On the Mean Convexity of a Space-and-Time Neighborhood of Generic Singularities Formed by Mean Curvature flow

Abstract

We consider a nondegenerate case of generic singularity when the limit cylinder of the rescaled mean curvature flow is ${\mathbb {R}}^3 \times {\mathbb {S}}^1$. We obtain a detailed description of a possibly small, but fixed, neighborhood of the singularity, up to (and including) the blowup time, and find that it is mean convex and the singularity is isolated. In the considered nondegenerate cases, we confirm the conjectures of mean convexity and isolation of singularities.

Proof of (4.26)

The proof of (4.26) is easier than controlling $\Big \Vert (100+|y|)^{-3+|k|} \Vert P_{\theta ,\ge 2} \partial _{\theta }^{3-|k|}\nabla _{y}^{k}\chi _{Y}v\Vert _{L_{\theta }^2}\Big \Vert _{\infty }$, $|k|=0,1,2,$ in the previous sections because the wanted decay rate is significantly slower, and hence is easier to obtain. Thus here we only sketch the proof.

We start with recalling some relevant results from [18]. We proved that, if $\tau _0$ is sufficiently large, then for $\tau \ge \tau _0$, v can be decomposed as

$$\begin{aligned} \begin{aligned}&v(y,\ \theta ,\tau )=V_{a(\tau ), B(\tau )}(y)+\Omega _1(\tau )\cdot y+\Omega _2(\tau )\cdot y \cos \theta + \Omega _3(\tau )\cdot y \sin \theta \\&\quad +\alpha _1(\tau ) \cos \theta +\alpha _2(\tau ) \sin \theta +w(y,\ \theta ,\tau ), \end{aligned} \end{aligned}$$

(A.1)

such that $\chi _{R}w$ is orthogonal to the same 18 functions listed in (3.12). Recall the definition of R from (3.6).

We take the following results from [18]:

Proposition A.1

There exists a small constant $\delta$ and a large time $\tau _1$ such that when $|y|\le (1+\epsilon )R(\tau )$, and $\tau \ge \tau _1$, and for $|k|+l=1,\ldots ,4$,

$$\begin{aligned} \left| v(\cdot ,\tau )-\sqrt{2}\right| ,\ |\nabla _{y}^{k}\partial _{\theta }^{l} v(\cdot ,\tau )|\le \delta . \end{aligned}$$

(A.2)

The scalar functions a, $\alpha _l$, $l=1,2,$ the three-dimensional-vector-valued functions $\Omega _k,$ $k=1,2,3,$ and the $3\times 3$-symmetric-matrix-valued function B satisfy the estimates in (3.13)–(3.15), and for the function w there exists a constant C such that,

$$\begin{aligned} \begin{aligned} \Vert \langle y\rangle ^{-3}\chi _{R}w(\cdot ,\tau )\Vert _{\infty }\le&C \kappa (\epsilon )R^{-4}(\tau ),\\ \Vert \langle y\rangle ^{-2}\nabla _{y}^{m}\partial _{\theta }^{n}\chi _{R}w(\cdot ,\tau )\Vert _{\infty }\le&C \kappa (\epsilon )R^{-3}(\tau ),\ |m|+n=1,\\ \Vert \langle y\rangle ^{-1}\nabla _{y}^{m}\partial _{\theta }^{n}\chi _{R}w(\cdot ,\tau )\Vert _{\infty }\le&C \kappa (\epsilon )R^{-2}(\tau ),\ |m|+n=2. \end{aligned} \end{aligned}$$

(A.3)

Now we prepare for proving (4.26) by making the technique of maximum principle applicable. Recall that the equation for $\chi _{R}w,$ derived in [18], is,

$$\begin{aligned} \partial _{\tau }\chi _{R}w=&-L \ \chi _{Y}w +\chi _{R}\left[ F+G+N_1(v)+N_2(\eta )\right] +\mu _{R}(w), \end{aligned}$$

