A local estimate for the mean curvature flow

Abstract

We establish a pointwise estimate of $\vert A\vert$ along the mean curvature flow in terms of the initial geometry and the $\vert HA\vert$ bound. As corollaries we obtain the blowup rate estimate of $\vert HA\vert$ and an extension theorem with respect to $\vert HA\vert$.

1 Introduction

Let ${\textbf{x}}_0:\Sigma ^n\rightarrow {\mathbb {R}}^{n+1}$ be a complete smooth immersed hypersurface without boundary and a family of immersions ${\textbf{x}}(x,t):\Sigma ^n\times [0,T)\rightarrow {\mathbb {R}}^{n+1}$ be a solution to the equation

$$\begin{aligned} \partial _{t}{\textbf{x}}=-H{\textbf{n}},\quad {\textbf{x}}(0)={\textbf{x}}_0, \end{aligned}$$

which is called a mean curvature flow with the singular time T. If $\Sigma$ is a closed embedded hypersurface, then the flow develops a singularity at $T<\infty$ and $\sup _{\Sigma _t}\vert A\vert \rightarrow \infty$ as $t\rightarrow T$ according to Huisken [1].

Since the finite-time singularity for a compact mean curvature flow is characterized by the blowup of the second fundamental form, it is of great interest to express this criterion in terms of some simpler quantity. A natural conjecture is the blowup of the mean curvature H, which is proposed as an open problem in [2]. The case of $n=2$ was confirmed by Li–Wang [3]. However, in [4] Stolarski showed that for general cases $n\ge 7$ the mean curvature does not necessarily blow up at the finite singular time.

Hence we turn to consider some alternative conditions for general dimensions $n\ge 2$ which may be stronger than the mean curvature bound. In [5] Cooper proved the HA tensor also blows up at time T. In [5,6,7], Cooper and Le-Sesum proved that the mean curvature blows up under the assumption of some slow blowup rate of the second fundamental form. Some extension results under integral conditions also can be seen in Le–Sesum [8] and Xu–Ye–Zhao [9].

Note that similar blowup and extension results have been studied for Ricci flow as well. In [10] Hamilton proved that the Riemann curvature tensor blows up at the finite singular time. In [11] Sesum proved the blowup of the Ricci curvature. In [12,13,14], Wang, Chen-Wang and Kotschwar-Munteanu-Wang arrived at estimates on curvature growth in terms of the Ricci curvature.

The explict local estimate in Kotschwar–Munteanu–Wang [14] has some precedent on a gradient shrinking soliton in [15] that a bound on Ric implies a polynomial growth bound on Rm. The feasibility lies in the observation that the second order derivatives of Ric appear as time-derivative of Rm, i.e.,

$$\begin{aligned} \partial _{t}Rm=c\nabla ^2 Ric, \end{aligned}$$

which helps to yield a differential inequality on integrations. This equation follows from the fact that Ric describes the metric evolution along a Ricci flow. In a similar way HA describes the metric evolution of the mean curvature flow and plays the role of Ric by

$$\begin{aligned} \partial _{t}\vert A\vert ^2=2\big (\nabla ^{2}H\cdot A+ H\cdot \text{ tr }(A^3)\big ). \end{aligned}$$

In the present paper, we follow the techniques on integration estimates from [14] and establish the following local $L^\infty$ estimate of A in terms of the initial geometry and the $\vert HA\vert$ bound along the flow.

Theorem 1.1

Fix $x_0\in {\mathbb {R}}^{n+1}$ and $r>0$. Let ${\textbf{x}}:\Sigma ^n\times [t_0,t_1]\rightarrow {\mathbb {R}}^{n+1}$ be a complete smooth mean curvature flow satisfying the uniform bound

$$\begin{aligned} \sup _{B(x_0,r)\cap \Sigma _t}\vert HA\vert (\cdot ,t)\le K(t),\quad \forall \, t\in [t_0,t_1]. \end{aligned}$$

Then for any $q>n+2$ there exist positive constants $C=C(n,r,t_1-t_0,q,K(t_0))$ and $c=c(n,q)$ such that for any $t\in [t_0,t_1]$

$$\begin{aligned}{} & {} \sup _{B(x_0,r/2)\cap \Sigma _t}\vert A\vert \\{} & {} \quad \le C\Big (1+\Vert A\Vert _{L^{q}(B(x_0,2r)\cap \Sigma _{t_0})}^{q}\Big )^c\Big (1+\text{ Vol}_{g(t_0)}(B_{2r,t})\Big )^c \left( \int _{t_0}^{t}e^{\int _{t_0}^{s}cK}ds\right) ^{c}, \end{aligned}$$

where $B_{2r,t}=B(x_0,2r+n^{1/4}\int _{t_0}^{t}\sqrt{K})\cap \Sigma _{t_0}$.

This local estimate provides a new proof of the blowup of $\vert HA\vert$ in [5] and extends the estimates for the Ricci flow in terms of Ric in [12,13,14] to an estimate for the mean curvature flow in terms of $\vert HA\vert$.

One of its direct corollaries is the following extension theorem as well as a blowup estimate of $\vert HA\vert$ at the first finite singular time. This result generalizes Theorem 1.2 of [7] and Theorem 5.1 of [5] and can be seen as another version of Theorem 1.1 of [12] and Theorem 2 of [14].

Theorem 1.2

There exists a positive constant $\epsilon =\epsilon (n)$ satisfying the following properties. Let ${\textbf{x}}:\Sigma ^n\times [0,T)\rightarrow {\mathbb {R}}^{n+1}$ be a complete smooth mean curvature flow with $T<\infty$. Suppose each time slice $\Sigma _t$ has bounded second fundamental form. If

$$\begin{aligned} \sup \limits _{\Sigma _t}\vert HA\vert \le \frac{\epsilon }{T-t},\quad \forall \,t\in [0,T), \end{aligned}$$

then

$$\begin{aligned} \limsup _{t\rightarrow T}\sup \limits _{\Sigma _t}\vert A\vert (\cdot ,t)\le C(n,T,\Sigma _0)<\infty , \end{aligned}$$

which implies the flow can be extended past time T. Conversely, if the flow blows up at time T, then

$$\begin{aligned} \limsup _{t\rightarrow T}\Big ((T-t)\sup _{\Sigma _t}\vert HA\vert \Big )\ge \epsilon . \end{aligned}$$

The organization of this paper is as follows. In Sect. 2 we recall some basic results on mean curvature flow. In Sect. 3 we develop $L^p$ estimate in terms of initial data and $\vert HA\vert$ bound, following the argument in [14]. In Sect. 4 we establish the $L^\infty$ estimate by Moser iteration as in [8] and finish the extension theorem. In Sect. 5 we estimate the blowup rate of $\vert HA\vert$, using the $L^{\infty }$ estimate and the blowup estimate of $\vert A\vert$.

