Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II

Abstract

We continue the study, initiated by the first two authors in [15], of Type-II curvature blow-up in mean curvature flow of complete noncompact hypersurfaces embedded in Euclidean space. In particular, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics near the “vanishing” time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the rate ${(T-t)}^{-1}$. (2) In a neighbourhood of the tip, the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infinity, the hypersurface approaches a collapsing cylinder at an exponential rate.

Introduction

This paper continues the investigation by the first two authors [15] concerning the Type-II curvature blow-up in mean curvature flow (MCF) of noncompact hypersurfaces embedded in Euclidean space.

Given a one-parameter family of embeddings (or more generally, immersions) $\phi (t):M^n\to {\mathbb{R}}^{n+1}$, $t_0<t<t_1$, of n-dimensional hypersurfaces in the Euclidean space, MCF is defined by the following evolution equation

$${\partial }_t\phi (p,t)=\overrightarrow{H},p\in M^n,t_0⩽t<t_1,$$

which geometrically deforms the hypersurface in the direction of its mean curvature vector $\overrightarrow{H}$.

In local coordinates, the MCF equation (1.1) is a (weakly) parabolic PDE system whose short-time existence and uniqueness is well-known. Although the flow has smoothing property in short time, it can develop singularities over larger time scales for many initial data. For example, under MCF and in finite time, any closed convex hypersurface develops a “spherical singularity” [13], whereas hypersurfaces close to a round cylinder develop a “cylindrical singularity” [10].

Let $M_t:=\phi (t)(M^n)$ be the hypersurface under MCF at time t and $h(p,t)$ the second fundamental form of $M_t$ at p. Suppose MCF of $M_t$ becomes singular at time $t=T<\infty$. Then this finite-time singularity is called Type-I if

$$\underset{p\in M_t}{\sup }|h(p,t)|{(T-t)}^{1/2}⩽C$$

for some finite constant C, and it is called Type-II (more precisely, Type-IIa¹) if ${\sup }_{M_t}|h(\cdot ,t)|$ blows up at a faster rate.

Examples of Type-I MCF solutions are plentiful in all dimensions. For example, MCF of any closed embedded curve in the plane always becomes convex [11] and then forms a Type-I round singularity [9]. In dimension two or higher, typical Type-I examples include the round sphere, the round cylinder, and hypersurfaces in suitable open sets around them [13], [10]. In contrast, MCFs which develop Type-II singularities are more difficult to specify and are typically expected to appear if the behaviour of the flow undergoes a “phase change”. To explain what we mean by such a phase change, we consider the following scenarios.

Consider a one-parameter family of rotationally symmetric n-spheres ($n⩾2$) embedded in ${\mathbb{R}}^{n+1}$ with the parameter controlling the extent to which the equator is tightly cinched. Depending on the amount of cinching, MCF starting from a 2-sphere in this family has the following behaviours: (i) For very loose cinching, the flow converges to the shrinking round sphere with its usual (global) Type I singularity [13]. (ii) For very tight cinching, the equator shrinks more rapidly than the two “dumbbell” hemispheres, and forms a (local) Type-I “neckpinch” modelled locally by a cylinder [14]. Scenarios (i) and (ii) represent different behaviours of MCF. As the cinching parameter varies from “very loose” to “very tight”, we expect: (iii) at some “threshold” parameter in between, MCF forms a finite-time singularity that is not Type-I, and hence Type-II. The existence of scenario (iii) is justified by Angenent, Altschuler and Giga [1]. The quantitative precise asymptotics for such Type-II solutions have been obtained by Angenent and Velázquez in [3].

For each integer $m⩾3$, Angenent and Velázquez [3] construct a (mean convex) rotationally symmetric MCF on an n-sphere ($n⩾2$) (centred at the origin) shrinking to a point (the origin) in a “non-convex” fashion in finite time T. If m is even, the solution has reflexive symmetry across the equator and corresponds to the aforementioned scenario (iii); if m is odd, the solution looks like an asymmetric “dumbbell” and we refer the reader to [3].