(A.4)

where the operator L is defined as

$$\begin{aligned} L:=-\Delta _{y}+\frac{1}{2}y\cdot \nabla _{y}-\frac{1}{2}-V_{a,B}^{-2}, \end{aligned}$$

(A.5)

the functions F, G, $N_1$, and $N_2$ are defined in (4.1), and $\mu _{R}(w)$ is defined as

$$\begin{aligned} \mu _{R}(w):=\frac{1}{2}\left( y\cdot \nabla _{y}\chi _{R}\right) w+\left( \partial _{\tau }\chi _{R}\right) w-\left( \Delta _{y}\chi _{R}\right) w-2\nabla _{y}\chi _{R}\cdot \nabla _{y}w. \end{aligned}$$

Impose the operator $P_{\theta ,\ge 2}$ on both sides of (A.4) and use that $P_{\theta ,\ge 2}(F+G)=0$ to find

$$\begin{aligned} \partial _{\tau }(P_{\theta ,\ge 2}\chi _{R}w)=&-L(P_{\theta ,\ge 2}\chi _{Y}w) +P_{\theta ,\ge 2}\left[ N_1(\eta )\chi _{R}+N_2(v)\chi _{R}+\mu _{R}(w)\right] . \end{aligned}$$

(A.6)

To facilitate later discussion we define a function ${\tilde{\Phi }}_3$:

$$\begin{aligned} {{\tilde{\Phi }}}_3(y,\tau ):=(100+|y|^2)^{-3}\Vert P_{\theta ,\ge 2}\partial _{\theta }^3 \chi _{R}w(y,\cdot ,\tau )\Vert _{L^{2}_{\theta }}^2. \end{aligned}$$

We derive a governing equation for it from (A.6),

$$\begin{aligned} \begin{aligned} \partial _{\tau }{{\tilde{\Phi }}}_3=&-(L_3+W_3){{\tilde{\Phi }}}_3-2 (100+|y|^2)^{-3}\left[ V_{a,B}^{-2}\Vert P_{\theta ,\ge 2}\partial _{\theta }^4 \chi _{R}w\Vert _{L^{2}_{\theta }}^2+\Vert P_{\theta ,\ge 2}\partial _{\theta }^3\nabla _{y} \chi _{R}w\Vert _{L^{2}_{\theta }}^2\right] \\&+2(100+|y|^2)^{-3}D. \end{aligned} \end{aligned}$$

(A.7)

Here $L_3$ is a differential operator, and $W_3$ is a multiplier, defined as

$$\begin{aligned} \begin{aligned} L_3:=&-\Delta +\frac{1}{2}y\cdot \nabla _{y}-2(100+|y|^2)^{-3} \left( \nabla _y (100+|y|^2)^{3}\right) \cdot \nabla _{y},\\ W_3:=&-1+\frac{3|y|^2}{100+|y|^2}-\frac{18}{100+|y|^2}-\frac{24|y|^2}{(100+|y|^2)^2}-2V_{a,B}^{-2}, \end{aligned} \end{aligned}$$

(A.8)

and the term D is defined as

$$\begin{aligned} D:=&\Big \langle P_{\theta ,\ge 2}\partial _{\theta }^3 \chi _{R}w, \ \partial _{\theta }^{3}\left( \chi _{R}\left( N_{1}(v)+N_{2}(\eta )\right) +\mu _{R}(w)\right) \Big \rangle _{\theta }. \end{aligned}$$

We claim that D satisfies the estimate, when $\tau$ is sufficiently large,

$$\begin{aligned} \begin{aligned} (100+|y|^2)^{-3} D&\le \frac{1}{50}(100+|y|^2)^{-3}\left[ \Vert P_{\theta ,\ge 2}\partial _{\theta }^4 \chi _{R}w\Vert _{L^{2}_{\theta }}^2+\Vert P_{\theta ,\ge 2}\partial _{\theta }^3\nabla _{y} \chi _{R}w\Vert _{L^{2}_{\theta }}^2\right] \\&\quad +C\delta ^2\kappa ^2(\epsilon )R^{-8}. \end{aligned} \end{aligned}$$

