The space of asymptotically conical self-expanders of mean curvature flow

Abstract

We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders—that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone—are generic in a certain sense.

Appendices

Appendix A: Estimates for the heat equation on $\mathbb {R}^n$

We first state a well-known maximum principle for parabolic equations on $\mathbb {R}^n$.

Proposition A.1

If $w\in C^0(\mathbb {R}^n\times [0,T])$ has continuous spatial derivatives up to the second order and continuous time derivative and satisfies

$$\begin{aligned} \left\{ \begin{array}{cc} \partial _t w-\sum \limits _{i,j=1}^n a^{ij}\partial _{x_ix_j}^2 w-\sum \limits _{i=1}^n b^i\partial _{x_i}w-cw\le 0 &{} \text{ in } \mathbb {R}^n\times (0,T)\\ w(x,0)\le 0 &{} \text{ for } x\in \mathbb {R}^n \end{array} \right. \end{aligned}$$

where for some $\lambda >0$,

$$\begin{aligned} \sum _{i,j=1}^n a^{ij}(x,t)\xi _i\xi _j \ge 0 \text{ and } \sum _{i,j=1}^n |a^{ij}(x,t)|+\sum _{i=1}^n |b^i(x,t)|+|c(x,t)| \le \lambda , \end{aligned}$$

then $w\le 0$ on $\mathbb {R}^n\times [0,T]$.

Proof

For each $l>0$ we define

$$\begin{aligned} w_l(x,t)=e^{-2\lambda t} w(x,t)-l(1+|x|^2)^{\frac{1}{2}}. \end{aligned}$$

Then, by direct computations,

$$\begin{aligned} \partial _t w_l-\sum _{i,j=1}^n a^{ij}\partial _{x_ix_j}^2 w_l-\sum _{i=1}^n b^i \partial _{x_i} w_l+(2\lambda -c) w_l \le C(n,\lambda ) l. \end{aligned}$$

Clearly, $w_l(x,0)\le 0$, and as $w\in C^0(\mathbb {R}^n\times [0,T])$, $w_l(x,t)\rightarrow -\infty$ when (x, t) approaches infinity.

Thus, there is a point $(x_l,t_l)\in \mathbb {R}^n\times [0,T]$ such that

$$\begin{aligned} \sup _{(x,t)\in \mathbb {R}^n\times [0,T]} w_l(x,t)=w_l(x_l,t_l). \end{aligned}$$

If $t_l>0$, then

$$\begin{aligned} \partial _t w_l(x_l,t_l)\ge 0, \, \partial _{x_i} w_l(x_l,t_l)=0, \text{ and } \sum _{i,j=1}^n a^{ij}(x_l,t_l)\partial _{x_ix_j}^2 w_l(x_l,t_l)\le 0. \end{aligned}$$

As $|c|\le \lambda$, it follows that $w_l(x_l,t_l)\le C l \lambda ^{-1}$. If $t_l=0$, then $w_l(x_l,t_l)\le 0$. Hence,

$$\begin{aligned} \sup _{(x,t)\in \mathbb {R}^n\times [0,T]} w_l(x,t)\le C l \lambda ^{-1}. \end{aligned}$$

Now passing $l\rightarrow 0$, we get

$$\begin{aligned} \sup _{(x,t)\in \mathbb {R}^n\times [0,T]} w(x,t)\le 0, \end{aligned}$$

proving the claim. $\square$

We now prove Lemma 5.5 for the heat equation on $\mathbb {R}^n$.

Proposition A.2

Let $\beta \in (0,1)$. Given $h\in C^0((0,1); C^\beta (\mathbb {R}^n))$ the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{cc} \partial _t w-\Delta w=h &{} \text{ in } \mathbb {R}^n\times (0,1)\\ \displaystyle \lim _{(x^\prime ,t)\rightarrow (x,0)} w(x^\prime ,t)=0 &{} \text{ for } x\in \mathbb {R}^n \end{array} \right. \end{aligned}$$

