Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds

Introduction

Background and notation

A plane curvature flow is a one-parameter family of smooth closed curves

which evolves in the direction of its inner unit normal $N$ with speed $V$:

Here $T=X_s$ is the unit tangent, $s$ is arc length, $V$ may depend on the signed curvature $\kappa$, position parameter $u$, tangential angle $\theta$ and time parameter $t$, and the orientation is chosen so that

where $R_{\pi/2}$ denotes the counterclockwise rotation matrix in $\mathbb{R}^2$ through an angle $\pi/2$. The rotation index of a smooth closed curve $X$ is

Suppose $X$ has strictly positive curvature everywhere. We say that $X$ is convex if $m=1$, and locally convex if $m\ge2$. A locally convex curve with $m\ge2$ is in general immersed and may have self-intersections. Its Euclidean length and signed area are

For an immersed curve, $A$ is the algebraic area and may be positive, zero, or negative. Since $\kappa>0$, the tangential angle $\theta$ is a global parameter on $S_m$ and $\kappa\,ds=d\theta$. We now recall some terminology related to the blow-up phenomenon for the power-type curvature flow. For $\alpha>0$, a power-type curvature flow generally takes the form of

where $\Psi(\theta)>0$ is an anisotropic function and $\lambda(t)$ is a nonlocal term related to some geometric quantities such as length and area. We call the flow anisotropic if $\Psi$ is not a constant and isotropic otherwise. We say the flow (3) develops a singularity at time $T$ ($0<T\le\infty$) if the curvature $\kappa(\cdot,t)$ of the evolving curves $X_t$ satisfies

where $\kappa_{\max}(t):=\max_{u\in S_m}\kappa(u,t)$. For $\alpha>0$ and $T<\infty$, the singularity is called Type-I if

and Type-II if

The classical curve shortening flow (CSF) is given by

Gage–Hamilton [14] proved that every convex embedded closed curve evolving by CSF shrinks to a point in finite time, with a Type-I singularity. Then Grayson [17] extended the round-point theorem to the case of arbitrary embedded closed curves. For immersed curves the singularity theory is much richer. Angenent [6] studied singularity formation for the CSF and showed that, in the locally convex setting, both Type-I and Type-II singularities can occur. Type-I blow-up is related to homothetically shrinking solutions, the Abresch–Langer curves, whereas Type-II rescaling leads to a translating solution, the Grim Reaper (see [1, 6]).

A natural nonlinear extension is the power-type curvature flow

or, in the anisotropic setting,

For the anisotropic flow (8) with embedded convex initial curves, Andrews [3] developed a general theory describing how the exponent $\alpha$ affects the evolution. In particular, only a Type-I singularity occurs when $\alpha> 1/3$. Andrews [4] further studied the instability of the flow (8) for $0 < \alpha< 1/3$ and showed that the isoperimetric ratio may become unbounded, leading to a Type-II singularity. Such behavior may also occur at the critical exponent $\alpha= 1/3$ for suitable anisotropies, whereas the isotropic critical case has only a Type-I singularity. See also Poon–Tsai [32] for the relation between the isoperimetric ratio and the blow-up type. In the isotropic setting $\Psi\equiv1$, Andrews [5] classified the closed homothetically shrinking solutions of (7). Other related curvature flows with forcing terms were studied by Chou–Zhu [9], who considered the anisotropic flow $\beta(\theta)V = g(\theta)\kappa+ F$, where $\beta, g > 0$ are smooth $2\pi$-periodic functions and $F$ is a constant. In the shrinking case, for embedded convex initial curves, they proved finite-time extinction with Type-I blow-up. Moreover, the rescaled curves converge smoothly along a subsequence to a homothetically shrinking solution of the unforced flow $\beta(\theta)V = g(\theta)\kappa$. For immersed locally convex curves moving by (7), Poon–Tsai [31] treated the fast regime $\alpha> 1$ and proved that finite-time blow-up is of Type-I. Lin–Poon–Tsai [22] studied the slow regime $0 < \alpha\le1$ and analyzed both Type-I and Type-II singularities. Moreover, Type-I blow-up is associated with homothetic solutions, while under a special symmetry assumption, a Type-II blow-up develops a translator profile. For the anisotropic flow (8), Poon–Tsai [32] investigated the corresponding degenerate parabolic equation. Under the Type-I assumption, they established subsequential convergence to self-similar profiles, and under the Type-II assumption, they showed that, in the case where the equation defines an embedded convex flow, the blow-up profile near a maximum point is a translating solution of (8). For the anisotropic flow (8) with $\alpha> 1$, Wang–Wang [39] proved that for immersed locally convex closed initial curves, any finite-time singularity is of Type-I. They also established subsequential convergence of the rescaled curvature and further obtained full-time convergence under suitable symmetry assumptions on both the anisotropy and the initial curve.