2 Preliminaries

Let ${\textbf{x}}(p,t):\Sigma ^n\rightarrow {\mathbb {R}}^{n+1}$ be a family of smooth immersions. $\{(\Sigma ^n,{\textbf{x}}(\cdot ,t)),0\le t< T\}$ is called a mean curvature flow if ${\textbf{x}}$ satisfies

$$\begin{aligned} \partial _{t}{\textbf{x}}=-H{\textbf{n}},\quad \forall \,t\in [0,T), \end{aligned}$$

(2.1)

where we denote by $A=(h_{ij})$ the second fundamental form and by $H=g^{ij}h_{ij}$ the mean curvature. Sometimes we also write $\Sigma _t$ as ${\textbf{x}}(t)$ for short.

Some equations are listed here for later calculations. See [1] or [2] for details.

Lemma 2.1

(Sect. 3 of [1]). Along the mean curvature flow,

$$\begin{aligned} \partial _{t} d\mu= & {} -H^2 d\mu ,\nonumber \\ \partial _{t}\vert A\vert ^2= & {} 2(\nabla _{e_i}\nabla _{e_j}H\cdot A_{ij}+ HA_{kl}A_{lm}A_{mk}), \end{aligned}$$

(2.2)

$$\begin{aligned} 2\vert \nabla H\vert ^2= & {} (\Delta -\partial _{t})H^2+2H^2\vert A\vert ^2, \end{aligned}$$

(2.3)

$$\begin{aligned} 2\vert \nabla A\vert ^2= & {} (\Delta -\partial _{t})\vert A\vert ^2+2\vert A\vert ^4. \end{aligned}$$

(2.4)

By maximum principle the second fundamental form blows up at least at a rate of 1/2, which holds for noncompat cases as well.

Lemma 2.2

(Proposition 2.4.6 of [2]). Suppose the flow (2.1) blows up at the finite singular time T and each time slice $\Sigma _t$ has bounded second fundamtental form. Then

$$\begin{aligned} \sup _{\Sigma _t}\vert A\vert \ge \frac{1}{\sqrt{2(T-t)}}. \end{aligned}$$

On a hypersurface we also have the Sobolev inequality, i.e., the Michael-Simon inequality. See [1, 16].

Lemma 2.3

(Lemma 5.7 of [1]). Let f be a nonnegative Lipschitz function with compact support on a hypersurface $\Sigma ^n\subset {\mathbb {R}}^{n+1}$. Then there exists a positive constant $c=c(n)$ such that

$$\begin{aligned} \left( \int _{\Sigma }\vert f\vert ^{\frac{n}{n-1}}d\mu \right) ^{\frac{n-1}{n}}\le c_n \int _{\Sigma }(\vert \nabla f\vert +\vert H\vert f)d\mu . \end{aligned}$$

From $L^p$ estimate to $L^\infty$ estimate we require the process of Moser iteration which depends on the Michael-Simon inequality, i.e, Lemma 2.3. We conclude the following result from Lemma 5.2 in [8].

Lemma 2.4

(Moser iteration). Let ${\textbf{x}}:\Sigma ^n\times [t_0,t_1]\rightarrow {\mathbb {R}}^{n+1}$ be a smooth mean curvature flow. Consider the differential inequality

$$\begin{aligned} (\partial _{t}-\Delta )v\le fv,\quad v\ge 0. \end{aligned}$$

Fix $x_0\in {\mathbb {R}}^{n+1}$ and $r>0$. For any $q>n+2$ and $\beta \ge 2$ there exists a constant $C=C(n,r,t_1-t_0,q,\beta )$ such that for any $t\in [t_0,t_1]$

$$\begin{aligned}{} & {} \Vert v\Vert _{L^{\infty }(D_{t,r}')}\\{} & {} \quad \le C\big (1+\Vert f\Vert _{L^{q/2}(D_{t,r})}\big )^{\frac{qn^2}{\beta (q-n-2)}} \big (1+\Vert H\Vert _{L^{n+2}(D_{t,r})}^{n+2}\big )^{\frac{qn^3}{\beta (n+2)(q-n-2)}}\Vert v\Vert _{L^{\beta }(D_{t,r})}, \end{aligned}$$

where

$$\begin{aligned} D_{t,r}:= & {} \bigcup _{t_0\le s\le t}\big (B(x_0,r)\cap \Sigma _s\big ), \\ D_{t,r}':= & {} \bigcup _{(t_0+t)/2\le s\le t}\big (B(x_0,r/2)\cap \Sigma _s\big ). \end{aligned}$$

Proof

Without loss of generality, we assume $t_0=0$, $t=1$ and $r=1$. Set

$$\begin{aligned} C_0=1+\Vert f\Vert _{L^q(D_{t,r})},\quad C_1=\big (1+\Vert H\Vert _{L^{n+2}(D_{t,r})}^{n+2}\big )^{\frac{n}{n+2}},\quad \nu =\frac{n+2}{2q-(n+2)}, \end{aligned}$$

where $q>\frac{n+2}{2}$. According to the proof of Lemma 5.2 of [8] we have for $\beta \ge 2$,

$$\begin{aligned} \Vert v\Vert _{L^{\infty }(D'_{t,r})}\le C_b\Vert v\Vert _{L^{\beta }(D_{t,r})}, \end{aligned}$$

where

$$\begin{aligned} C_b= & {} C_b(n,q,\beta ,C_0,C_1)=\left( 4\left( \frac{n+2}{n}\right) ^{1+\nu }C_z\beta ^{1+\nu }\right) ^{\frac{n^2}{\beta }}, \\ C_z= & {} C_z(n,q,C_0,C_1)=16\cdot 100^{1+\nu }c_nC_a. \end{aligned}$$

According to the proof of Lemma 4.1 of [8] we have

$$\begin{aligned} C_a=C_a(n,q,C_0,C_1)=(2c_nC_0C_1)^{1+\nu }. \end{aligned}$$

As a conclusion,

$$\begin{aligned} C_b= & {} \big (C(n,q,\beta )C_z\big )^{\frac{n^2}{\beta }} =\big (C(n,q,\beta )C_a\big )^{\frac{n^2}{\beta }} =C(n,q,\beta )(C_0C_1)^{\frac{n^2(1+\nu )}{\beta }}\\= & {} C(n,q,\beta )\big (1+\Vert f\Vert _{L^q(D_{t,r})}\big )^{\frac{2qn^2}{\beta (2q-n-2)}} \big (1+\Vert H\Vert _{L^{n+2}(D_{t,r})}^{n+2}\big )^{\frac{2qn^3}{\beta (n+2)(2q-n-2)}}. \end{aligned}$$

$\square$

3 $L^p$ estimate

Throughout this section we use c to denote a nonnegative constant depending only on n and p and we use $c_n$ to denote a nonnegative constant depending only on n, which may change from line to line.