(A.9)

Suppose the claim holds, then we put it back to (A.7) and observe some obvious cancellations. This, together with the facts $\Vert P_{\theta ,\ge 2}\partial _{\theta }^4 \chi _{R}w\Vert _{L^{2}_{\theta }}^2\ge 4 \Vert P_{\theta ,\ge 2}\partial _{\theta }^3 \chi _{R}w\Vert _{L^{2}_{\theta }}^2$ and that $V_{a,B}^{-2}=\frac{1}{2}+{\mathcal {O}}(\tau ^{-\frac{1}{2}})$ if $|y|\le (1+\epsilon )R$, implies that, for some $C>0,$

$$\begin{aligned} \partial _{\tau }{{\tilde{\Phi }}}_3\le -(L_3+\frac{1}{2}){{\tilde{\Phi }}}_3+C \delta ^2 \kappa ^2(\epsilon ) R^{-8}. \end{aligned}$$

(A.10)

Before applying the maximum principle we need to check the boundary conditions. The cutoff function $\chi _{R}$ in the definition of ${{\tilde{\Phi }}}_3$ makes ${{\tilde{\Phi }}}_3(y,\tau )=0$ when $|y|\ge (1+\epsilon )R(\tau )$.

Apply the maximum principle to have that, for some $C_1>0,$

$$\begin{aligned} {\tilde{\Phi }}_3(\tau )\le e^{-\frac{1}{2}(\tau -\tau _1)}{{\tilde{\Phi }}}_3(\tau _1)+C_1 \delta ^2 \kappa ^2(\epsilon ) R^{-8}. \end{aligned}$$

(A.11)

This, together with ${{\tilde{\Phi }}}_3(\tau _1)\le \delta$ if $\tau \ge \tau _0$ implied by (A.2), implies the desired estimate, provided that $\tau$ is sufficiently large.

To complete the proof we need to prove the claim (A.9).

1.1 Proof of (A.9)

We decompose D into two terms

$$\begin{aligned} D:= D_{1}+D_{2}, \end{aligned}$$

(A.12)

where $D_{1}$ is defined as

$$\begin{aligned} D_{1}:= \langle P_{\theta ,\ge 2}\partial _{\theta }^3 \chi _{R}w, \ \partial _{\theta }^{3} \chi _{R}\left( N_{1}+N_{2}\right) \rangle _{\theta }=-\langle P_{\theta ,\ge 2}\partial _{\theta }^4 \chi _{R}w, \ \chi _{R} \partial _{\theta }^{2} \left( N_{1}+N_{2}\right) \rangle _{\theta }, \end{aligned}$$

here we integrate by parts in $\theta$ in the second step, and $D_2$ is defined as

$$\begin{aligned} D_2:=\langle P_{\theta ,\ge 2}\partial _{\theta }^3 \chi _{R}w, \ \mu _{R}(\partial _{\theta }^{3}w)\rangle _{\theta }. \end{aligned}$$

For $D_1$ we compute directly, use (A.2) and $\partial _{\theta }v=\partial _{\theta }\eta$ to obtain

$$\begin{aligned} \chi _{R}|\partial _{\theta }^{2}N_{1}|\lesssim&\delta \chi _{R}\sum _{0\le l\le 2} \left[ |\partial _{\theta }^{l}\nabla _{y}\eta |^2+|\partial _{\theta }^{l+1}\eta |\right] , \end{aligned}$$

(A.13)

$$\begin{aligned} \chi _{R}\Vert \partial _{\theta }^2 N_2\Vert _{L_{\theta }^2}\lesssim&\delta \chi _{R}\sum _{l=1,2,3,4}\Vert \partial _{\theta }^l \eta \Vert _{L_{\theta }^2}. \end{aligned}$$

(A.14)

Next we estimate the terms on the right-hand side.