We next turn to nonlocal curvature flows. Gage [13] introduced the area-preserving curvature flow

for embedded convex curves and established global existence together with smooth convergence to a circle. A power-type generalization

was studied for embedded convex curves by Tsai–Wang [37]. Other important isotropic nonlocal flows include the length-preserving flow due to Ma–Zhu [26] and the gradient flow of the isoperimetric ratio proposed by Jiang–Pan [20]. For embedded convex initial curves, these flows typically possess global solutions and converge smoothly to a circle.

For immersed curves, finite-time singularities can occur under suitable initial conditions. For the immersed area-preserving curvature flow

Escher–Ito [12] proved finite-time singularity formation when $A(0)<0$, or when $m \ge2$ and $L^2(0)<4m\pi A(0)$. Nagasawa–Nakamura [28] studied several nonlocal curvature flows for plane curves with general rotation number, including the area-preserving flow (11). They obtained convergence results for global solutions, together with finite-time blow-up criteria, upper estimates for the blow-up time, and lower estimates for the blow-up rate. For the power-type area-preserving flow (10), Šešum–Tsai–Wang [33] studied immersed locally convex curves and proved finite-time singularity formation under the explicit conditions

They also obtained convergence to an $m$-fold circle for suitable global solutions. Lippoth [23] obtained further blow-up results for general immersed closed curves under (11), including a sufficient condition for finite-time curvature blow-up when the initial signed area is positive and the rotation index is one. He also proved curvature blow-up for initial curves with zero signed area. Furthermore, people are also interested in the type and the profile of singularities. In the isotropic case, Mäder-Baumdicker [27] studied Gage’s area-preserving curvature flow with Neumann free boundary conditions and, under suitable assumptions, proved that any finite-time singularity must be of Type-II. Moreover, for convex initial curves, she employed Hamilton’s rescaling technique to show that the asymptotic profile is a Grim Reaper or a half Grim Reaper. Dittberner [11] investigated isotropic curvature flows with global forcing terms, including Gage’s area-preserving curvature flow, for embedded curves satisfying suitable geometric conditions. She proved global existence for a class of embedded curves by first ruling out Type-I singularities and then excluding Type-II singularities using a distance comparison principle. More recently, Anada–Ishiwata–Ushijima [2] considered the isotropic power-type area-preserving flow (10) for $0<\alpha\le1$ with a special class of highly symmetric Abresch–Langer type immersed initial curves, and proved Type-II blow-up.

A natural problem is what happens if we consider nonlocal curvature flows in the anisotropic setting. Let $\sigma\in C^\infty(\mathbb{R})$ be a positive $2m\pi$-periodic function satisfying

Motivated by Green–Osher [18] and Ševčovič–Yazaki [34], we define the anisotropic curvature and anisotropic length by

The associated Wulff curve is parametrized by

This parametrization defines a closed, convex curve when $m=1$, and a closed locally convex curve when $m\ge2$, which may have self-intersections or be multiply covered. Its Euclidean curvature is $\varphi^{-1}$ and its anisotropic curvature is identically one. We denote its signed area by $|W_\sigma|$. Direct calculation gives