Theorem 3.1

Fix $x_0\in {\mathbb {R}}^{n+1}$ and $r>0$. Let ${\textbf{x}}:\Sigma ^n\times [t_0,t_1]\rightarrow {\mathbb {R}}^{n+1}$ be a complete smooth mean curvature flow satisfying the bound

$$\begin{aligned} \sup _{B(x_0,r)\cap \Sigma _{t}}\vert HA\vert (\cdot ,t)\le K(t),\quad \forall \, t\in [t_0,t_1], \end{aligned}$$

where K(t) is nondecreasing. Then for any $p\ge 2$ there exist positive constants $c=c(n,p)$ such that for any $t\in [t_0,t_1]$,

$$\begin{aligned} \int _{B(x_0,r/2)\cap \Sigma _t}\vert A\vert ^p\le & {} \left( K(t_0)^{-1}\int _{B(x_0,r)\cap \Sigma _{t_0}}\vert A\vert ^{p+2}(t_0) +c\int _{B(x_0,r)\cap \Sigma _{t_0}}\vert A\vert ^{p}(t_0) \right. \\{} & {} \left. +K(t_0)^{-1}r^{-(p+2)}\text{ Vol}_{g(t_0)}(B_{r,t})\right) \cdot e^{\int _{t_0}^{t}cK}, \end{aligned}$$

where $B_{r,t}=B(x_0,r+n^{1/4}\int _{t_0}^{t}\sqrt{K})\cap \Sigma _{t_0}$.

Proof

Let $\phi (x,t)$ be a nonnegative smooth function with compact support which will be determined later. Note that $\vert H\vert \vert A\vert \le K$. By Eq. (2.2) we have

$$\begin{aligned}{} & {} \partial _{t}\int _{\Sigma _t}\vert A\vert ^p\phi \\{} & {} \quad \le \int _{\Sigma _t}\partial _t\vert A\vert ^p\phi +\int _{\Sigma _t}\vert A\vert ^p\partial _t\phi \\{} & {} \quad = p\int _{\Sigma _t}\vert A\vert ^{p-2}\phi (\nabla _{e_i}\nabla _{e_j}H\cdot A_{ij}+HA_{kl}A_{lm}A_{mk})+\int _{\Sigma _t}\vert A\vert ^p\partial _t\phi \\{} & {} \quad \le c\int _{\Sigma _t}\vert A\vert ^{p-2}\vert \nabla H\vert \vert \nabla A\vert \phi +c\int _{\Sigma _t}\vert A\vert ^{p-1}\vert \nabla H\vert \vert \nabla \phi \vert +c\int _{\Sigma _t}\vert H\vert \vert A\vert ^{p+1}\\{} & {} \quad \quad +\int _{\Sigma _t}\vert A\vert ^p\partial _t\phi \\{} & {} \quad \le \frac{c}{K}\int _{\Sigma _t}\vert A\vert ^p\vert \nabla H\vert ^2\phi +cK\int _{\Sigma _t}\vert A\vert ^{p-4}\vert \nabla A\vert ^2\phi +cK\int _{\Sigma _t}\vert A\vert ^{p}\phi \\{} & {} \qquad +\,cK\int _{\Sigma _t}\vert A\vert ^{p-2}\phi ^{-1}\vert \nabla \phi \vert ^2+\int _{\Sigma _t}\vert A\vert ^p \partial _{t}\phi . \end{aligned}$$

By Eq. (2.3) we have

$$\begin{aligned}{} & {} \int _{\Sigma _t}\vert A\vert ^{p}\vert \nabla H\vert ^2\phi \\{} & {} \quad = \frac{1}{2}\int _{\Sigma _t}\vert A\vert ^p\phi (\Delta -\partial _t)H^2+\int _{\Sigma _t}H^2\vert A\vert ^{p+2}\phi \\{} & {} \quad \le c\int _{\Sigma _t}\vert H\vert \vert A\vert ^{p-1}\vert \nabla H\vert \vert \nabla A\vert \phi +c\int _{\Sigma _t}\vert H\vert \vert A\vert ^p\vert \nabla H\vert \vert \nabla \phi \vert \\{} & {} \qquad -\frac{1}{2}\partial _t\int _{\Sigma _t}H^2\vert A\vert ^p\phi +\frac{1}{2}\int _{\Sigma _t}H^2\partial _t(\vert A\vert ^p\phi )+\int _{\Sigma _t}H^2\vert A\vert ^{p+2}\phi \\{} & {} \quad = c\int _{\Sigma _t}\vert H\vert \vert A\vert ^{p-1}\vert \nabla H\vert \vert \nabla A\vert \phi +c\int _{\Sigma _t}\vert H\vert \vert A\vert ^p\vert \nabla H\vert \vert \nabla \phi \vert \\{} & {} \qquad -\frac{1}{2}\partial _t\int _{\Sigma _t}H^2\vert A\vert ^p\phi +c\int _{\Sigma _t}H^2\vert A\vert ^{p-2}(\nabla _{e_i}\nabla _{e_j}H\cdot A_{ij}+HA_{kl}A_{lm}A_{mk})\phi \\{} & {} \qquad +\frac{1}{2}\int _{\Sigma _t}H^2\vert A\vert ^p\partial _t\phi +\int _{\Sigma _t}H^2\vert A\vert ^{p+2}\phi \\{} & {} \quad \le c\int _{\Sigma _t}\vert H\vert \vert A\vert ^{p-1}\vert \nabla H\vert \vert \nabla A\vert \phi +c\int _{\Sigma _t}\vert H\vert \vert A\vert ^p\vert \nabla H\vert \vert \nabla \phi \vert \\{} & {} \qquad +\,c\int _{\Sigma _t}\vert H\vert \vert A\vert ^{p-1}\vert \nabla H\vert ^2\phi +c\int _{\Sigma _t}H^2\vert A\vert ^{p-2}\vert \nabla H\vert \vert \nabla A\vert \phi +c\int _{\Sigma _t}\vert H\vert ^3\vert A\vert ^{p+1}\phi \\{} & {} \qquad -\frac{1}{2}\partial _t\int _{\Sigma _t}H^2\vert A\vert ^p\phi +\frac{1}{2}\int _{\Sigma _t}H^2\vert A\vert ^p\partial _t\phi +c\int _{\Sigma _t}H^2\vert A\vert ^{p+2}\phi \\{} & {} \quad \le c\int _{\Sigma _t}\vert H\vert \vert A\vert ^{p-1}\vert \nabla H\vert \vert \nabla A\vert \phi +c\int _{\Sigma _t}\vert H\vert \vert A\vert ^p\vert \nabla H\vert \vert \nabla \phi \vert \\{} & {} \qquad -\frac{1}{2}\partial _t\int _{\Sigma _t}H^2\vert A\vert ^p\phi +\frac{1}{2}\int _{\Sigma _t}H^2\vert A\vert ^p\partial _t\phi +c\int _{\Sigma _t}H^2\vert A\vert ^{p+2}\phi . \end{aligned}$$