For the first term on the right-hand side of (A.13), we decompose $\eta$ as in (4.4), use (A.2), and change the order of $\nabla _y$ and $\chi _R$ to find

$$\begin{aligned} \chi _{R} |\partial _{\theta }^{l}\nabla _y \eta |^2\lesssim (1+|y|)^2 \tau ^{-1}+\sum _{m=\pm 1}|\nabla _{y}\chi _{R}w_{m}||\nabla _{y}w_{m}|+\delta |\nabla _{y}\chi _{R}|+\delta |\partial _{\theta }^{l}\nabla _{y}P_{\theta ,\ge 2}\chi _{R}w|. \end{aligned}$$

(A.15)

For the second term on the right-hand side we have

$$\begin{aligned} \begin{aligned}&(100+|y|^2)^{-\frac{3}{2 }}\sum _{m=\pm 1}\left\| (\nabla _{y}\chi _{R}w_{m})(\nabla _{y}w_{m})\right\| _{L_{\theta }^2}\\&\quad \lesssim \Vert \langle y\rangle ^{-2}\nabla _{y}\chi _{R}w_{m}\Vert _{\infty }\left[ \Vert \langle y\rangle ^{-1}\nabla _y \chi _{R}w_{m}\Vert _{\infty }+\Vert 1_{\le (1+\epsilon )R}\langle y\rangle ^{-1}\nabla _y(1-\chi _{R}) w_{m}\Vert _{\infty }\right] \\&\quad \lesssim \Vert \langle y\rangle ^{-2}\nabla _{y}\chi _{R}w\Vert _{\infty }\left[ R^2 \Vert \langle y\rangle ^{-3}\nabla _y \chi _{R}w\Vert _{\infty }+\Vert 1_{\le (1+\epsilon )R}\langle y\rangle ^{-1}\nabla _y(1-\chi _{R}) w\Vert _{\infty }\right] \\&\quad \lesssim \kappa (\epsilon ) R^{-4}\left( \kappa (\epsilon )R^{-1}+\delta \right) , \end{aligned} \end{aligned}$$

(A.16)

where in the second step, to control $\nabla _y w$ we inserted $1=\chi _{R}+(1-\chi _{R})$ before w, and applied the estimates in (A.3) in the last step; we also used that the function $\nabla _y(1-\chi _{R}) w$ is supported by the set $\{y\ |\ |y|\ge R\}$ to obtain $\langle y\rangle ^{-1}\lesssim R^{-1}$. For the third term, the function $|\nabla _{y}\chi _{R}|=R^{-1}\left| \chi ^{'}\left( \frac{|y|}{R}\right) \right|$ is supported by the set $\{ y\ \big | \ |y|\ge R\},$ thus,

$$\begin{aligned} \langle y\rangle ^{-3}|\nabla _{y}\chi _{R}|\lesssim \kappa (\epsilon )R^{-4}. \end{aligned}$$

(A.17)

Return to (A.15), the estimates above and that $\Vert \partial _{\theta }f\Vert _{L_{\theta }^2}\le \Vert \partial _{\theta }^2f\Vert _{L_{\theta }^2}$ for any smooth periodic function f imply

$$\begin{aligned} (100+|y|^2)^{-\frac{3}{2 }}\chi _{R}\Big \Vert |\partial _{\theta }^{l}\nabla _y \eta |^2\Big \Vert _{L_{\theta }^2}\lesssim \delta (100+|y|^2)^{-\frac{3}{2}} \Vert \nabla _{y}\partial _{\theta }^3 P_{\theta ,\ge 2} \chi _{R}w\Vert _{L_{\theta }^2}+\delta \kappa (\epsilon )R^{-4}. \end{aligned}$$

(A.18)

The second term on the right-hand side of (A.13) can be controlled more easily since $\chi _{Y}$ and $\partial _{\theta }$ commute, thus we just state the result, but skip the proof,

$$\begin{aligned} (100+|y|^2)^{-\frac{3}{2 }}\chi _{R}\left\| |\partial _{\theta }^{l+1} \eta |^2\right\| _{L_{\theta }^2} \lesssim \delta (100+|y|^2)^{-\frac{3}{2}} \Vert \partial _{\theta }^4\chi _{R}P_{\theta ,\ge 2} w\Vert _{L_{\theta }^2}+\delta \kappa (\epsilon )R^{-4}. \end{aligned}$$