The flow $V=\kappa_\sigma$ is the gradient flow of the anisotropic interfacial energy $\int_X\sigma\,ds$; see Taylor–Cahn [36]. If we consider the gradient flow of the surface energy with the area-preserving constraint, then the following anisotropic version of Gage’s area-preserving curvature flow is obtained:

Wang–Wang [38] proved that for embedded smooth convex closed initial curves, the flow {(xref)}(1.17) exists globally and converges smoothly, up to translation, to the boundary of the homothetic Wulff shape $\sqrt{A(0)/|W_\sigma|}W_\sigma$. Liu–Tsai–Wang [24] considered a power-type extension

and established analogous global existence and smooth convergence results for embedded convex closed initial curves. There are several other related anisotropic nonlocal or constrained curve flows. We refer, for example, to Beneš–Yazaki–Kimura [7], Pan–Yang [30], Gao–Zhang [15], Li–Wang [21], Liu–Yang–Zhang [25], Zhao–Yang–Mao–Fang [41], Sun [35], and Ye–Wang [40]. See also the survey of Giga–Požár [16].

The present paper focuses on generalized anisotropic Gage area-preserving flows with slow curvature speeds, corresponding to the regime $0<\alpha\le1$ in {(xref)}(1.18):

Since

Jensen’s inequality gives

The flow preserves the signed area and decreases the anisotropic length. We first show that every finite-time singularity with $\inf_{t<T}L(t)>0$ is of Type-II, and we determine its local blow-up profile by applying Hamilton’s rescaling method. We then provide sufficient conditions on immersed locally convex initial curves that guarantee finite-time singularity formation. In contrast to Anada–Ishiwata–Ushijima [2], we impose no symmetry condition, and the anisotropy may be general under {(xref)}(1.13).

Main results

Our first two theorems concern the type and local profile of a finite-time singularity under a positive length lower bound.

The length lower bound in Theorem 1.1 cannot be omitted in general, already in the isotropic case $\alpha=1$. In a recent preprint, Cernomazov [8 Main Result A, Remarks 1.7 and 2.1] constructed noncircular locally convex shrinkers of the isotropic flow (1.11). These have zero signed area and shrink by a factor proportional to $\sqrt{T-t}$, so $L(t)\to0$ and the singularity is of Type-I.

Since the signed area is preserved and an immersed closed plane curve satisfies $L^2 \ge4\pi|A|$, the lower bound for $L(t)$ is automatic whenever $A(0)\ne0$.

Our next theorem identifies the curvature profile near a Type-II blow-up point.

The limiting curvature profile in Theorem 1.2 is realized, after the following rescaling, by a translating solution of the corresponding anisotropic flow $V=\kappa^\alpha$. Put $q_j=\kappa_\sigma^\alpha(\theta_j,t_j)$, $Q_j=q_j^{1/\alpha}$ and $p=1+1/\alpha$, and let $\tau=q_j^p(t-t_j)$ denote the rescaled time. Define

Spatial dilation by $Q_j$ divides curvature by $Q_j$ and preserves the tangent direction. Thus (1.24) is precisely the convergence of the anisotropic curvature of $\widetilde{X}_j(\cdot,0)$ to $(\cos\xi)^{1/\alpha}$. Moreover,

On compact subintervals of $(-\pi/2,\pi/2)$ the denominator converges to a positive function. Since $\widetilde{X}_j(0,0)=0$, integration gives $C^1_{\mathrm{loc}}$ convergence to

The positive factor in $\partial_\xi X_\infty$ shows that its unit tangent and normal are $T_\infty=T(\theta_*+\xi)$ and $N_\infty=N(\theta_*+\xi)$. Using $ds_\infty/d\xi=\varphi(\theta_*+\xi)/(\cos\xi)^{1/\alpha}$ and $\partial_\xi T_\infty=N_\infty$, we obtain