By Cauchy’s inequality we have

$$\begin{aligned}{} & {} \int _{\Sigma _t}\vert A\vert ^{p}\vert \nabla H\vert ^2\phi \\{} & {} \quad \le \frac{1}{2}\int _{\Sigma _t}\vert A\vert ^p\vert \nabla H\vert ^2\phi +c\int _{\Sigma _t}H^2\vert A\vert ^{p-2}\vert \nabla A\vert ^2\phi +c\int _{\Sigma _t}H^2\vert A\vert ^p\phi ^{-1}\vert \nabla \phi \vert ^2\\{} & {} \qquad -\frac{1}{2}\partial _t\int _{\Sigma _t}H^2\vert A\vert ^p\phi +\frac{1}{2}\int _{\Sigma _t}H^2\vert A\vert ^p\partial _t\phi +c\int _{\Sigma _t}H^2\vert A\vert ^{p+2}\phi \\{} & {} \quad \le -\frac{1}{2}\partial _{t}\int _{\Sigma _t}H^2\vert A\vert ^{p}\phi +cK^2\int _{\Sigma _t}\vert A\vert ^{p-4}\vert \nabla A\vert ^2\phi +cK^2\int _{\Sigma _t}\vert A\vert ^{p}\phi \\{} & {} \qquad +\frac{1}{2}\int _{\Sigma _t}\vert A\vert ^p\vert \nabla H\vert ^2\phi +cK^2\int _{\Sigma _t}\vert A\vert ^{p-2}\phi ^{-1}\vert \nabla \phi \vert ^2+\frac{1}{2}\int _{\Sigma _t}H^2\vert A\vert ^p\partial _{t}\phi , \end{aligned}$$

and then

$$\begin{aligned}{} & {} \frac{c}{K}\int _{\Sigma _t}\vert A\vert ^p\vert \nabla H\vert ^2\phi \\{} & {} \quad \le -\frac{c}{K}\partial _{t}\int _{\Sigma _t}H^2\vert A\vert ^p\phi +cK\int _{\Sigma _t}\vert A\vert ^{p-4}\vert \nabla A\vert ^2\phi +cK\int _{\Sigma _t}\vert A\vert ^p\phi \\{} & {} \qquad +\,cK\int _{\Sigma _t}\vert A\vert ^{p-2}\phi ^{-1}\vert \nabla \phi \vert ^2+\frac{c}{K}\int _{\Sigma _t}H^2\vert A\vert ^p\partial _{t}\phi . \end{aligned}$$

By Eq. (2.4) we have for $p\ge 4$,

$$\begin{aligned}{} & {} \int _{\Sigma _t}\vert A\vert ^{p-4}\vert \nabla A\vert ^2\phi \\{} & {} \quad = \frac{1}{2}\int _{\Sigma _t}\vert A\vert ^{p-4}(\Delta -\partial _t)\vert A\vert ^2\phi +\int _{\Sigma _t}\vert A\vert ^p\phi \\{} & {} \quad = -\frac{1}{2}\int _{\Sigma _t}\nabla (\vert A\vert ^{p-4}\phi )\cdot \nabla \vert A\vert ^2-\frac{1}{2}\int _{\Sigma _t}\vert A\vert ^{p-4}\phi \cdot \partial _t\vert A\vert ^2+\int _{\Sigma _t}\vert A\vert ^p\phi \\{} & {} \quad \le -(p-4)\int _{\Sigma _t}\vert A\vert ^{p-4}\vert \nabla \vert A\vert \vert ^2\phi +\int _{\Sigma _t}\vert A\vert ^{p-3}\vert \nabla A\vert \vert \nabla \phi \vert \\{} & {} \qquad -c\partial _{t}\int _{\Sigma _t}\vert A\vert ^{p-2}\phi +c\int _{\Sigma _t}\vert A\vert ^{p-2}\partial _{t}\phi +\int _{\Sigma _t}\vert A\vert ^{p}\phi \\{} & {} \quad \le -c\partial _{t}\int _{\Sigma _t}\vert A\vert ^{p-2}\phi +\frac{1}{2}\int _{\Sigma _t}\vert A\vert ^{p-4}\vert \nabla A\vert ^2\phi +c\int _{\Sigma _t}\vert A\vert ^{p-2}\phi ^{-1}\vert \nabla \phi \vert ^2\\{} & {} \qquad +\,c\int _{\Sigma _t}\vert A\vert ^{p-2}\partial _{t}\phi +\int _{\Sigma _t}\vert A\vert ^p\phi , \end{aligned}$$

and then

$$\begin{aligned}{} & {} K\int _{\Sigma _t}\vert A\vert ^{p-4}\vert \nabla A\vert ^2\phi \\{} & {} \quad \le -cK\partial _{t}\int _{\Sigma _t}\vert A\vert ^{p-2}\phi +2K\int _{\Sigma _t}\vert A\vert ^{p}\phi +cK\int _{\Sigma _t}\vert A\vert ^{p-2}\phi ^{-1}\vert \nabla \phi \vert ^2\\{} & {} \qquad +\,cK\int _{\Sigma _t}\vert A\vert ^{p-2}\partial _{t}\phi . \end{aligned}$$

Combining the results above together we have for $p\ge 4$,

$$\begin{aligned}{} & {} \partial _{t}\int _{\Sigma _t}\vert A\vert ^{p}\phi \nonumber \\{} & {} \quad \le -\frac{c}{K}\partial _{t}\int _{\Sigma _t}H^2\vert A\vert ^p\phi +cK\int _{\Sigma _t}\vert A\vert ^{p-4}\vert \nabla A\vert ^2\phi +cK\int _{\Sigma _t}\vert A\vert ^p\phi \nonumber \\{} & {} \qquad +\,cK\int _{\Sigma _t}\vert A\vert ^{p-2}\phi ^{-1}\vert \nabla \phi \vert ^2 +\frac{c}{K}\int _{\Sigma _t}H^2\vert A\vert ^p\partial _{t}\phi +\int _{\Sigma _t}\vert A\vert ^p\partial _{t}\phi \nonumber \\{} & {} \quad \le -\frac{c}{K}\partial _{t}\int _{\Sigma _t}H^2\vert A\vert ^{p}\phi -cK\partial _{t}\int _{\Sigma _t}\vert A\vert ^{p-2}\phi +cK\int _{\Sigma _t}\vert A\vert ^p\phi \nonumber \\{} & {} \qquad +\,cK\int _{\Sigma _t}\vert A\vert ^{p-2}\phi ^{-1}\vert \nabla \phi \vert ^2 +\int _{\Sigma _t}\vert A\vert ^p\vert \partial _{t}\phi \vert +cK\int _{\Sigma _t}\vert A\vert ^{p-2}\vert \partial _{t}\phi \vert .\nonumber \\ \end{aligned}$$