The geometric-analytic features of an Angenent-Velázquez solution can be summarized as follows: (1) At each pole (the “tip”) of the sphere (an umbilical point), the curvature blows up at the Type-II rate ${(T-t)}^{-(1-1/m)}$ and the singularity model there is the bowl soliton, which is the unique (up to rigid motion) translating soliton that is rotationally symmetric and strictly convex [12]. (2) Near the equator (the “neck”), the curvature blows up at the Type-I rate and the singularity model there is the shrinking soliton. (3) Between each pole and the equator, the solution is approximately given by rotating the profile (with the x-axis being the axis of rotation) $u^2+K{x}^m=2(n-1)(T-t)$, $t\in [t_0,T)$, for some positive constant K. These examples are all believed to be “rare”, as is reflected by the fact that their Type-II curvature blow-up rates are discrete and quantized. Indeed, by the fundamental work of Colding and Minicozzi [8], we know these solutions are non-generic. We note that, integrating ${(T-t)}^{-(1-1/m)}$ in t, the tip moves by a finite distance over the time interval $[t_0,T)$.

Having established the existence of compact MCF solutions with Type-II curvature blow-up, it is natural to seek noncompact counterparts, as first realized in [15] by the first two authors of this paper. More precisely, for each real number $\gamma >1/2$, we have constructed MCF of noncompact rotationally symmetric embedded hypersurfaces that are complete convex graphs over a shrinking ball and asymptotically approach a shrinking cylinder near spatial infinity. Such a mean curvature flow solution exhibits the following behaviour near the “vanishing” time T: (1) The highest curvature, concentrated at the tip of the hypersurface (an umbilical point), blows up at the rate ${(T-t)}^{-(\gamma +1/2)}$ where $\gamma >1/2$, and the singularity model there is the bowl soliton. (2) Near spatial infinity, the hypersurface approaches a collapsing cylinder at a power decay rate dependent on the parameter γ. (3) Between the tip and the cylindrical end, the solution is approximately given by rotating the profile (with the x-axis being the axis of rotation) $u^2+K{x}^{\frac{1}{1/2-\gamma }}=2(n-1)(T-t)$, $t\in [t_0,T)$, for some positive constant² K.

The Isenberg-Wu solutions and the Angenent-Velázquez solutions share similar geometric features—in particular, in both cases, the solutions join a translating soliton to a shrinking soliton. Yet they are different in terms of the topology of the hypersurfaces and the geometric-analytic features. In particular, the noncompact Isenberg-Wu solutions seem to be much more “abundant” than the compact Angenent-Velázquez ones, as is reflected by the fact that their Type-II curvature blow-up rates form a continuum $(1,\infty )$ and we have an open set of solutions for each $\gamma >1/2$. We note that, integrating ${(T-t)}^{-(\gamma +1/2)}$ in t and because $\gamma >1/2$, the tip moves by an infinite distance over the finite time interval $[t_0,T)$, so the MCF solution disappears at spatial infinity at T, exactly when the asymptotic cylinder collapses to a line. The general behaviour, but not the precise asymptotics of such solutions is studied in [17].

An inspection of the Angenent-Velázquez solutions and the Isenberg-Wu solutions immediately raises the following question: does there exist a Type-II MCF solution with curvature blow-up rate ${(T-t)}^{-1}$? The existence is suggested by taking the appropriate limit of the parameter in either construction:

$$\underset{m\to \infty }{\lim }{(T-t)}^{-1+\frac{1}{m}}=\underset{\gamma \to \frac{1}{2}+}{\lim }{(T-t)}^{-(\gamma +1/2)}={(T-t)}^{-1}.$$

Further motivation comes from the differences which have been observed between the Type-II solutions in Ricci flow on compact manifolds and those seen on noncompact manifolds. On compact manifolds Σ, all the examples that have been found [5] have “quantized” blowup rates of $\underset{x\in \mathrm{\Sigma }}{\sup }|\mathrm{Rm}(x,t)|\sim {(T-t)}^{\frac{2}{k}-2}$ for integers $k⩾3$ (here T is the time of the first singularity). By contrast, for noncompact manifolds Σ, the known examples [18] have a continuous spectrum of blowup rates: $\underset{x\in \mathrm{\Sigma }}{\sup }|\mathrm{Rm}(x,t)|\sim {(T-t)}^{-\lambda -1}$ for all $\lambda ⩾1$. The borderline Type-II rate ${(T-t)}^{-2}$ (letting $\lambda =1$ or $k\to \infty$) in Ricci flow can be realized on a noncompact manifold. Correspondingly, the borderline Type-II rate ${(T-t)}^{-1}$ in MCF can be expected on a noncompact hypersurface. In this paper, we confirm this expectation.