Set $e=N(\theta_*)$. Since $e\cdot N_\infty=\cos\xi$, the family

satisfies the normal-velocity law

Indeed, its full velocity in the fixed-angle parametrization is $\kappa_{\sigma,\infty}^{\alpha}N_\infty-(\kappa_{\sigma,\infty}^{\alpha})_\xi T_\infty=\cos\xi N_\infty+\sin\xi T_\infty=e$. Thus it is a translating solution of the anisotropic curvature flow $V=\kappa_\sigma^\alpha$, up to tangential reparametrization. Since $0<\alpha\leq1$, its arc length diverges at both endpoints of the angle interval. When $\alpha=1$ and $\sigma\equiv1$, a rigid motion gives $X_\infty(\xi,\tau)=(\xi,-\log\cos\xi)+\tau(0,1)$, the classical Grim Reaper.

We now give sufficient conditions on the initial curve that guarantee finite-time singularity formation.

Remark 1.4 (Initial curves near the Wulff curve). For every $\sigma$ satisfying (1.13) and every $m\geq2$, the second condition in (1.29) is satisfied by locally convex closed initial curves arbitrarily close to $\partial W_\sigma$ in $C^\infty(\mathbb{S}_m)$. The strict inequality persists under sufficiently small smooth perturbations, so these examples need not be symmetric. The construction is given in Subsection 5.2.

Both conditions in (1.29) imply $A(0)\ne0$. Therefore Theorems 1.3 and 1.1 immediately give the following result.

Corollary 1.5. If $X_0$ satisfies either condition in (1.29), then the solution of (1.19) develops a finite-time Type-II singularity.

The paper is organized as follows. Section 2 gives the evolution equations, the monotonicity of geometric quantities, the gradient estimate and a concentration estimate near points of maximal curvature. Section 3 proves Theorem 1.1 by showing that the Type-I assumption would force the Euclidean length to vanish. Section 4 applies Hamilton’s rescaling method and proves the compactness, nonlocal-error, and Harnack estimates needed to identify the blow-up profile in Theorem 1.2. Finally, Section 5 establishes uniform curvature bounds for global solutions whose anisotropic length has a positive limit, identifies the limiting Wulff curve, and proves the finite-time singularity criteria in Theorem 1.3. It also gives the construction described in Remark 1.4.

Preliminaries

Evolution equations

For a locally convex solution of (19), one can use the tangential angle to parameterize it. Put $v(\theta,t):=\kappa_{\sigma}^{\alpha}(\theta,t)$ and $p=1+1/\alpha\ge2$. A tangential velocity changes only the parametrization, not the geometric image of the curve; see Chou–Zhu [10] and Gage [13]. If $V$ and $G$ are the normal and tangential velocities, then $\partial_t\theta=\partial_sV+\kappa G$. Choosing $G=-v_\theta$ with $V=v-\lambda(t)$ therefore keeps $\theta$ independent of time and gives

At fixed angle the curvature satisfies $\kappa_t=\kappa^2(V_{\theta\theta}+V)$. Since $\varphi$ is independent of time and $v=(\varphi\kappa)^\alpha$, this gives

where

We also have

In particular, the Euclidean length satisfies

This follows either by differentiating (33) and using periodicity, or from the first variation of length. Notice that $\lambda$ is the arc-length average of $v$, so $0<\lambda\le v_{\max}$.

Short-time existence and preservation of local convexity follow by the standard quasilinear parabolic argument used for anisotropic nonlocal curvature flows; see [10, 24]. In particular, the solution can be continued beyond any finite time $T$, provided its curvature remains uniformly bounded on $[0,T)$. Indeed, if $v\le M$, then $\lambda\le M$ and the spatial minimum satisfies

in the lower Dini derivative sense. Thus $v$ stays uniformly positive on every bounded time interval, and the equation remains uniformly parabolic when the curvature is bounded.

Preserved area and non-increasing anisotropic length

Proof. The standard variation formulas give

by the definition of $\lambda$. Also

Since $\int_{X_t}\left(\kappa_\sigma^\alpha-\lambda\right)\,ds=0$,

The integrand is nonnegative because $s\mapsto s^\alpha$ is increasing on $(0,\infty)$, which proves (2.6).