(3.1)

Consider a smooth decreasing function $\eta$, which equals 1 on [0, r/2] and vanishes on $[r,\infty )$, satisfying $\vert \eta '\vert \le 3/r$. For any $0<\delta <1$ we set $\psi :=\eta ^{1/\delta }$ such that

$$\begin{aligned} \vert \psi '\vert \le \frac{3}{\delta r}\psi ^{1-\delta }. \end{aligned}$$

Now we choose $\phi :=\psi (\vert x-x_0\vert )$. Then

$$\begin{aligned}{} & {} \phi ^{-1}\vert \nabla \phi \vert ^2\le \psi ^{-1}(\psi ')^2\le \frac{9}{\delta ^2r^2}\phi ^{1-2\delta }, \\{} & {} \vert \partial _{t}\phi \vert =\vert \psi '\vert \vert \partial _{t}(\vert x-x_0\vert ) \vert \le \vert \psi '\vert \vert H\vert \le \frac{3}{\delta r}\vert H\vert \phi ^{1-\delta }. \end{aligned}$$

Take $\delta =\frac{1}{p}$. By Young’s inequality we have

$$\begin{aligned} \int _{\Sigma _t}\vert A\vert ^{p-2}\phi ^{-1}\vert \nabla \phi \vert ^2\le & {} cr^{-2}\int _{\Sigma _t}\vert A\vert ^{p-2}\phi ^{1-\frac{2}{p}} \le c\int _{\Sigma _t}\vert A\vert ^p\phi +c\int _{\Sigma _t\cap \text{ supp }\phi }r^{-p}\\\le & {} c\int _{\Sigma _t}\vert A\vert ^p\phi +cr^{-p}\text{ Vol}_{g(t)}\big (B(x_0,r)\cap \Sigma _{t}\big ), \end{aligned}$$

and

$$\begin{aligned} \int _{\Sigma _t}\vert A\vert ^{p-2}\vert \partial _{t}\phi \vert\le & {} cr^{-1}\int _{\Sigma _t}\vert A\vert ^{p-1}\phi ^{1-\frac{1}{p}} \le c\int _{\Sigma _t}\vert A\vert ^p\phi +c\int _{\Sigma _t\cap \text{ supp }\phi }r^{-p}\\\le & {} c\int _{\Sigma _t}\vert A\vert ^p\phi +cr^{-p}\text{ Vol}_{g(t)}\big (B(x_0,r)\cap \Sigma _{t}\big ), \end{aligned}$$

and

$$\begin{aligned} \int _{\Sigma _t}\vert A\vert ^{p}\vert \partial _{t}\phi \vert\le & {} cr^{-1}\int _{\Sigma _t}\vert H\vert \vert A\vert ^{p}\phi ^{1-\frac{1}{p}} \le cr^{-1}K\int _{\Sigma _t}\vert A\vert ^{p-1}\phi ^{1-\frac{1}{p}}\\\le & {} cK\int _{\Sigma _t}\vert A\vert ^p\phi +cKr^{-p}\text{ Vol}_{g(t)}\big (B(x_0,r)\cap \Sigma _{t}\big ). \end{aligned}$$

Back to (3.1), we obtain

$$\begin{aligned}{} & {} \partial _{t}\int _{\Sigma _t}\vert A\vert ^p\phi +\frac{c}{K}\partial _{t}\int _{\Sigma _t}H^2\vert A\vert ^p\phi +cK\partial _{t}\int _{\Sigma _t}\vert A\vert ^{p-2}\phi \nonumber \\{} & {} \quad \le cK\int _{\Sigma _t}\vert A\vert ^p\phi + cKr^{-p}\text{ Vol}_{g(t)}\big (B(x_0,r)\cap \Sigma _{t}\big ). \end{aligned}$$

(3.2)

If we set

$$\begin{aligned} U(t)=\int _{\Sigma _t}\vert A\vert ^p\phi +\frac{c}{K}\int _{\Sigma _t}H^2\vert A\vert ^p\phi +cK\int _{\Sigma _t}\vert A\vert ^{p-2}\phi , \end{aligned}$$

then actually it becomes

$$\begin{aligned} U'\le & {} -\frac{cK'}{K^2}\int _{\Sigma _t}H^2\vert A\vert ^p\phi +cK'\int _{\Sigma _t}\vert A\vert ^{p-2}\phi +cK\int _{\Sigma _t}\vert A\vert ^p\phi \\{} & {} +\,cKr^{-p}\text{ Vol}_{g(t)}\big (B(x_0,r)\cap \Sigma _{t}\big )\\\le & {} (K'/K+cK)U +cKr^{-p}\text{ Vol}_{g(t)}\big (B(x_0,r)\cap \Sigma _{t}\big ). \end{aligned}$$

Since

$$\begin{aligned} \partial _{t}\vert x-x_0\vert \le \vert H\vert \le n^{1/4}\sqrt{K} \end{aligned}$$

and

$$\begin{aligned} \partial _{t}d\mu =-H^2d\mu , \end{aligned}$$

we know

$$\begin{aligned}{} & {} B(x_0,r)\cap \Sigma _t\subset B\left( x_0,r+n^{1/4}\int _{t_0}^{t}\sqrt{K}\right) \cap \Sigma _{t_0}:=B_{r,t}, \\{} & {} \text{ Vol}_{g(s)}\big (B(x_0,r)\cap \Sigma _{s}\big )\le \text{ Vol}_{g(t_0)}(B_{r,t}),\quad \forall \,s\in [t_0,t]. \end{aligned}$$

