Generic Regularity of Level Set Flows with Spherical Singularity

Abstract

The sphere is well-known as the only generic compact shrinker for mean curvature flow (MCF). In this paper, we characterize the generic dynamics of MCFs with a spherical singularity. In terms of the level set flow formulation of MCF, we establish that generically the arrival time function of level set flow with spherical singularity has at most $C^2$ regularity.

Appendices

Appendix A: Action of Translations and Dilations

In Euclidean space $\mathbb {R}^{n+1}$, the translations form a group denoted by $\mathbb {R}^{n+1}$, and the dilations form a semigroup denoted by $\mathbb {R}_+$. Each of these spaces carries a natural (Euclidean) norm. Let $(U,\alpha )$ be an element in $\mathbb {R}^{n+1}\times \mathbb {R}_+$. For any point or hypersurface, $(U,\alpha )$ acts by first dilating the point/hypersurface by $\alpha$ and then translating it by the vector U. Suppose $\Sigma$ is a hypersurface in $\mathbb {R}^{n+1}$ and $\Gamma$ is a graph of the function u over $\Sigma$, which means $\Gamma = {x+u(x){\textbf{n}}(x) : x \in \Sigma }$, where ${\textbf{n}}$ is the unit normal vector field on $\Sigma$. Let $|U|=1$. We want to understand how the action of $(\beta U, \alpha )$ on $\Gamma$ would change this graph. Suppose $(\beta U, \alpha )(\Gamma ) = {\widetilde{\Gamma }}$. We assume that $|(\alpha U, \alpha )|$ is sufficiently small such that ${\widetilde{\Gamma }}$ is also the graph of a function w over $\Sigma$. In other words, we have

$$\begin{aligned} \Gamma =\{x+u(x){\textbf{n}}(x):x\in \Sigma \}, \mathrm {\ and\ } {\widetilde{\Gamma }}=\{x+w(x){\textbf{n}}(x):x\in \Sigma \}. \end{aligned}$$

The action of $(U,\alpha )$ indicates that

$$\begin{aligned} \{\alpha (x+u(x){\textbf{n}}(x))+U:x\in \Sigma \}=\{x+w(x){\textbf{n}}(x):x\in \Sigma \}. \end{aligned}$$

We will be interested in the case that $\Sigma$ is a sphere. In this case, we can assume $x={\textbf{n}}(x)$, and the second fundamental form is the identity. Then $\alpha (r+u(x_0)){\textbf{n}}(x_0)+\beta U=x_{\alpha ,\beta }+w_{\alpha ,\beta }(x_{\alpha ,\beta }){\textbf{n}}(x_{\alpha ,\beta })$. We have the following control on Q-norm.

Lemma A.1

There exists $\varepsilon _0>0$ with the following significance. When $\Vert u\Vert _{C^1}\le \varepsilon _0$, $|\alpha -1|\le \varepsilon _0$ and $\beta <\varepsilon _0$, we have

$$\begin{aligned} \left\| w_{\alpha ,\beta }-u-\left[ (\alpha -1)+ \beta \langle U,x\rangle \right] \right\| _{Q}\le C(|\alpha -1|^2+|\beta |^2+\Vert u\Vert _{C^1}(|\alpha -1|+|\beta |)),\nonumber \\ \end{aligned}$$

(A.1)

where C is a uniform constant.

Proof

It is clear that $x_{1,0}=x$ and $w_{1,0}(x)=u(x_0)$. Taking derivative of $\alpha$ and U we obtain the following: at $\alpha =1,\beta =0$,

$$\begin{aligned} {{\,\mathrm{\partial }\,}}_{\alpha }x_{\alpha ,\beta }=0, \ {{\,\mathrm{\partial }\,}}_\beta x_{\alpha ,\beta }=\frac{U^\top }{1+u}. \end{aligned}$$

(A.2)

and

$$\begin{aligned} {{\,\mathrm{\partial }\,}}_{\alpha } w_{\alpha ,\beta } = 1+u,\ {{\,\mathrm{\partial }\,}}_{\beta } w_{\alpha ,\beta } =\langle U,x_0\rangle - \frac{\langle \nabla u,U\rangle }{1+u}. \end{aligned}$$

(A.3)

Then we have

$$\begin{aligned} w_{\alpha ,\beta }=u+(1+u)(\alpha -1)+ \left( \langle U,x_0\rangle - \frac{\langle \nabla u,U\rangle }{1+u}\right) \beta +O((\alpha -1)^2+\beta ^2).\nonumber \\ \end{aligned}$$