A gradient estimate

The following gradient estimate will be used in both the global argument and the blow-up analysis.

Proof. Fix $t>0$ and suppose the maximum of $\Psi$ on $S_m\times[0,t]$ is attained at $(\theta_*,t_*)$ with $t_*>0$. At this point

If $v_\theta(\theta_*,t_*)=0$, then $\Psi=v^2$ and there is nothing to prove. Otherwise

at $(\theta_*,t_*)$. Differentiating (31) and evaluating at this point gives

where $p\geq2$ has been used and the non-positive terms have been discarded in the last line. Hence

Thus $\Psi\leq C_\varphi v^2$ at every interior spacetime maximum with $v_\theta\ne0$, and (36) follows.

Proof. By Lemma 2.2,

where $C_I:=\max_{\mathbb{S}_m}(v^2+v_\theta^2)(\theta,0)$ depends only on the initial data. Integrating from $\theta$ to $\theta_*$ and choosing $\delta=\varepsilon/\sqrt{C_\varphi}$ gives (2.8).

Type-II singularities

This section proves Theorem 1.1. Assume for contradiction that the singularity is of Type-I. Set

The Type-I assumption and Lemma 2.2 imply

At a spatial maximum of $v$, equation (31) gives, in the upper Dini derivative sense,

Since $T$ is a curvature blow-up time, integration toward $T$ gives

for $t$ sufficiently close to $T$. Therefore

The assumption $\inf_{t<T}L(t)>0$ and (21) give a uniform bound for $\lambda$. From (2.1) and (3.2),

Hence there exists a continuous limiting map $X_T:\mathbb{S}_m\to\mathbb{R}^2$ satisfying

Proof. Suppose not. Then there are $t_j \uparrow T$ such that

By (3.2), after passing to a subsequence,

uniformly on $\mathbb{S}_m$, where $U_\infty\ge0$ is Lipschitz. By (3.4), it is not identically zero, while it has at least one zero. Let $I=(a,b)$ be a connected component of $\{U_\infty>0\}$.

Put $r_j=(T-t_j)^{1/(\alpha+1)}$. Since

the length of $X(\cdot,t_j)$ restricted to any compact subinterval of $I$ tends to zero. Hence $X_T$ is constant on $I$. Write $X_T\equiv x_*$ there. Define

By (3.5), $|Y_j|\le C$ on $I$, and

Since $T_\theta=N$,

Integrating (3.6) over $[a',b']\Subset I$ and using $|Y_j|\le C$, we have

Passing to the limit gives

where $C$ is independent of $a'$ and $b'$. Letting $a'\downarrow a$ and $b'\uparrow b$, we obtain

But $U_\infty$ is Lipschitz and vanishes at an endpoint of $I$, so near that endpoint

Since $1/\alpha\ge1$, the integral in (3.7) must diverge, a contradiction. □

Proof of Theorem 1.1. By Lemma 3.1,

Thus

contradicting $\inf_{t<T}L(t)>0$. Hence Type-I is impossible and the singularity is of Type-II. □

Hamilton rescaling and the local profile

We now assume that a Type-II singularity occurs at $T<\infty$ and that (1.23) holds.

Hamilton’s rescaling method

Following Hamilton’s rescaling method [19], choose $s_j \uparrow T$ such that

and set

Choose $(\theta_j,t_j)\in S_m\times[0,T_j]$ so that

Then

Since $v$ is bounded on compact time intervals in $[0,T)$, we have $t_j\to T$. After taking a subsequence, assume $t_j\uparrow T$ and put