Then for any $s\in [t_0,t]$,

$$\begin{aligned}{} & {} \partial _{s}\Big (e^{-\int _{t_0}^s(K'/K+cK)}U(s)\Big )\\{} & {} \quad \le cKe^{-\int _{t_0}^s(K'/K+cK)}r^{-p}\text{ Vol}_{g(t_0)}(B_{r,t})\\{} & {} \quad \le (cK+K'/K)e^{-\int _{t_0}^s(K'/K+cK)}r^{-p}\text{ Vol}_{g(t_0)}(B_{r,t})\\{} & {} \quad =\partial _{s}\Big (-e^{-\int _{t_0}^s(K'/K+cK)}\Big )r^{-p}\text{ Vol}_{g(t_0)}(B_{r,t}), \end{aligned}$$

i.e.,

$$\begin{aligned} \partial _{s}\Big (e^{-\int _{t_0}^s(K'/K+cK)}\big (U(s)+r^{-p}\text{ Vol}_{g(t_0)}(B_{r,t})\big )\Big )\le 0, \end{aligned}$$

which implies

$$\begin{aligned} U(s)\le & {} e^{\int _{t_0}^s(K'/KcK)}\Big (U(t_0)+r^{-p}\text{ Vol}_{g(t_0)}(B_{r,t})\Big )\nonumber \\= & {} K(s)/K(t_0)\Big (U(t_0)+r^{-p}\text{ Vol}_{g(t_0)}(B_{r,t})\Big )e^{\int _{t_0}^s{cK}},\quad \forall \,s\in [t_0,t].\nonumber \\ \end{aligned}$$

(3.3)

In particular, we focus on the third term of U to see that for $p\ge 4$,

$$\begin{aligned}{} & {} cK(t)\int _{B(x_0,r/2)\cap \Sigma _t}\vert A\vert ^{p-2}\\{} & {} \quad \le K(t)/K(t_0)\left( \int _{B(x_0,r)\cap \Sigma _{t_0}}\vert A\vert ^{p}(t_0) +cK(t_0)\int _{B(x_0,r)\cap \Sigma _{t_0}}\vert A\vert ^{p-2}(t_0)\right. \\{} & {} \qquad \left. +\,r^{-p}\text{ Vol}_{g(t_0)}(B_{r,t})\right) \cdot e^{\int _{t_0}^{t}cK}. \end{aligned}$$

In other words, for $p\ge 2$,

$$\begin{aligned} \int _{B(x_0,r/2)\cap \Sigma _t}\vert A\vert ^{p}\le & {} \left( K(t_0)^{-1}\int _{B(x_0,r)\cap \Sigma _{t_0}}\vert A\vert ^{p+2}(t_0) +c\int _{B(x_0,r)\cap \Sigma _{t_0}}\vert A\vert ^{p}(t_0)\right. \\{} & {} \left. +\,K(t_0)^{-1}r^{-(p+2)}\text{ Vol}_{g(t_0)}(B_{r,t})\right) \cdot e^{\int _{t_0}^{t}cK}. \end{aligned}$$

Similarly, we can focus on the first term instead to see that for $p\ge 4$,

$$\begin{aligned} \int _{B(x_0,r/2)\cap \Sigma _t}\vert A\vert ^p\le & {} \left( K(t_0)^{-1}\int _{B(x_0,r)\cap \Sigma _{t_0}}\vert A\vert ^{p}(t_0) +c\int _{B(x_0,r)\cap \Sigma _{t_0}}\vert A\vert ^{p-2}(t_0)\right. \nonumber \\{} & {} \left. +\,K(t_0)^{-1}r^{-p}\text{ Vol}_{g(t_0)}(B_{r,t})\right) \cdot K(t)e^{\int _{t_0}^{t}cK}. \end{aligned}$$

(3.4)

$\square$

4 $L^\infty$ estimate and extension theorem

Combining Theorem 3.1 and Lemma 2.4 we obtain the following local estimate.

Theorem 4.1

$(L^\infty$ estimate). Fix $x_0\in {\mathbb {R}}^{n+1}$ and $r>0$. Let ${\textbf{x}}:\Sigma ^n\times [t_0,t_1]\rightarrow {\mathbb {R}}^{n+1}$ be a complete smooth mean curvature flow satisfying the bound

$$\begin{aligned} \sup _{B(x_0,r)\cap \Sigma _t}\vert HA\vert (\cdot ,t)\le K(t),\quad \forall \, t\in [t_0,t_1], \end{aligned}$$

where K(t) is nondecreasing. Then for any $q>n+2$ there exist positive constants $C=C(n,r,t_1-t_0,q,K(t_0))$ and $c=c(n,q)$ such that for any $t\in [t_0,t_1]$,

$$\begin{aligned} \sup _{D_{t,r}'}\vert A\vert \le C\left( 1+\Vert A\Vert _{L^{q+2}(B(x_0,2r)\cap \Sigma _{t_0})}\right) ^c\Big (1+\text{ Vol}_{g(t_0)}(B_{2r,t})\Big )^c \left( \int _{t_0}^{t}e^{\int _{t_0}^{s}cK}ds\right) ^{c}, \end{aligned}$$

where $B_{2r,t}=B(x_0,2r+n^{1/4}\int _{t_0}^{t}\sqrt{K})\cap \Sigma _{t_0}.$

Proof

Take $\beta =\frac{n+2}{2}$. Applying Lemma 2.4 to

$$\begin{aligned} (\partial _{t}-\Delta )\vert A\vert ^2=-2\vert \nabla A\vert ^2+2\vert A\vert ^4\le 2\vert A\vert ^2\cdot \vert A\vert ^2 \end{aligned}$$

yields

$$\begin{aligned}{} & {} \sup \limits _{D_{t,r}'}\vert A\vert \nonumber \\{} & {} \quad \le C\Big (1+\Vert A\Vert _{L^{q}(D_{t,r})}\Big )^{\frac{qn^2}{\beta (q-n-2)}} \Big (1+\Vert A\Vert _{L^{n+2}(D_{t,r})}^{n+2}\Big )^{\frac{qn^3}{2\beta (n+2)(q-n-2)}}\Vert A\Vert _{L^{2\beta }(D_{t,r})}\nonumber \\{} & {} \quad \le C\Big (1+\Vert A\Vert _{L^{q}(D_{t,r})}\Big )^{\frac{2qn^2}{(n+2)(q-n-2)}} \Big (1+\Vert A\Vert _{L^{n+2}(D_{t,r})}\Big )^{1+\frac{qn^3}{(n+2)(q-n-2)}}\nonumber \\{} & {} \quad \le C\Big (1+\Vert A\Vert _{L^{q}(D_{t,r})}\Big )^{\frac{2qn^2}{(n+2)(q-n-2)}} \Big (1+\Vert A\Vert _{L^{q}(D_{t,r})}\text{ Vol }(D_{t,r})^{\frac{1}{n+2}-\frac{1}{q}}\Big )^{1+\frac{qn^3}{(n+2)(q-n-2)}}\nonumber \\{} & {} \quad \le C\Big (1+\Vert A\Vert _{L^{q}(D_{t,r})}\Big )^{\frac{2qn^2}{(n+2)(q-n-2)}} \Big (1+\Vert A\Vert _{L^{q}(D_{t,r})}\Big )^{1+\frac{qn^3}{(n+2)(q-n-2)}}\nonumber \\{} & {} \qquad \Big (1+\text{ Vol }(D_{t,r})^{\frac{1}{n+2}-\frac{1}{q}}\Big )^{1+\frac{qn^3}{(n+2)(q-n-2)}}\nonumber \\{} & {} \quad \le C\Big (1+\Vert A\Vert _{L^q(D_{t,r})}\Big )^{1+\frac{qn^2}{q-n-2}} \Big (1+\text{ Vol}_{g(t_0)}(B_{2r,t})\Big )^{\frac{q-n-2}{q(n+2)}+\frac{n^3}{(n+2)^2}} , \end{aligned}$$

