# Deforming Locally Convex Curves into Curves of Constant $k$-order Width

Laiyuan Gao Horst Martini Deyan Zhang The corresponding author.

Abstract A nonlocal curvature flow is introduced to evolve locally convex
curves in the plane. It is proved that this flow with any initial locally
convex curve has a global solution, keeping the local convexity and the
elastic energy of the evolving curve, and that, as the time goes to infinity,
the curve converges to a smooth, locally convex curve of constant $k$-order
width. In particular, the limiting curve is a multiple circle if and only if
the initial locally convex curve is $k$-symmetric.

Keywords curvature flow, locally convex curve, curves of constant $k$-order
width, Blaschke-Lebesgue problem.

Mathematics Subject Classification (2020)  52A10, 53A04, 53E10, 35K15

##  1 Introduction

Let $X_{0}:S^{1}\rightarrow\mathbb{E}^{2}$ be a $C^{2}$, immersed and closed
curve in the Euclidean plane. If its relative curvature $\kappa$ is positive
everywhere, then $X_{0}$ is called a _locally convex curve_. If $X_{0}$ is
also embedded, then it is called a _convex curve_.

In this paper a new curvature flow is established to evolve locally convex
curves into curves of constant $k$-order width. This work is motivated by the
following series of studies. Let
$X:S^{1}\times[0,\omega)\rightarrow\mathbb{E}^{2}$ be a family of smooth and
locally convex curves in the plane, with $s$ and $\theta$ denoting the arc
length parameter and the tangent angle, respectively. Since
$\frac{d\theta}{ds}$ equals the curvature $\kappa(s)>0$ for all $s$, the angle
$\theta$ can be used as a parameter. For every $\theta$, $p(\theta,t)=-\langle
X(\theta,t),N(\theta,t)\rangle$ is called the value of the _support function_
, where $N(\theta,t)$ is the unit normal. Gao and Pan studied in [7] a
curvature flow for convex curves given by

$\left\\{\begin{aligned} &\frac{\partial X}{\partial
t}(\theta,t)=\left(w(\theta,t)-\eta(\theta,t)\right)N(\theta,t),\\\
&X(\theta,0)=X_{0}(\theta),\quad(\theta,t)\in[0,2\pi]\times[0,T_{\max}),\end{aligned}\right.$
(1.1)

where $w(\theta,t)=p(\theta,t)+p(\theta+\pi,t)$ is the width function;
$\eta(\theta,t)=\rho(\theta,t)+\rho(\theta+\pi,t)$ and
$\rho(\theta,t)=\frac{1}{\kappa(\theta,t)}$ is the radius of curvature. They
proved that this flow drives the evolving curve to a limiting convex curve of
constant width, if the initial curve satisfies a $1/3$ curvature pinching
condition. Later, this result was generalized by Gao and Zhang [8] for the
evolution of convex hypersurfaces in higher dimensional Euclidean space.
Another generalized model was presented by Fang in the paper [6]. He replaced
$w$ and $\eta$ in the equation (1.1) by the _$k$ -order width function_
$w_{k}(\theta)=p(\theta)+p\left(\theta+\frac{2\pi}{k}\right)+\cdots+p\left(\theta+\frac{2(k-1)\pi}{k}\right)$
and
$\eta_{k}=\rho(\theta)+\rho\left(\theta+\frac{2\pi}{k}\right)+\cdots+\rho\left(\theta+\frac{2(k-1)\pi}{k}\right)$,
respectively, where $k\geq 2$ is a positive integer. He proved that under a
$\frac{2k-1}{k-1}$ curvature pinching condition the curvature flow deforms an
initial convex curve into a limiting curve of constant $k$-order width.