Following the set up in [15], in this paper we consider mean curvature flow of rotationally symmetric hypersurfaces embedded in Euclidean space. For any point $(x_0,x_1,\dots ,x_n)\in {\mathbb{R}}^{n+1}$ for $n⩾2$, we write

$$x=x_0,r=\sqrt{x_1^2+\cdots +x_n^2}.$$

A noncompact hypersurface Γ is said to be rotationally symmetric if

$$\mathrm{\Gamma }=\{(x_0,x_1,\dots ,x_n):r=u(x),a⩽x<\infty \}.$$

The rotational symmetry is preserved along MCF, for example, by the obvious curvature bound in our consideration and the standard uniqueness result.

We assume that u is strictly concave so that the hypersurface Γ is convex and that u is strictly increasing with $u(a)=0$ and with $\underset{x\nearrow \infty }{\lim }u(x)=r_0$, where $r_0$ is the radius of the cylinder. The function u is assumed to be smooth, except at $x=a$. Note that this particular non-smoothness of u is a consequence of the choice of the (cylindrical-type) coordinates; in fact, as seen below, if the time-dependent flow function $u(x,t)$ is inverted in a particular way, this irregularity is removed. We label the point where $u=0$ the tip of the surface.

We focus our attention on the class of complete hypersurfaces that are rotationally symmetric, (strictly) convex,³ smooth graphs over a ball and asymptotic to a cylinder. One readily verifies that embeddings with these properties are preserved by MCF (see for example [17]). Representing the evolving hypersurface ${\mathrm{\Gamma }}_t$ by the profile of rotation, i.e., the graph of $r=u(x,t)$, then the function u satisfies the PDE,

$$u_t=\frac{u_{xx}}{1+u_x^2}-\frac{n-1}{u}.$$

We introduce the following scaled time and space parameters, and the scaled profile function:

$$\tau =-\log (T-t),y=x+a\log (T-t),\varphi (y,\tau )=u(x,t){(T-t)}^{-1/2},$$

where $a>0$ is to be chosen later.

Under the rescaled parameters, (1.2) for $u(x,t)$ is transformed to the following PDE for $\varphi (y,\tau )$:

$${{\partial }_{\tau }|}_y\varphi =\frac{e^{-\tau }{\varphi }_{yy}}{1+e^{-\tau }{\varphi }_y^2}+a{\varphi }_y+\frac{\varphi }{2}-\frac{(n-1)}{\varphi },$$

where ${{\partial }_{\tau }|}_y$ means taking the partial derivative for the variable τ with respect to the coordinates $(y,\tau )$; in other words, with y fixed. This notation appears repeatedly throughout this paper. We readily note that equation (1.3) admits the constant solution $\varphi \equiv \sqrt{2(n-1)}$, which corresponds to the collapsing cylinder (a shrinking soliton).

Because our hypersurface is assumed to be a complete convex graph over a ball, it is useful to invert the coordinates and work with

$$y(\varphi ,\tau )=y(\varphi (y,\tau ),\tau ).$$

This inversion can be done because the hypersurface under consideration is a convex graph over a ball. In terms of $y(\varphi ,\tau )$, the equation corresponding to mean curvature flow (equivalent to equation (1.3) and also (1.2)) is

$${{\partial }_{\tau }|}_{\varphi }y=\frac{y_{\varphi \varphi }}{1+e^{\tau }{y}_{\varphi }^2}+(\frac{(n-1)}{\varphi }-\frac{\varphi }{2})y_{\varphi }-a.$$

Our main result is the following.

Theorem 1.1

For any choice of an integer $n⩾2$ and for any real number $a>0$, there exists a family $\mathcal{G}$ of n-dimensional, smooth, complete noncompact, rotationally symmetric, strictly convex hypersurfaces in ${\mathbb{R}}^{n+1}$ such that the MCF evolution ${\mathit{\Gamma }}_t$ starting at each hypersurface $\mathit{\Gamma }\in \mathcal{G}$ is trapped in a shrinking cylinder, escapes at spatial infinity while the cylinder becomes singular at $T<\infty$, and has the following precise asymptotic properties near the vanishing time T of ${\mathit{\Gamma }}_t$:

• The highest curvature occurs at the tip of the hypersurface ${\mathit{\Gamma }}_t$, and it blows up at the precise Type-II rate

  $$\underset{p\in M_t}{\sup }|h(p,t)|\sim {(T-t)}^{-1}\mathit{\text{as}}t\nearrow T.$$

• Near the tip, the Type-II blow-up of ${\mathit{\Gamma }}_t$ converges to a translating soliton which is a higher-dimensional analogue of the “Grim Reaper”

  $$y(e^{-\tau /2}z,\tau )=y(0,\tau )+e^{-\tau }(\frac{\tilde{P}\left(az\right)}{a}+o(1))\mathit{\text{as}}\tau \nearrow \infty$$

  uniformly on compact z intervals, where $z=\varphi e^{\tau /2}$ and $\tilde{P}$ is defined in equation (2.7).

• Away from the tip and near spatial infinity, the Type-I blow-up of ${\mathit{\Gamma }}_t$ approaches the cylinder at the rate

  $$2(n-1)-{\varphi }^2\sim e^{-y/a}\mathit{\text{as}}y\nearrow \infty .$$

In particular, the solution⁴ constructed has the asymptotics predicted by the formal solution described in Section 2.

Comparing with [15], we see that the definitions of τ and ϕ remain the same but that of y has changed from $y=x{(T-t)}^{\gamma -1/2}$ in [15] to $y=x+a\log (T-t)$ in the present paper. Indeed, to capture the borderline case $\gamma =1/2$, taking the limit $\gamma \to \frac{1}{2}$ in [15] is insufficient. The new scaling of y, on the other hand, is natural because in [15] the asymptotic cylindricality is measured precisely by $2(n-1)-{\varphi }^2\sim y^{{(1/2-\gamma )}^{-1}}$ and if we let $\gamma \to 1/2$, then we expect $2(n-1)-{\varphi }^2$ to decay faster than any arbitrarily large power of y; i.e., we have exponential decay in y, as is captured by the asymptotic property (3) of Theorem 1.1. The new scaling of y implies changes in the rescaled PDEs for MCF; e.g., equations (1.3) and (1.4), cf. the same-numbered equations in [15]. In particular, we note that the change occurs in the first-order term in equation (1.3), or equivalently in the zeroth-order term in equation (1.3). This suggests that the method of construction in [15] is still applicable.

The proof of this theorem is based on matched asymptotic analysis and barrier arguments for nonlinear PDE. While the analysis is intricate, this method is powerful and has been successfully applied in a number of studies of Type-I and Type-II singularities which develop both in Ricci flow [6], [5], [18] and in MCF [3], [15]. The proof proceeds in the following steps: (1) By considering rotationally symmetric hypersurfaces, we reduce the MCF equation to a quasilinear parabolic PDE for a scalar function. (2) Applying matched asymptotic analysis, we formally construct approximate solutions to the rescaled versions of this PDE. (3) For each such approximate solution, we construct subsolutions and supersolutions which, if carefully patched, form barriers for the rescaled PDE. These barriers carry information of the approximate solution for times very close to the vanishing time T. (4) Once we have shown (using a comparison principle) that any solution starting from initial data between the barriers does stay between them up to time T, and once we have determined that such initial data sets do exist, we can conclude that there are MCF solutions whose behaviours are predicted by the barriers. Near spatial infinity, the barriers give us precise measure of the asymptotic cylindricality of a solution. At the tip, the barriers have Type-II speeds (cf. Section 2), which implies that on average any MCF solution in between is also Type-II. However, this alone does not imply the stronger convergence result of Type-II blow-up. To prove the strong convergence result stated in property (2) of the theorem, we rely on Lemma 5.5.

This paper is organized as follows. Section 2 describes the construction of formal solutions using the method of formal matched asymptotics. In Section 3, we use these formal solutions to construct the corresponding supersolutions and subsolutions to the rescaled PDE. The supersolutions and subsolutions are ordered and patched to create the barriers to the rescaled PDE in Section 4; a comparison principle for the subsolutions and supersolutions is also proved there. In Section 5, we use these results to complete the proof of our main theorem.

We thank Dan Knopf for helpful discussion on this project and the anonymous referee for valuable comments on the manuscript. J. Isenberg is partially supported by the grant PHY-1707427 of the National Science Foundation; H. Wu thanks the support by Discovery Early Career Researcher Award DE180101348 of the Australian Research Council; Z. Zhang thanks the support by Future Fellowship FT150100341 of the Australian Research Council.