(4.1)

where

$$\begin{aligned} C= & {} C(n,r,t_1-t_0,q), \\ \text{ Vol }(D_{t,r}):= & {} \int _{t_0}^{t}\text{ Vol}_{g(s)}(B(x_0,r)\cap \Sigma _s)ds, \\ B_{2r,t}= & {} B(x_0,2r+n^{1/4}\int _{t_0}^{t}\sqrt{K})\cap \Sigma _{t_0}. \end{aligned}$$

It is derived from Theorem 3.1 that for $q>n+2$

$$\begin{aligned} \Vert A\Vert _{L^{q}(D_{t,r})}^{q}\le & {} \Big (K(t_0)^{-1}\Vert A\Vert _{L^{q+2}(B(x_0,2r)\cap \Sigma _{t_0})}^{q+2} +c\Vert A\Vert _{L^{q}(B(x_0,2r)\cap \Sigma _{t_0})}^{q}\\{} & {} +\,K(t_0)^{-1}(2r)^{-(q+2)}\text{ Vol}_{g(t_0)}(B_{2r,t})\Big ) \cdot \int _{t_0}^{t}e^{\int _{t_0}^{s}cK}ds, \end{aligned}$$

where $c=c(n,q)$. Note that

$$\begin{aligned}{} & {} K(t_0)^{-1}\Vert A\Vert _{L^{q+2}(B(x_0,2r)\cap \Sigma _{t_0})}^{q+2} +c\Vert A\Vert _{L^{q}(B(x_0,2r)\cap \Sigma _{t_0})}^{q}\\{} & {} \qquad +\,K(t_0)^{-1}(2r)^{-(q+2)}\text{ Vol}_{g(t_0)}(B_{2r,t})\\{} & {} \quad \le K(t_0)^{-1}\Vert A\Vert _{L^{q+2}(B(x_0,2r)\cap \Sigma _{t_0})}^{q+2} +c\Vert A\Vert _{L^{q+2}(B(x_0,2r)\cap \Sigma _{t_0})}^{q}\text{ Vol}_{g(t_0)}(B_{2r,t})^{\frac{2}{q+2}}\\{} & {} \qquad +\,K(t_0)^{-1}(2r)^{-(q+2)}\text{ Vol}_{g(t_0)}(B_{2r,t})\\{} & {} \quad \le C(n,r,q,K(t_0))\Big (1+\Vert A\Vert _{L^{q+2}(B(x_0,2r)\cap \Sigma _{t_0})}\Big )^{q+2}\Big (1+\text{ Vol}_{g(t_0)}(B_{2r,t})\Big ). \end{aligned}$$

The final coefficient is

$$\begin{aligned}{} & {} C\cdot \Big (1+\Vert A\Vert _{L^{q+2}(B(x_0,2r)\cap \Sigma _0)}\Big )^{(q+2)(\frac{1}{q}+\frac{n^2}{q-n-2})}\\{} & {} \quad \cdot \Big (1+\text{ Vol}_{g(t_0)}(B_{2r,t})\Big )^{\frac{1}{q}+\frac{n^2}{q-n-2}+\frac{q-n-2}{q(n+2)}+\frac{n^3}{(n+2)^2}}, \end{aligned}$$

where $C=C(n,r,t_1-t_0,q,K(t_0))$. Back to (4.1), we have the local $L^{\infty }$ estimate

$$\begin{aligned} \sup _{D_{t,r}'}\vert A\vert \le C\Big (1+\Vert A\Vert _{L^{q+2}(B(x_0,2r)\cap \Sigma _{t_0})}\Big )^c\Big (1+\text{ Vol}_{g(t_0)}(B_{2r})\Big )^c \left( \int _{t_0}^{t}e^{\int _{t_0}^{s}cK}ds\right) ^{c}. \end{aligned}$$

$\square$

As an application of the local estimate above, one immediately gets the following extension theorem about HA.

Corollary 4.2

Let ${\textbf{x}}:\Sigma ^n\times [0,T)\rightarrow {\mathbb {R}}^{n+1}$ be a complete smooth mean curvature flow. Suppose each time slice $\Sigma _t$ has bounded $\vert HA\vert$. There exists a positive constant $C=C(n,T,K,V,E,q)$ such that if

• (1) $\vert HA\vert$ satisfies

  $$\begin{aligned} \sup _{t\in [0,T)}\sup _{\Sigma _t}\vert HA\vert (\cdot ,t)\le K<\infty ; \end{aligned}$$

• (2) the initial data satisfies a uniform volume bound

  $$\begin{aligned} \sup _{x\in \Sigma _0}\text{ Vol}_{g(0)}(B(x,1+n^{1/4}T\sqrt{K})\cap \Sigma _0)\le V<\infty ; \end{aligned}$$

• (3) the initial data satisfies an integral bound

  $$\begin{aligned} \sup _{x\in \Sigma _0}\Vert A\Vert _{L^{q+2}(B(x,1)\cap \Sigma _0)}\le E<\infty \end{aligned}$$

  for some $q>n+2$,

then

$$\begin{aligned} \limsup _{t\rightarrow T}\sup \limits _{\Sigma _t}\vert A\vert (\cdot ,t)\le C<\infty . \end{aligned}$$

In particular, the flow can be extended past time T.

Proof

It suffices to use $B_{2r,t}\subset B_{2r,T}$ and take $r=1$ in Theorem 4.1. $\square$

5 Blowup estimate of $\vert HA\vert$

In this section we derive a blowup estimate of $\vert HA\vert$ from Theorem 4.1 and Lemma 2.2, which also implies a blowup estimate of mean curvature.