(A.19)

To complete our treatment of (A.13), we use (A.18) and (A.19) to obtain

$$\begin{aligned} \begin{aligned}&(100+|y|^2)^{-\frac{3}{2}}\chi _{R}\Vert \partial _{\theta }^{2}\chi _{R}N_{1}(v)\Vert _{L_{\theta }^2}\\&\quad \lesssim \delta (100+|y|^2)^{-\frac{3}{2}}\left[ \Vert \nabla _{y}\partial _{\theta }^3 P_{\theta ,\ge 2} \chi _{R}w\Vert _{L_{\theta }^2}+ \Vert \partial _{\theta }^4\chi _{R}P_{\theta ,\ge 2} w\Vert _{L_{\theta }^2}\right] +\delta \kappa (\epsilon )R^{-4}. \end{aligned} \end{aligned}$$

(A.20)

Apply similar techniques on (A.14) to find that,

$$\begin{aligned}&(100+|y|^2)^{-\frac{3}{2}}\chi _{R}\Vert \partial _{\theta }^{2}\chi _{R}N_{2}\Vert _{L_{\theta }^2} \lesssim \delta (100+|y|^2)^{-\frac{3}{2}} \Vert \partial _{\theta }^4\chi _{R}P_{\theta ,\ge 2} w\Vert _{L_{\theta }^2}+\delta \kappa (\epsilon )R^{-4}. \end{aligned}$$

(A.21)

Return to the definition of $D_1$ in (A.12). The preliminary estimates in (A.13) and (A.14), the estimates (A.20), (A.21), and Young’s inequality imply that, for some $C>0,$

$$\begin{aligned} \begin{aligned}&(100+|y|^2)^{-3}|D_{1}|\\&\quad \le \frac{1}{100}(100+|y|^2)^{-3}\left[ \Vert P_{\theta ,\ge 2}\partial _{\theta }^4 \chi _{R}w\Vert _{L^{2}_{\theta }}^2+ \Vert P_{\theta ,\ge 2}\partial _{\theta }^{3}\chi _{R}N_{2}(\eta )\Vert _{L^{2}_{\theta }}^2\right] +C\delta ^2 \kappa ^2 (\epsilon )R^{-8}. \end{aligned} \end{aligned}$$

(A.22)

For $D_{2}$, the term $\frac{1}{2}(y\nabla _{y}\chi _{R})$ in the definition of $\mu _{R}(\partial _{\theta }^3P_{\theta ,\ge 2} w)$ is of order ${\mathcal {O}}(1)$, but has a favorable nonpositive sign. This makes

$$\begin{aligned} D_{2}\le \left\langle P_{\theta ,\ge 2}\partial _{\theta }^3 \chi _{R}w, \ P_{\theta ,\ge 2}\partial _{\theta }^3\left( \big (\partial _{\tau }\chi _{R}\big )w-\left( \Delta _{y}\chi _{R}\right) w-2\nabla _{y}\chi _{R}\cdot \nabla _{y} w\right) \right\rangle _{\theta }. \end{aligned}$$

(A.23)

Its decay rate is generated by the derivatives of $\chi _{R}$ and that they are supported by the set $\left\{ y \big | \ |y|\ge R\right\}$, similar to (A.17) above. These together with (A.2) and Young’s inequality imply, for some $C>0,$

$$\begin{aligned} (100+|y|^2)^{-3} D_{2}\le \frac{1}{100} {{\tilde{\Phi }}}_3+C\delta ^2 \kappa ^2 (\epsilon )R^{-8}. \end{aligned}$$

(A.24)

Take the estimates in (A.22), (A.24) to (A.12), and obtain the desired estimate (A.9).