Here $Q_j$ is the spatial dilation factor. Define

Then

On the rescaled interval $[-q_j^pt_j,H_j)$, whose endpoints tend to $-\infty$ and $+\infty$, respectively, the maximality in (4.1) gives

whenever the rescaled flow is defined and $\tau<H_j$. In particular,

Space-time compactness

Proof. Set

which depends only on the initial curve. Since $(v_j)_\theta=v_\theta/q_j$, Lemma 2.2 gives

where $t=t_j+\tau/q_j^p$. The maximality in (45) gives, for $0\leq s\leq t<T_j$,

Taking the power $2/p$ and using $H_j,q_j\to\infty$ proves (50). □

Proof. By (21) and (23), $\eta_j\leq q_j^{-1}(K_\sigma/\ell_0)^\alpha$, where $\ell_0=\inf_{t<T}L(t)>0$. Write (48) as

where

Lemma 4.1 gives $|F_j|+|G_j|\leq C_R$. Fix an arbitrary $\theta_0\in S_m$. For $0<h\leq1$, let $I_h=(\theta_0-h,\theta_0+h)$ and define $A_{j,h}(\tau)=(2h)^{-1}\int_{I_h}v_j(\theta,\tau)\,d\theta$, using periodic extension if necessary. The spatial Lipschitz bound and integration of the equation give, respectively,

where $\Delta=|\tau_2-\tau_1|$. Therefore

For $0<\Delta\leq1$, take $h=\sqrt{\Delta}$. For $\Delta\geq1$, use $|v_j(\theta_0,\tau_2)-v_j(\theta_0,\tau_1)|\leq2\|v_j\|_\infty\leq C_R\sqrt{\Delta}$. This proves (51). □

Disappearance of the nonlocal term

Define the rescaled Euclidean length and the rescaled numerator of the nonlocal term by

Then

Proof. Since $\mathcal{L}_j\ge Q_j\inf_{t<T}L(t)\to\infty$, Jensen’s inequality gives

uniformly on compact rescaled time intervals. Differentiating (52), using (48) and periodic integration by parts, we obtain

Indeed, the powers of $v_j$ cancel in the first differentiation, while in the second they leave the factor $v_j$:

The right-hand sides of (55) and (56) are uniformly bounded on each compact rescaled time interval by Lemma 4.1. The quotient rule therefore gives

uniformly there as well.

A Harnack estimate

Since $v_j>0$, define

A direct calculation using (48) gives

Proof. Let $m_j(\tau)=\min_{\mathbb{S}_m}z_j(\cdot,\tau)$. At a spatial minimum of $z_j$, equation (58) and Lemmas 4.1 and 4.3 give, in the lower Dini derivative sense,

where $p\ge2$ has been used, $\varepsilon_j(R)=C_R(\lVert\eta_j\rVert_\infty^2+\lVert\eta_j'\rVert_\infty)\to0$, and the norms are taken on $[-R,R]$. The function

satisfies $y_j'\le y_j^2-\varepsilon_j(R)$ and tends to $-\infty$ as $\tau\downarrow-R$. Comparison gives $m_j\ge y_j$, proving (4.15). Integrating $\partial_\tau\log v_j=z_j$ and exponentiating gives (4.16), with $c_j(R)=\sqrt{\varepsilon_j(R)}$.

Identification of the profile

Proof of Theorem 1.2. After passing to a subsequence,

Set $u_j(\xi,\tau)=v_j(\theta_j+\xi,\tau)$ and $b_j(\xi)=\alpha\varphi(\theta_j+\xi)^{-1}$. Then $b_j\to b_\infty:=\alpha\varphi(\theta_*+\xi)^{-1}$ smoothly. Lemmas 4.1 and 4.2, followed by Arzelà–Ascoli and a diagonal argument, give

By (49),

Fix $\tau_1<\tau_2$ and $R>\max\{|\tau_1|,|\tau_2|\}+1$. Passing to the limit in (60), with $R$ fixed, yields

Letting $R\to\infty$ gives

In particular, (4.19) implies

Let $I$ be the connected component of $\{u_\infty(\cdot,0)>0\}$ containing $0$. By (4.20) and local uniform convergence, $u_j$ has a uniform positive lower bound on each $K \Subset I \times[\delta,R]$, where $K \Subset I$ and $0<\delta<R$. Standard interior parabolic estimates and Lemma 4.3 yield

The limit satisfies

and is smooth there by interior regularity. By (4.20), we have $w:=(u_\infty)_\tau\geq0$. Differentiating (4.23) gives

This equation is uniformly parabolic on compact subsets of $I\times(0,\infty)$, with locally bounded coefficients. Since $w\geq0$ and $w(0,\tau)=0$ for every $\tau>0$, the strong maximum principle gives $w\equiv0$ on $I\times(0,\infty)$.