(A.4)

Similarly, taking the gradient we get

$$\begin{aligned} \nabla w(x) = \left( \frac{D x_{\alpha ,\beta }}{Dx}\right) ^{-1} \alpha \nabla u(x), \end{aligned}$$

where $\frac{D x_{\alpha ,\beta }}{Dx}$ is the differential of $x_{\alpha ,\beta }$ at $\alpha =1$, $\beta =0$. From the local expansion

$$\begin{aligned} \frac{D x_{\alpha ,\beta }}{Dx}=\textrm{Id} +\beta \frac{\nabla U^\top }{1+u} -\beta \frac{U\otimes \nabla u}{(1+u)^2} +O((\alpha -1)^2+\beta ^2), \end{aligned}$$

we can see that

$$\begin{aligned} \begin{aligned} \nabla w(x) =&\nabla u(x) +(\alpha -1)\nabla u(x)+ \beta \nabla \langle U,x\rangle +\\&+ O(|\alpha -1|^2+|\beta |^2+\Vert u\Vert _{C^1}(|\alpha -1|+|\beta |)). \end{aligned} \end{aligned}$$

(A.5)

Pointwise we have the Taylor expansion (A.4) and (A.5). Then we integrate the square and use Cauchy-Schwarz inequality to get the desired inequality. $\square$

Next, we prove Lemma 5.4. The main idea is to show that if the ${\mathcal {X}}_2$ part of a function u has a large proportion in $\Vert u\Vert$, then the ${\mathcal {X}}_+$ part has a small proportion, hence the centering map is also small. Otherwise, the centering map will enhance the ${\mathcal {X}}_+$ part a lot, then the invariant cone argument (Theorem 3.1) shows that the graph RMCF grows exponentially, which is a contradiction to the definition of a centering map.

Let us recall the setting. Suppose $M_t$ is a graph of function f over $\mathbb {S}^n$, and suppose $\widetilde{M_t}$ is a graph of function u over $\mathbb {S}^n$. Suppose both $\Vert f\Vert _{C^2}$ and $\Vert u\Vert _{C^2}$ are sufficiently small. Throughout the rest of this section, $\Vert \cdot \Vert$ is the Q-norm. Suppose $\Vert \pi _2(u-f)\Vert \ge (1-\varepsilon )\Vert u-f\Vert$ for some sufficiently small $\varepsilon$. Now we apply the centering map to $\widetilde{M_t}$, and suppose after the centering map ${\mathcal {C}}(\widetilde{M_t})$ is a graph of a function w(t) over $\mathbb {S}^n$.

Lemma A.2

(Lemma 5.4) In the above setting, we have

$$\begin{aligned} \Vert \pi _2(w-f)\Vert \ge (1-2\varepsilon )\Vert w-f\Vert . \end{aligned}$$

Proof

By triangle inequality, $\Vert \pi _2(w-f)\Vert \ge \Vert \pi _2(u-f)\Vert -\Vert \pi _2(u-w)\Vert _{L^2}$, $\Vert w-f\Vert \le \Vert (u-f)\Vert +\Vert u-w\Vert$. Thus

$$\begin{aligned} \begin{aligned} \Vert \pi _2(w-f)\Vert \ge&(1-\varepsilon )(\Vert w-f\Vert -\Vert u-w\Vert )-\Vert \pi _2(u-w)\Vert \\ \ge&(1-\varepsilon )\Vert w-f\Vert -2\Vert u-w\Vert . \end{aligned} \end{aligned}$$

Then we only need to show $\Vert u-w\Vert \le \frac{\varepsilon }{2}\Vert w-f\Vert$. Suppose the centering map is given by $(\alpha ,\beta U)$. Lemma A.1 implies that when $|\alpha -1|$ and $|\beta |$ and $\Vert u\Vert _{C^1}$ are sufficiently small,

$$\begin{aligned} \Vert \pi _+(u-w)\Vert \ge (|\alpha -1|+|\beta |)(1-2\varepsilon _0). \end{aligned}$$

Then we have

$$\begin{aligned} \Vert \pi _+(w-f)\Vert \ge (|\alpha -1|+|\beta |)(1-2\varepsilon _0) - \Vert \pi _+ (u-f)\Vert . \end{aligned}$$