(A.1)

has a unique solution in $C^0((0,1); C^{2,\beta }(\mathbb {R}^n))$. Moreover, w satisifes

$$\begin{aligned} \sup _{0<t<1} \sum _{i=0}^2 t^{\frac{i-2}{2}} \Vert \nabla ^i w(\cdot ,t)\Vert _{\beta } \le \nu (n,\beta ) \sup _{0<t<1} \Vert h(\cdot ,t)\Vert _\beta . \end{aligned}$$

Proof

On $\mathbb {R}^n\times (0,1)$ we define

$$\begin{aligned} w(x,t)=\int _0^t \int _{\mathbb {R}^n} h(y,s) \Phi (x-y,t-s) \, dyds, \end{aligned}$$

where $\Phi (x-y,t-s)=(4\pi (t-s))^{-\frac{n}{2}} e^{-\frac{|x-y|^2}{4(t-s)}}$. It follows from a direct calculation—cf. [17, pp. 263–264]—that $\partial _t w-\Delta w=h$.

Observe that $|w(x,t)|\le t\Vert h\Vert _0$, and that

$$\begin{aligned} |w(x,t)-w(x^\prime ,t)|&\le \int _0^t \int _{\mathbb {R}^n} |h(x-y,s)-h(x^\prime -y,s)| \Phi (y,t-s) \, dyds \\&\le t \sup _{0<s<1} [h(\cdot ,s)]_\beta |x-x^\prime |^\beta . \end{aligned}$$

Thus,

$$\begin{aligned} \sup _{0<t<1} t^{-1} \Vert w(\cdot ,t)\Vert _{\beta } \le \sup _{0<t<1}\Vert h(\cdot ,t)\Vert _\beta . \end{aligned}$$

Next we use [17, Chapter 4, (2.5)] to estimate

$$\begin{aligned} |\partial _{x_i} w(x,t)| \le \Vert h\Vert _0\int _0^t \int _{\mathbb {R}^n} |\partial _{x_i}\Phi (x-y,t-s)| \, dyds \le C(n) \sqrt{t} \Vert h\Vert _0. \end{aligned}$$

Moreover,

$$\begin{aligned} |\partial _{x_i} w(x,t)-\partial _{x_i} w(x^\prime ,t)|&\le \int _0^t \int _{\mathbb {R}^n} |h(x-y,s)-h(x^\prime -y,s)| |\partial _{y_i}\Phi (y,t-s)| \, dyds \\&\le C(n)\sqrt{t}\sup _{0<s<1} [h(\cdot ,s)]_\beta |x-x^\prime |^\beta . \end{aligned}$$

Thus,

$$\begin{aligned} \sup _{0<t<1} t^{-\frac{1}{2}} \Vert \partial _{x_i} w(\cdot ,t)\Vert _{\beta } \le C(n) \sup _{0<t<1} \Vert h(\cdot ,t)\Vert _{\beta }. \end{aligned}$$

Next we use [17, Chapter 4, (1.9) and (2.5)] to estimate

$$\begin{aligned} |\partial _{x_ix_j}^2 w(x,t)|&\le \int _0^t \int _{\mathbb {R}^n} |h(y,s)-h(x,s)| |\partial _{x_ix_j}^2\Phi (x-y,t-s)| \, dyds \\&\le \sup _{0<s<1} [h(\cdot ,s)]_\beta \int _0^t \int _{\mathbb {R}^n} |x-y|^\beta |\partial _{x_ix_j}^2\Phi (x-y,t-s)| \, dyds \\&\le C^{\prime }(n,\beta )\sup _{0<s<1} [h(\cdot ,s)]_\beta . \end{aligned}$$

This together with the Hölder estimate [17, pp. 276–277] of $\partial _{x_ix_j}^2 w$ gives

$$\begin{aligned} \sup _{0<t<1} \Vert \partial _{x_ix_j}^2 w(\cdot ,t)\Vert _\beta \le C^{\prime }(n,\beta ) \sup _{0<t<1} \Vert h(\cdot ,t)\Vert _\beta . \end{aligned}$$

Hence, we have proved that w is a classical solution to the problem (A.1) satisfying

$$\begin{aligned} \sup _{0<t<1} \sum _{i=0}^2 t^{\frac{i-2}{2}} \Vert \nabla ^i w(\cdot ,t)\Vert _\beta \le C^{\prime \prime }(n,\beta ) \sup _{0<t<1} \Vert h(\cdot ,t)\Vert _\beta . \end{aligned}$$