It follows that $u_\infty$ is independent of $\tau$ and $(u_\infty)_{\xi\xi}+u_\infty=0$ on $I$. Since $u_\infty(0)=1$ and $(u_\infty)_\xi(0)=0$, we obtain $u_\infty(\xi,\tau)=\cos\xi$. By continuity the identity also holds at $\tau=0$, and maximality of the positive component gives $I=(-\pi/2,\pi/2)$. Hence (4.18) gives

Finally, the gradient estimate gives

Taking $\xi=0$ shows that $v(\theta_*,t_j)/q_j\to1$. Therefore

uniformly on the same interval. Taking the power $1/\alpha$ and using $v=(\varphi\kappa)^\alpha$ proves (1.24) and (1.25), and completes the proof. □

Remark 4.5. Theorems 1.1 and 1.2 give the Type-II conclusion without the symmetry assumption in the isotropic work of Anada–Ishiwata–Ushijima [2], and also allow anisotropy, under the stated length lower bound. For the anisotropic flow of Poon–Tsai [32 Theorem 4.7 and Corollary 4.9], the nonlocal error is absent. Their geometric Type-II result is stated for embedded convex curves, whereas the same profile argument here applies to immersed locally convex curves without symmetry. These conclusions concern selected blow-up subsequences and do not assert uniqueness of the blow-up direction.

Global compactness and finite-time singularity

Global compactness and proof of Theorem 1.3

We now turn to the sufficient conditions for finite-time singularity formation. The following convergence result will be used to prove Theorem 1.3.

Proof. By (5.1) and positivity of $\sigma$,

Thus (1.21) gives a positive time-independent upper bound for $\lambda$. Write $\lambda\le\Lambda$ and set

Lemma 2.1 gives $\mathcal{D}\ge0$ and $\int_0^\infty\mathcal{D}(t)\,dt=L_\sigma(0)-L_\sigma(\infty)<\infty$. For any fixed $d_0>0$, there are $t_j\in[(j-1)d_0,jd_0]$ with $\mathcal{D}(t_j)\to0$. By the first variation of anisotropic length,

Consequently, for $F(t):=\int_{S_m}v\,d\theta$,

for all sufficiently large $j$, where $B$ is independent of $d_0$.

To extend the bound to all times, put $a=\alpha\varphi^{-1}$. By (2.2), periodic integration by parts and $\lambda\ge0$ give

Since $v_s=(v^{1/\alpha}/\varphi)v_\theta$, $d\theta=(v^{1/\alpha}/\varphi)\,ds$ and $p-1=1/\alpha$, we obtain

The one-dimensional Gagliardo–Nirenberg inequality gives

where all norms are taken along $X_t$ and the constant is uniform by (73). Taking the power $2p$ and applying Young’s inequality yields

where $C_\epsilon>0$ is a time-independent constant. Using $\|v\|_{L^p}^p=\int_{S_m}\varphi v\,d\theta\le\varphi_{\max}F$ and taking $\epsilon$ sufficiently small to absorb the derivative term in (75), we obtain

Thus there is a number $\delta_0>0$, independent of $j$, such that a fixed bound for $F(t_j)$ implies a fixed bound for $F$ on $[t_j,t_j+\delta_0]$. Choosing $d_0=\delta_0/2$ makes these intervals overlap, and hence