Theorem 5.1

(HA-blowup). There exists a positive constant $\epsilon =\epsilon (n)$ satisfying the following properties. Let ${\textbf{x}}:\Sigma ^n\times [0,T)\rightarrow {\mathbb {R}}^{n+1}$ be a complete smooth mean curvature flow with $T<\infty$. Suppose each time slice $\Sigma _t$ has bounded second fundamental form. If the flow blows up at time T, then

$$\begin{aligned} \limsup _{t\rightarrow T}\Big ((T-t)\sup _{\Sigma _t}\vert HA\vert \Big )\ge \epsilon . \end{aligned}$$

Conversely, if

$$\begin{aligned} \sup \limits _{\Sigma _t}\vert HA\vert \le \frac{\epsilon }{T-t},\quad \forall \, t\in [0,T), \end{aligned}$$

then

$$\begin{aligned} \limsup _{t\rightarrow T}\sup \limits _{\Sigma _t}\vert A\vert (\cdot ,t)\le C(n,T,\Sigma _0)<\infty , \end{aligned}$$

which implies the flow can be extended past time T.

Proof

Assume that the flow blows up at time T and there exist $t_0\in [0,T)$ and $\epsilon >0$ such that

$$\begin{aligned} \sup \limits _{\Sigma _t}\vert HA\vert < \frac{\epsilon }{T-t},\quad \forall \,t\in [t_0,T). \end{aligned}$$

Actually we find a smooth mean curvature flow ${\textbf{x}}:\Sigma ^n\times [t_0,T)\rightarrow {\mathbb {R}}^{n+1}$ with a $\vert HA\vert$ bound

$$\begin{aligned} K(t)=\frac{\epsilon }{T-t}. \end{aligned}$$

For t close to T,

$$\begin{aligned} \int _{t_0}^{t}c_nK= & {} \int _{t_0}^{t}\frac{c_n\epsilon }{T-s}ds=c_n\epsilon \log \left( \frac{T-t_0}{T-t}\right) , \\ \int _{t_0}^{t}e^{\int _{t_0}^{s}c_nK}ds= & {} \int _{t_0}^{t}\left( \frac{T-t_0}{T-s}\right) ^{c_n\epsilon }ds =\frac{(T-t_0)^{c_n\epsilon }}{1-c_n\epsilon }\Big ((T-t_0)^{1-c_n\epsilon }-(T-t)^{1-c_n\epsilon }\Big ). \end{aligned}$$

On the other hand, by Lemma 2.2 we know

$$\begin{aligned} \sup \limits _{\Sigma _t}\vert A\vert \ge \frac{1}{\sqrt{2}}(T-t)^{-\frac{1}{2}}. \end{aligned}$$

Note that $\Sigma _{t_0}$ has bounded geometry. If $c_n\epsilon <1$, then the integral $\int _{t_0}^{t}e^{\int _{t_0}^{s}c_nK}ds$ is bounded and the flow can be extended past time T by Theorem 4.1. This actually proves the second part. If $1\le c_n\epsilon \le \frac{3}{2}$, then

$$\begin{aligned} (T-t)^{-\frac{1}{2}} \gg C(n,T,\Sigma _{t_0},\epsilon )(T-t)^{-(c_n\epsilon -1)} \end{aligned}$$

as $t\rightarrow T$. In a word, the choice of $\epsilon \le \epsilon (n)$ causes a contradiction. This completes the proof of the first part. $\square$

Remark that Theorem 5.1 certainly works for the closed cases. The type-I blowup is optimal since the standard sphere $S^n\hookrightarrow {\mathbb {R}}^{n+1}$ satisfies $\vert HA\vert =\frac{n}{2(T-t)}$.

Corollary 5.2

(H-blowup). Let ${\textbf{x}}:\Sigma ^n\times [0,T)\rightarrow {\mathbb {R}}^{n+1}$ be a complete smooth mean curvature flow with a finite singular time T. Suppose each time slice $\Sigma _t$ has bounded second fundamental form. If

$$\begin{aligned} \limsup _{t\rightarrow T}\Big ((T-t)^{\lambda }\sup _{\Sigma _t}\vert A\vert \Big )<\infty \end{aligned}$$

for some $\lambda \in [\frac{1}{2},1)$, then we have the blowup estimate of mean curvature

$$\begin{aligned} \limsup _{t\rightarrow T}\Big ((T-t)^{1-\lambda }\sup _{\Sigma _t}\vert H\vert \Big )>0. \end{aligned}$$

Proof

For otherwise for any $\varepsilon >0$ one finds $t_{\varepsilon }$ such that

$$\begin{aligned} \sup _{\Sigma _t}\vert H\vert \le \varepsilon (T-t)^{\lambda -1},\quad \forall \,t\in [t_{\varepsilon },T). \end{aligned}$$

By the assumption there exist nonnegative constants $t_1\in [0,T)$ and

$$\begin{aligned} C:=\limsup _{t\rightarrow T}\Big ((T-t)^{\lambda }\sup _{\Sigma _t}\vert A\vert \Big )<\infty \end{aligned}$$

such that

$$\begin{aligned} \sup _{\Sigma _t}\vert A\vert \le C(T-t)^{-\lambda },\quad \forall \,t\in [t_{1},T). \end{aligned}$$

Hence we have

$$\begin{aligned} \sup _{\Sigma _t}\vert HA\vert \le C\varepsilon (T-t)^{-1},\quad \forall \,t\in [\max \{t_{\varepsilon },t_1\},T). \end{aligned}$$

Note the constant $\epsilon =\epsilon (n)$ in Theorem 5.1. Choosing $\varepsilon$ such that $C\varepsilon <\epsilon$ causes a contradiction according to Theorem 5.1. $\square$

Remark that by Theorem 5.1 of [5] Cooper proved the blowup of mean curvature under the same assumption in Corollary 5.2 and by Theorem 1.2 of [7] Le-Sesum proved the case of $\lambda =\frac{1}{2}$. Hence Theorem 5.1 and Corollary 5.2 can be seen as generalizations of these results.

Corollary 5.3

Let ${\textbf{x}}:\Sigma ^n\times [0,T)\rightarrow {\mathbb {R}}^{n+1}$ be a complete smooth mean curvature flow with a finite singular time T. Suppose each time slice $\Sigma _t$ has bounded second fundamental form. If

$$\begin{aligned} \limsup _{t\rightarrow T}\Big ((T-t)^{\lambda }\sup _{\Sigma _t}\vert H\vert \Big )<\infty \end{aligned}$$

for some $\lambda \in [0,\frac{1}{2})$, then we have the blowup estimate

$$\begin{aligned} \limsup _{t\rightarrow T}\Big ((T-t)^{1-\lambda }\sup _{\Sigma _t}\vert A\vert \Big )>0. \end{aligned}$$

In particular, $t=T$ is a type-II singularity.

Proof

By the same argument used in the proof of Corollary 5.2 we obtain the result. $\square$