Moreover, observe that for $(x,t)\in \mathbb {R}^n\times (0,1)$ and $0<\delta <1-t$,

$$\begin{aligned} w(x,t+\delta )-w(x,t)&=\int _0^t \int _{\mathbb {R}^n} (h(y,s+\delta )-h(y,s))\Phi (x-y,t-s) \, dyds \\&\quad +\int _{-\delta }^0 \int _{\mathbb {R}^n} h(y,s+\delta ) \Phi (x-y,t-s) \, dyds. \end{aligned}$$

By the preceding discussions, if $h\in C^0((0,1); C^\beta (\mathbb {R}^n))$, then $w\in C^0((0,1); C^{2,\beta }(\mathbb {R}^n))$. The uniqueness follows from Proposition A.1. $\square$

Appendix B: Notation guide

Section 1.

$\mathbf {H}_\Sigma$:

The mean curvature vector of $\Sigma$

$\mathbf {n}_\Sigma$:

The unit normal of $\Sigma$

$\mathbf {x}$:

The position vector

$\mathbf {x}^\perp$:

The normal component of the position vector

${\mathcal {H}^n}$:

The n-dimensional Hausdorff measure

${\mathcal {ACE}^{k,\alpha }_n(\Gamma )}$:

The space of equivalence classes of $C^{k,\alpha }_*$-asymptotically conical embeddings of $\Gamma$ into $\mathbb {R}^{n+1}$ whose images are self-expanders.

Section 2.1.

$B^n_R(x), B_R(x)$:

The open ball in $\mathbb {R}^n$ of radius R and center x

$\bar{B}^n_R(x), \bar{B}_R(x)$:

The closed ball in $\mathbb {R}^n$ of radius R and center x

$B^n_R, B_R$:

The open ball in $\mathbb {R}^n$ of radius R and center origin

$\bar{B}^n_R, \bar{B}_R$:

The closed ball in $\mathbb {R}^n$ of radius R and center origin

${\mathcal {L}{[}\mathcal {C}}{]}$:

The link of cone $\mathcal {C}$

${\mathcal {C}{[}\sigma {]}}$:

The cone over $\sigma \subset \mathbb {S}^n$

Section 2.2.

$\nabla _\Sigma$:

The covariant derivative on $\Sigma$

$d_\Sigma$:

The geodesic distance on $\Sigma$

$B_R^\Sigma (p)$:

The open geodesic ball in $\Sigma$ of radius R and center $p\in \Sigma$

$\tau ^{\Sigma }_{p,q}$:

The parallel transport along the unique minimizing geodesic in $\Sigma$ from p to q

$\Vert f\Vert _{l;\Omega }, \Vert f \Vert _{l}, \Vert f \Vert _{l,0}$:

The $C^l$ norm for function f on $\Omega$

$C^l(\Omega ), C^{l,0}(\Omega )$:

The space of functions on $\Omega$ with finite $C^{l}$ norm

${[}f{]}_{\beta ;\Omega }, {[}f{]}_{\beta }$:

The Hölder semi-norm with exponent $\beta$ for function f on $\Omega$

${[}T{]}_{\beta ;\Omega }, {[}T{]}_{\beta }$:

The Hölder semi-norm with exponent $\beta$ for tensor field T on $\Omega$

$\Vert f\Vert _{l,\beta ;\Omega }, \Vert f\Vert _{l,\beta }$:

The $C^{l,\beta }$ norm for function f on $\Omega$

$\Vert f\Vert _{\beta ;\Omega },\Vert f\Vert _{\beta }=\Vert f\Vert _{0,\beta }$:

The $C^\beta$ norm for function f on $\Omega$

$C^{l,\beta }(\Omega )$:

The space of functions on $\Omega$ with finite $C^{l,\beta }$ norm

$C^\beta (\Omega )=C^{0,\beta }(\Omega )$:

The space of functions on $\Omega$ with finite $C^\beta$ norm

$\Vert f\Vert _{l;\Omega }^{(d)}, \Vert f \Vert _{l}^{(d)}, \Vert f\Vert _{l,0}^{(d)}$:

The $(1+|\mathbf {x}|)^d$-weighted $C^l$ norm for function f on $\Omega$

$C^{l}_d(\Omega ), C^{l,0}_d(\Omega )$:

The space of functions on $\Omega$ with finite $\Vert \cdot \Vert ^{(d)}_l$ norm

${[}f{]}_{\beta ;\Omega }^{(d)}, {[}f{]}_{\beta }^{(d)}$:

The $(1+|\mathbf {x}|)^d$-weighted Hölder semi-norm with exponent $\beta$ for function f on $\Omega$

${[}T{]}_{\beta ;\Omega }^{(d)}, {[}T{]}_{\beta }^{(d)}$:

The $(1+|\mathbf {x}|)^d$-weighted Hölder semi-norm with exponent $\beta$ for tensor field T on $\Omega$

$\Vert f\Vert _{l,\beta ;\Omega }^{(d)}, \Vert f\Vert _{l,\beta }^{(d)}$:

The $(1+|\mathbf {x}|)^d$-weighted $C^{l,\beta }$ norm for function f on $\Omega$

$\Vert f\Vert _{\beta ;\Omega }^{(d)},\Vert f\Vert _{\beta }^{(d)}=\Vert f\Vert _{0,\beta }^{(d)}$:

The $(1+|\mathbf {x}|)^d$-weighted $C^{\beta }$ norm for function f on $\Omega$

$C^{l,\beta }_d(\Omega )$:

The space of functions on $\Omega$ with finite $\Vert \cdot \Vert _{l,\beta }^{(d)}$ norm

$C^{\beta }_d(\Omega )=C^{0,\beta }_{d}(\Omega )$:

The space of functions on $\Omega$ with finite $\Vert \cdot \Vert _{\beta }^{(d)}$ norm

$X(\Omega ;\mathbb {R}^M)$:

The space of maps from $\Omega$ to $\mathbb {R}^M$ with finite X norm

Section 2.3.

${\mathcal {C}_R}$:

The cone $\mathcal {C}$ outside the closed ball $\bar{B}_R$

${\mathscr {E}_d^{\mathrm {H}}[\varphi ]}$:

The homogeneous extension of degree d of $\varphi$ where $\varphi$ is a map from the link of a cone to $\mathbb {R}^M$

$\mathrm {tr}{[}\mathbf {f}{]}$:

The trace of $\mathbf {f}$ where $\mathbf {f}$ is a homogeneous map from a cone to $\mathbb {R}^M$

$\mathrm {tr}_\infty ^d{[}\mathbf {g}{]}$:

The trace at infinity of $\mathbf {g}$ where $\mathbf {g}$ is an asymptotically homogeneous of degree d map from a cone to $\mathbb {R}^M$

$C^{l,\beta }_{d,\mathrm {H}}(\mathcal {C}_R;\mathbb {R}^M)$:

The subspace of $C^{l,\beta }_d(\mathcal {C}_R;\mathbb {R}^M)$ consisting of elements that are asymptotically homogeneous

$C^{l,\beta }_{d,0}(\mathcal {C}_R;\mathbb {R}^M)$:

The subspace of $C^{l,\beta }_{d,\mathrm {H}}(\mathcal {C}_R;\mathbb {R}^M)$ consisting of elements with trace at infinity equal to zero

Section 2.4.

$\Sigma _{f}$:

The $\mathbf {v}$-graph of function f

$\pi _{\mathbf {v}}$:

The projection map onto a hypersurfaces along transverse section $\mathbf {v}$ on the hypersurface

${\mathscr {E}_{\mathbf {v}}{[}\mathbf {f}{]}}$:

The $\mathbf {v}$-extension of $\mathbf {f}$ where $\mathbf {f}$ is a map from a hypersurface to $\mathbb {R}^M$ and $\mathbf {v}$ is a transverse section on the hypersurface

Section 2.5.