Because w is the graph of a hypersurface after the centering map, $\Vert \pi _+(w-f)\Vert \le \varepsilon \Vert (w-f)\Vert$, since otherwise, Theorem 3.1 would imply that the cone ${\mathcal {K}}^+_\varepsilon$ is preserved under the dynamics and $\pi _+(w-f)$ grows exponentially hence changes the spacetime location of the singularity, which is a contradiction to the definition of the centering map. So we get the estimate

$$\begin{aligned}|\alpha -1|+|\beta |\le \frac{1}{1-2\varepsilon _0} (\varepsilon \Vert (w-f)\Vert +\Vert \pi _+(u-f)\Vert ). \end{aligned}$$

Lemma A.1 also implies that

$$\begin{aligned} \Vert (u-w)\Vert \le (1+2\varepsilon _0) (|\alpha -1|+|\beta |). \end{aligned}$$

From $\Vert \pi _2(u-f)\Vert \ge (1-\varepsilon )\Vert u-f\Vert$, we also know that $\Vert \pi _+(u-f)\Vert \le \frac{\varepsilon }{1-\varepsilon }\Vert u-f\Vert$. Combining all the ingredients above shows that $\Vert \pi _+(w-f)\Vert \le \varepsilon \Vert (w-f)\Vert$. $\square$

Appendix B: Denseness of Image of the Fundamental Solution

Let $M_t, t\in [0,T]$ be an RMCF. Let ${\mathcal {T}}(t,0):\ L^2(M_0)\rightarrow L^2(M_t)$ be the fundamental solution to the variational equation $\partial _t u=L_{M_t}u$ and ${\mathcal {T}}^*(0,t):\ L^2(M_t)\rightarrow L^2(M_0)$ its adjoint solving the conjugate equation $\partial _t v=-L_{M_t}v+{\tilde{H}}^2 v,$ where ${\tilde{H}}=H-\langle x,\textbf{n}(x)\rangle /2$ is the rescaled mean curvature. The extra factor takes into account of the fact $\frac{d}{dt} d\mu _{M_t}=-\tilde{H}^2d\mu _{M_t}$. Denoting $\square _t=\partial _t -L_{M_t}$ and $\square ^*_{t}=-\partial _t -L_{M_t}+{\tilde{H}}^2,$ we get the Duhamel principle (c.f. Lemma 26.1 of [3])

$$\begin{aligned} \int _0^T dt \int _{M_t} (\square _t A(x,t) B(x,t)-A(x,t) \square ^*_t B(x,t))d\mu _{M_t}=\int _{M_t}A(x,t)B(x,t)d\mu _{M_t} \Big |_{t=0}^{t=T} \end{aligned}$$

for all $A,B:\ \cup _{t\in [0,T]} M_t\rightarrow \mathbb {R}$ that are $C^2$ in x and $C^1$ in t.

This implies that in particular $\langle v, {\mathcal {T}}(t,0) u\rangle _{L^2(M_t)}=\langle {\mathcal {T}}^*(0,t)v, u\rangle _{L^2(M_0)}$, where $u\in L^2(M_0)$ and $v\in L^2(M_t)$. We refer readers to [3, Chapter 26] for heat kernel on evolving manifolds.

We first give the proof of Lemma 5.2 following Lemma 5.2 of [11].

Proof of Lemma 5.2

Suppose $v\in L^2(M_t)$ satisfies $\langle v,{\mathcal {T}}(t,0)u\rangle =0$ for all $u\in L^2(M_0)$, where $\langle \cdot ,\cdot \rangle$ is the $L^2(M_t)$-inner product. Then we get $\langle {\mathcal {T}}^*(0,t)v,u\rangle =0$ for all $u\in L^2(M_0)$, which implies ${\mathcal {T}}^*(0,t)v=0$. We next denote $w(s)={\mathcal {T}}^*(t-s,t)v$ treating t as fixed. Thus we get $w(t)=0$ and $w(0)=v$ and w solves the equation $\partial _s w=L_{M_{t-s}} w-{\tilde{H}}^2_{M_{t-s}} w.$ Applying the following backward uniqueness theorem of Lions–Malgrange, we get $w(0)=v=0$. The proof is then complete.$\square$

Theorem B.1

([20])

• (1) Let $V\subset H$ be two Hilbert spaces such that the injection $V\subset H$ is continuous and V is dense in H.

• (2) Let A(t) be a self-adjoint operator such that the function $a(t,u,v):=\langle A(t)u,v\rangle _H$ is a semilinear form continuous on $V\times V$, is $C^1$ in $t\in [0,T]$, and there exists $C,\lambda , \alpha >0$ such that $\alpha \Vert v\Vert _V^2\le a(t,u,u)+\lambda \Vert u\Vert _H^2$.