To guarantee the global existence for the above curvature flows, some
curvature pinching condition of the initial curve or hypersurface is needed.
So a natural question is whether one can construct a proper curvature flow
which evolves every initial curve globally and drives the evolving curve into
the limiting curve. To settle this problem, we consider in this paper a new
curvature flow of locally convex curves. Let $X_{0}$ be a smooth, closed and
locally convex planar curve parameterized by the tangent angle $\theta$.
Denote by $m$ the _winding number_ of $X_{0}$. It equals the total curvature
divided by $2\pi$, i.e., $m=\frac{1}{2\pi}\int_{X_{0}}\kappa(s)ds$. For the
sake of brevity, we write the _elastic energy_ of the curve (see [10] and
[15]) as the integral

$\displaystyle E(X_{0}):=\int_{0}^{L_{0}}(\kappa_{0}(s))^{2}ds.$

Now we consider a curvature evolution problem for locally convex curves,
namely

$\left\\{\begin{aligned} \frac{\partial X}{\partial
t}(\theta,t)=\left(2w_{k}(\theta,t)-\rho_{k}(\theta,t)+f(t)\right)N(\theta,t),&\\\
X(\theta,0)=X_{0}(\theta),\quad(\theta,t)\in[0,2m\pi]\times[0,T_{\max}),\end{aligned}\right.$
(1.2)

where

$\displaystyle
w_{k}(\theta)=\sum_{i=0}^{k-1}p\left(\theta+\frac{2im\pi}{k}\right)~{}\text{and}~{}\rho_{k}(\theta)=\sum_{i=0}^{k-1}\rho\left(\theta+\frac{2im\pi}{k}\right),$
(1.3)

and the nonlocal term is defined by

$\displaystyle
f(t)=\frac{\int_{0}^{2m\pi}\kappa^{2}\frac{\partial^{2}\rho_{k}}{\partial\theta^{2}}\text{d}\theta-\int_{0}^{2m\pi}\kappa^{2}\rho_{k}\text{d}\theta}{\int_{0}^{2m\pi}\kappa^{2}\text{d}\theta}.$
(1.4)

Our main theorem is the following statement.

######  Theorem 1.1.

Let $X_{0}$ be a smooth, closed and locally convex planar curve. The flow
(1.2) has a global solution and keeps both the local convexity and the elastic
energy of the evolving curve. As time goes to infinity, the curve $X(\cdot,t)$
converges smoothly to a locally convex curve of constant $k$-order width. In
particular, the limiting curve is a multiple circle if and only if the initial
curve is $k$-symmetric.

Since some locally convex curves appear as self-similar solutions [1, 11] to
the classical Curve Shortening Flow, it is quite natural to consider curvature
flows for these curves. During the last years, Xiaoliu Wang and his
collaborators did some important research on this subject, see [16, 17, 18].
For more theories and applications of curvature flows of curves, one should
also consult the monograph [5] and suitable references therein.

######  Remark 1.2.

Comparing with models in the papers by Gao-Pan [7], Gao-Zhang [8] and Fang
[6], a complicated nonlocal term $f(t)$ is used in the flow (1.2) with the aim
to preserve the elastic energy of $X(\cdot,t)$. This property guarantees the
global existence of the flow. This term is motivated by the first author’s
recent work [10], where he introduces a new curvature flow to answer Yau’s
problem of evolving one curve to another in the case of locally convex curves.

######  Remark 1.3.

The original goal of this paper was to understand convex domains of (k-order)
constant width via curvature flows. In fact, convex curves (or convex domains)
of constant width and higher dimensional analogues are of special interest in
geometry. As far as we know, the famous related Blaschke-Lebesgue problem [2,
3, 4] for dimension $n\geq 3$ is still open. One may consult the monograph
[13] for more results on related topics.

This paper is organized as follows. In Section 2, short-time existence of the
flow (1.2) is proved. In Section 3, global existence is obtained. And in
Section 4, we prove convergence and the main theorem.

##  2 Short-time existence

Suppose $X:S^{1}\times[0,T)\rightarrow\mathbb{E}^{2}$ is a family of smooth,
closed and locally convex curves in the plane evolving according to the flow
(1.2). Usually, the tangent angle $\theta=\theta(s,t)$ varies as time goes. As
experts did in previous studies (see Proposition 1.1 in the paper [5]), we
consider the next flow instead of (1.2) such that $\theta$ is a variable
independent of time $t$:

$\left\\{\begin{aligned} &\frac{\partial\tilde{X}}{\partial
t}=\alpha(\theta,t)T(\theta,t)+\left(2\omega_{k}(\theta,t)-\rho_{k}(\theta,t)+f(t)\right)N(\theta,t),\\\
&\tilde{X}(\theta,0)=X_{0}(\theta),\quad(\theta,t)\in[0,2m\pi]\times[0,T_{\max}),\end{aligned}\right.$
(2.1)

where $\alpha$ is given by

$\displaystyle\alpha=-2\frac{\partial
w_{k}}{\partial\theta}+\frac{\partial\rho_{k}}{\partial\theta}.$

It follows from Proposition 1.1 in [5] that the solutions to (2.1) and (1.2)
are the same except altering the parametrization. So the short-time existence
of the flow (1.2) is equivalent to that of (2.1).