${\mathscr {E}^{\mathrm {H}}_{\mathbf {v},d}{[}\varphi {]}}$:

The $\mathbf {v}$-homogeneous extension of degree d of $\varphi$ where $\varphi$ is a map from the link of a cone to $\mathbb {R}^M$ and $\mathbf {v}$ is a tranverse section on the cone

Section 3.

${\mathcal {C}(\Sigma )}$:

The asymptotic cone of $\Sigma$

${\mathcal {L}(\Sigma )}$:

The link of the asymptotic cone of $\Sigma$

${\mathcal {C}_R(\Sigma )}$:

The asymptotic cone $C(\Sigma )$ outside the closed ball $\bar{B}_R$

${\mathcal {ACH}^{k,\alpha }_n}$:

The space of $C^{k,\alpha }_*$-asymptotically conical $C^{k,\alpha }$-hypersurfaces in $\mathbb {R}^{n+1}$

$\theta _{\mathbf {v};\Sigma ^\prime }$:

The inverse of $\pi _{\mathbf {v}}$ restricted to $\Sigma ^\prime$, a hypersurface outside a compact set

Section 3.1.

$\mathrm {tr}_\infty ^d{[}\mathbf {f}{]}$:

The trace at infinity of $\mathbf {f}$ when $\mathbf {f}$ is an asymptotically conical of degree d map from an asymptotically conical hypersurface into $\mathbb {R}^M$

$C^{l,\beta }_{d,\mathrm {H}}(\Sigma ;\mathbb {R}^M)$:

The subspace of $C^{l,\beta }_d(\Sigma ;\mathbb {R}^M)$ consisting of elements that are asymptotically homogeneous

$C^{l,\beta }_{d,0}(\Sigma ;\mathbb {R}^M)$:

The subspace of $C^{l,\beta }_{d,\mathrm {H}}(\Sigma ;\mathbb {R}^M)$ consisting of elements with trace at infinity equal to zero

Section 3.2.

${\mathcal {ACH}^{k,\alpha }_n(\Gamma )}$:

The space of $C^{k,\alpha }_*$-asymptotically conical embeddings of $\Gamma$ into $\mathbb {R}^{n+1}$

${\mathcal {C}{[}\mathbf {f}}{]}$:

The homogeneous extension of degree one of $\mathrm {tr}_\infty ^1[\mathbf {f}]$

$\mathbf {f}\sim \mathbf {g}$:

The asymptotically conical embeddings $\mathbf {f},\mathbf {g}$ are equivalent, provided $\mathbf {f}^{-1}\circ \mathbf {g}$ is a diffeomorphism that fixes infinity

Section 4.

$L_{\Sigma }$:

The Jacobi operator on $\Sigma$

$L_{\mathbf {v}}$:

The $\mathbf {v}$-Jacobi operator where $\mathbf {v}$ is a transverse section

Section 5.

${\mathscr {L}_\Sigma }$:

Certain Schrödinger operator on $\Sigma$ related to the Jacobi operator

${\mathcal {D}^{l,\beta }(\Sigma )}$:

Certain Banach space of functions on $\Sigma$

$\Vert f\Vert _{l,\beta }^*$:

The norm on $\mathcal {D}^{l,\beta }(\Sigma )$

Section 6.

${\mathcal {K}}$:

The kernel space of the Jacobi operator

${\mathcal {K}_{\mathbf {v}}}$:

The kernel space of the $\mathbf {v}$-Jacobi operator

r:

The distance to the origin restricted to a hypersurface

$\partial _r$:

The gradient of r

$\mathrm {tr}_\infty ^*{[}u{]}$:

The trace at infinity of Jacobi function u

${\mathscr {F}_{\mathbf {w}}{[}u{]}}$:

The leading term in the expansion of Jacobi function u

Section 7.

$\mathbf {H}{[}\mathbf {g}{]}$:

The mean curvature vector of embedding $\mathbf {g}$

$\mathbf {n}{[}\mathbf {g}{]}$:

The unit normal of embedding $\mathbf {g}$

$\mathbf {x}^\perp {[}\mathbf {g}{]}$:

The normal component of embedding $\mathbf {g}$

${\mathcal {B}_R(p;X)}$:

The open ball in Banach space X with radius R and center $p\in X$