If $v$ were not uniformly bounded, choose $(\theta_j,s_j)$ such that

Lemma 2.3 gives a fixed-length angle interval on which $v(\cdot,s_j)$ is comparable with this maximum. Thus $F(s_j)\to\infty$, contradicting (77). Hence

and Lemma 2.2 also gives $|v_\theta|\le C$. If $e_j:=v(\theta_j,t_j)\to0$, this Lipschitz bound would imply

since $1/\alpha\ge1$. Here $c,\delta>0$ are independent of $j$. This contradicts $L_\sigma(t_j)\le L_\sigma(0)$. Therefore

The same bounds hold for $\lambda$, since it is the arc-length average of $v$. The equation is now uniformly parabolic. Differentiating $J$ in (32) and integrating by parts, as in (56), gives

This formula, (34) and (73) bound $\lambda'=J'/L-\lambda L'/L$. Standard interior estimates then give time-independent bounds for $v_t$ and all spatial derivatives of $v$. Differentiating (74) therefore gives $|\mathcal{D}'(t)|\le C$. Since $\mathcal{D}$ is nonnegative and integrable, this bound implies $\mathcal{D}(t)\to0$. Moreover,

The last inequality follows from (5.10) and the mean value theorem applied to $s \mapsto s^{1/\alpha}$ on the common positive range of $v$ and $\lambda$. The uniform spatial derivative estimates and compactness therefore give

To identify the limit, use

Thus

and

Finally, the translated curves satisfy

smoothly. Since $\partial_\theta(-\sigma N+\sigma'T)=\varphi T$, the limit is a translation of $C_*^{-1}\partial W_\sigma$. Continuity and translation invariance of signed area, together with area preservation, give $A(0)=|W_\sigma|/C_*^2>0$, proving (5.3).

Proof of Theorem 1.3. Suppose first that $A(0)<0$ and the solution were global. A generalized isoperimetric inequality for immersed closed plane curves (see Osserman [29]) gives

Therefore

so (5.1) holds. Proposition 5.1 would then imply that the preserved signed area is positive, a contradiction.

Now assume

Again $A(0)>0$ and the generalized isoperimetric inequality gives (5.1). If the flow were global, Proposition 5.1 would give

Consequently,

Since $L_\sigma$ is non-increasing,

contradicting the assumption. Hence in either case the maximal existence time is finite. By the continuation criterion, the curvature blows up there.

Initial curves near the Wulff curve

We give the construction asserted in Remark 1.4.

For a locally convex immersion parametrized by its tangential angle, its support function relative to the origin is $h=-\langle X,N\rangle$. Then $h_\theta=\langle X,T\rangle$, so

Conversely, any smooth $2m\pi$-periodic function $h$ with $h+h_{\theta\theta}>0$ determines in this way a closed locally convex immersion of rotation index $m$. Here the support function is understood along the immersion, not as the support function of a convex body.

Put $B=|W_\sigma|$ and define

Then $\int_{S_m}\varphi f\,\mathrm{d}\theta=0$. For sufficiently small $|\varepsilon|$, the function $h_\varepsilon=\sigma+\varepsilon f$ satisfies $h_\varepsilon+h_{\varepsilon,\theta\theta}>0$, and hence defines a locally convex closed curve $X_\varepsilon=-h_\varepsilon N+(h_\varepsilon)_\theta T$. Periodic integration by parts gives

Since

we obtain

for all sufficiently small nonzero $\varepsilon$. Thus, for every $0<\alpha\leq1$, Corollary 1.5 gives a finite-time Type-II singularity for each such initial curve. Moreover, $X_\varepsilon\to-\sigma N+\sigma_\theta T$ in $C^\infty(S_m)$ as $\varepsilon\to0$. The strict inequality persists under sufficiently small smooth perturbations of each $X_\varepsilon$, so the examples need not be confined to a symmetric class.

Acknowledgements

This work is partially supported by the NSF of China 12471193, 12271092, 11871148, 11101078, the 2023 Qinglan Project of Jiangsu Province, and the Jiangsu Provincial Scientific Research Center of Applied Mathematics under Grant No. BK20233002.

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflict of interest statement

The authors state no conflict of interest.

References