• (3) Let u satisfy $u\in L^2((0,T), V)$,$u'\in L^2((0,T),H)$, $u\in \textrm{Dom}(A(t))$ for each $t\in (0,T)$, and $\partial _tu+A(t)u=0$, $u(T)=0$.

Then $u\equiv 0$ on [0, T].

In our case, we have $V=H^1(M_t)$, $H=L^2(M_t)$ and $A(s)=L_{M_{t-s}}-{\tilde{H}}^2_{M_{t-s}}$.

Proof of Lemma 5.3

We adapt the above proof as follows. Suppose $v\in H^1(M_t)$ satisfies $\langle v,{\mathcal {T}}(t,0)u\rangle _Q=0$ for all $u\in L^2(M_0)$, where $\langle \cdot ,\cdot \rangle _Q$ is the $Q(M_t)$-inner product. From the definition of the Q-norm,

$$\begin{aligned} \langle v_1,v_2\rangle _{Q(M_t)}:= & {} \frac{1}{2}(\Vert v_1+v_2\Vert ^2_{Q(M_t)}-\Vert v_1\Vert ^2_{Q(M_t)}-\Vert v_2\Vert ^2_{Q(M_t)})\\ {}= & {} \int _{M_t}(\nabla v_1\cdot \nabla v_2+\Lambda v_1v_2)e^{-\frac{|x|^2}{4}}d\mu . \end{aligned}$$

Thus we have $\langle (\Lambda -\Delta )v, {\mathcal {T}}(t,0)u\rangle =0$ where $\langle \cdot ,\cdot \rangle$ is the $L^2(M_t)$-inner product, and $(\Lambda -\Delta )v\in H^{-1}(M_t)$. Applying the adjoint heat kernel we get $\langle {\mathcal {T}}^*(0,t)((\Lambda -\Delta )v), u\rangle =0$. We define similarly $w(s)={\mathcal {T}}^*(s,t)((\Lambda -\Delta )v)$, which satisfies $w(t)=0$ and $w(0)=(\Lambda -\Delta )v$ and w solves the equation $\partial _s w=L_{M_{t-s}} w-{\tilde{H}}^2_{M_{t-s}} w.$ The above theorem of Lions-Malgrange implies that $w(0)=(\Lambda -\Delta )v=0$. Then Lax–Milgram implies that $v=0.$ This completes the proof.$\square$

Appendix C: Transplantation

Suppose $\Sigma$ and $\Sigma _1$ are embedded hypersurfaces and $\Sigma _1$ is a graph of function f over $\Sigma$, i.e. $\Sigma _1=\{x+f(x){\textbf{n}}(x):x\in \Sigma \}$. Then a function g defined on $\Sigma _1$ can be viewed as a function ${\bar{g}}$ defined on $\Sigma$ as follows: we define ${\bar{g}}(x)=g(x+f(x){\textbf{n}}(x))$. Such an identification is called transplantation in [25].

The following theorem was proved in Appendix of [25].

Theorem C.1

Let $\Sigma$ be a fixed embedded closed hypersurface. Then given$\varepsilon >0$, there exists a constant $\mu (\varepsilon )>0$ such that the following is true: Suppose $\Sigma _1$ is the graph $\{x+f{\textbf{n}}(x):x\in \Sigma \}$ over $\Sigma$, and $\Sigma _2$ is the graph $\{y+g{\textbf{n}}(y):y\in \Sigma _1\}$ over $\Sigma _1$, and $\Vert f\Vert _{C^4(\Sigma )}\le \mu$, $\Vert g\Vert _{C^{2,\alpha }(\Sigma _1)}\le \mu$, then $\Sigma _2$ is a graph of a function v on $\Sigma$, and

$$\begin{aligned} \Vert v-(f+g)\Vert _{C^{2,\alpha }(\Sigma )}\le \varepsilon \Vert g\Vert _{C^{2,\alpha }(\Sigma _1)}. \end{aligned}$$

Here we transplant g on $\Sigma _1$ to a function on $\Sigma$, and still use g to denote it.

For our purpose, we state the following theorem for the Q estimate. The proof is the same as the proof in [25], because in [25] actually a pointwise bound was proved.

Theorem C.2

Suppose the assumptions in Theorem C.1. Then we have

$$\begin{aligned} \Vert v-(f+g)\Vert _Q\le \varepsilon \Vert g\Vert _Q. \end{aligned}$$

Here we transplant g on $\Sigma _1$ to a function on $\Sigma$ and still use g to denote it.