.

Hence, $S_{T}(\mathcal{M})\setminus\mathop{\cup}\limits_{i=1}^{m}\{x_{i}\}$ is still compact and all remaining singularities are one $\mathbb{R}$-factor cylindrical. Now, by Theorem 3.1, for each $p\in S_{T}(\mathcal{M})\setminus\mathop{\cup}\limits_{i=1}^{m}\{x_{i}\}$, we can find $\delta^{\prime}>0$, such that the flow $\{M_{t}\cap B_{\delta^{\prime}}(p)\}$ is mean convex and any $(p^{\prime},t^{\prime})\in\mathcal{M}\cap\left(B(p,\delta^{\prime})\times(T-\delta^{\prime 2},T)\right)$ is $\varepsilon$-close to either a cylindrical shrinker or a bowl soliton. By compactness of $S_{T}(\mathcal{M})\setminus\mathop{\cup}\limits_{i=1}^{m}\{x_{i}\}$, we can find finitely many cylindrical singularities $\{p_{1},\dots,p_{k}\}\subset S_{T}(\mathcal{M})\setminus\mathop{\cup}\limits_{i=1}^{m}\{x_{i}\}$ and $\delta_{j}$ corresponding to $p_{j}$ for $j\in\{1,\cdots,k\}$, such that the above properties in statement 2 hold and $S_{T}(\mathcal{M})\setminus\mathop{\cup}\limits_{i=1}^{m}\{x_{i}\}\subset\cup_{j=1}^{k}B_{\delta_{j}}(p_{j})$.

Now, we choose $\delta=\min\{\delta_{1},\cdots,\delta_{m},\delta^{\prime}_{1},\cdots,\delta^{\prime}_{k}\}$ and take

$$\Omega_{M}=\mathop{\cup}\limits_{j=1}^{k}B_{\delta_{j}}(p_{j})$$

and

$$\Omega_{C}=\mathop{\cup}\limits_{i=1}^{m}B_{\delta}(x_{i}).$$

They satisfy the requirements in statement 1 and statement 2.

Let $\Omega_{L}=\Omega\setminus(\Omega_{M}\cup\Omega_{C})$. Note that $\Omega_{L}$ and $\mathop{\cup}\limits_{i=1}^{m}\left(\overline{B_{\delta}(x_{i})}\setminus B_{\frac{\delta}{2}}(x_{i})\right)$ are compact sets. By the local regularity theorem, we know the regularity scale $R$ is positive over their union. Combining this with the fact that regularity scale $R$ is $1$-Lipschitz, we can find some $\rho\in(0,\frac{\delta}{2})$, such that for $t\in(T-\delta^{2},T)$ and every $p\in M_{t}\cap\Omega_{L}$ or $p\in M_{t}\cap\Omega_{C}\setminus\mathop{\cup}\limits_{i=1}^{m}B_{\frac{\delta}{2}}(x_{i})$, we have

$$R(p,t)\geq\rho.$$

(5.5)

This completes the proof. ∎