Both the equations (1.2) and (2.1) are fully non-linear parabolic equation
systems. The main idea of the proof for short-time existence is to reduce
these complicated equations to a semi-linear system of the evolution equation
of $p$ and $\rho_{k}$.

Since the Frenet frame can be expressed as

$\displaystyle T=(\cos\theta,\sin\theta),\quad N=(-\sin\theta,\cos\theta),$

one gets the _Frenet formulae_

$\displaystyle\frac{\partial T}{\partial\theta}=N,\quad\frac{\partial
N}{\partial\theta}=-T.$

Set $\beta(\theta,t)=2w_{k}(\theta,t)-\rho_{k}(\theta,t)+f(t)$. Applying the
equations (1.14)-(1.17) in the book [5], one obtains from (2.1),

$\displaystyle\frac{\partial T}{\partial
t}=\left(\alpha\kappa+\frac{\partial\beta}{\partial
s}\right)N=\left(\alpha+\frac{\partial\beta}{\partial\theta}\right)\kappa N,$
(2.2) $\displaystyle\frac{\partial N}{\partial
t}=-\left(\alpha\kappa+\frac{\partial\beta}{\partial
s}\right)T=-\left(\alpha+\frac{\partial\beta}{\partial\theta}\right)\kappa T,$
(2.3) $\displaystyle\frac{\partial\theta}{\partial
t}=\alpha\kappa+\frac{\partial\beta}{\partial
s}=\left(\alpha+\frac{\partial\beta}{\partial\theta}\right)\kappa,$ (2.4)
$\displaystyle\frac{\partial\kappa}{\partial
t}=\kappa^{2}\left(\frac{\partial^{2}\beta}{\partial\theta^{2}}+\beta\right).$
(2.5)

By the choice of $\alpha$, both the Frenet frame $\\{T,N\\}$ and the tangent
angle $\theta$ are independent of the time:

$\displaystyle\frac{\partial T}{\partial t}\equiv 0,~{}~{}\frac{\partial
N}{\partial t}\equiv 0,~{}~{}\frac{\partial\theta}{\partial t}\equiv 0.$ (2.6)

So the support function satisfies

$\displaystyle\frac{\partial p}{\partial t}=-\frac{\partial}{\partial
t}\left\langle X,N\right\rangle=-(2w_{k}-\rho_{k}+f(t))=\rho_{k}-2w_{k}-f(t).$

Since

$\displaystyle\frac{\partial p}{\partial\theta}=-\left\langle\frac{\partial
X}{\partial\theta},N\right\rangle-\left\langle X,\frac{\partial
N}{\partial\theta}\right\rangle=\langle X,T\rangle,$

we have

$\displaystyle\frac{\partial^{2}p}{\partial\theta^{2}}=\left\langle\frac{\partial
X}{\partial s}\frac{\partial s}{\partial\theta},T\right\rangle=\rho-p.$

So one obtains

$\rho=\frac{\partial^{2}p}{\partial\theta^{2}}+p$ (2.7)

and

$\rho_{k}=\frac{\partial^{2}w_{k}}{\partial\theta^{2}}+w_{k}.$ (2.8)

Thus, the radius of curvature satisfies

$\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial t}(p+p_{\theta\theta})$ (2.9)
$\displaystyle=$ $\displaystyle\frac{\partial p}{\partial
t}+\frac{\partial^{2}}{\partial\theta^{2}}\left(\frac{\partial p}{\partial
t}\right)$ $\displaystyle=$
$\displaystyle\rho_{k}-2w_{k}-f(t)+\frac{\partial^{2}}{\partial\theta^{2}}\left(\rho_{k}-2w_{k}-f(t)\right)$
$\displaystyle=$
$\displaystyle\frac{\partial^{2}\rho_{k}}{\partial\theta^{2}}-\rho_{k}-f(t),$

and one also has the evolution equation of the $k$-width function:

$\frac{\partial w_{k}}{\partial
t}=-k(2w_{k}-\rho_{k}+f(t))=k\left(\frac{\partial^{2}w_{k}}{\partial\theta^{2}}-w_{k}-f(t)\right).$
(2.10)

Combining (2.8) and (2.10), one immediately obtains the evolution equation of
$\rho_{k}$:

$\displaystyle\frac{\partial\rho_{k}}{\partial t}(\theta,t)$ $\displaystyle=$
$\displaystyle
k\left(\frac{\partial^{2}\rho_{k}}{\partial\theta^{2}}(\theta,t)-\rho_{k}(\theta,t)-f(t)\right).$
(2.11)

In the evolution equation of $\rho_{k}$, the term $f(t)$ contains the function
$\rho$. One could not solve the the evolution equation of $\rho_{k}$ directly.
In order to get the _short-time existence_ of the flow, one needs to consider
the above equations as a system.

######  Lemma 2.1.

The nonlinear problem (2.1) is equivalent to the following system on the
domain $[0,2m\pi]\times[0,T_{\max})$,

$\left\\{\begin{aligned} &\frac{\partial p}{\partial
t}(\theta,t)=\rho_{k}(\theta,t)-2w_{k}(\theta,t)-f(t),\\\
&\frac{\partial\rho_{k}}{\partial
t}(\theta,t)=k\left(\frac{\partial^{2}\rho_{k}}{\partial\theta^{2}}(\theta,t)-\rho_{k}(\theta,t)-f(t)\right),\\\
&\frac{\partial w_{k}}{\partial
t}(\theta,t)=k\left(\frac{\partial^{2}w_{k}}{\partial\theta^{2}}(\theta,t)-w_{k}(\theta,t)-f(t)\right),\\\
&\frac{\partial\rho}{\partial
t}(\theta,t)=\frac{\partial^{2}\rho_{k}}{\partial\theta^{2}}(\theta,t)-\rho_{k}(\theta,t)-f(t),\end{aligned}\right.$
(2.12)

with initial values for $\theta\in[0,2m\pi]$,

$\displaystyle
p(\theta,0)=p_{0}(\theta),~{}~{}w_{k}(\theta,0)=w_{k0}(\theta),~{}~{}\rho_{k}(\theta,0)=\rho_{k0}(\theta),~{}~{}\rho(\theta,0)=\rho_{0}(\theta).$

###### Proof.

If $X(\cdot,t)$ is a family of locally convex curves evolving according to
(2.1), we immediately have evolution equations in (2.12). Suppose (2.12) has
smooth and positive solutions. Then one may construct a family of locally
convex curves by $p$ according to

$\displaystyle X(\theta,t)=\frac{\partial
p}{\partial\theta}(\theta,t)T(\theta)-p(\theta,t)N(\theta),$ (2.13)

where $T(\theta)$ and $N(\theta)$, parameterized by the tangent angle
$\theta$, form the Frenet frame of the curve at every point $X(\theta,t)$.
Therefore, the curve $X(\cdot,t)$ satisfies

$\displaystyle\frac{\partial X}{\partial t}=\frac{\partial^{2}p}{\partial
t\partial\theta}T-\frac{\partial p}{\partial
t}N=\frac{\partial}{\partial\theta}\left(\frac{\partial p}{\partial
t}\right)T-\frac{\partial p}{\partial t}N=\alpha
T+\left(2w_{k}-\rho_{k}+f\right)N.$

This is the evolution equation in (2.1). Thus we are done. ∎

######  Lemma 2.2.

The flow (2.1) has a unique and smooth solution on some time interval.

###### Proof.

According to Lemma 2.1, one needs show the system (2.12) has positive and
smooth solutions on some time interval. Denote by
$\eta_{k}(\theta,t):=\frac{\partial\rho_{k}}{\partial\theta}(\theta,t)$. Then
the function $\eta_{k}$ satisfies a linear equation

$\displaystyle\frac{\partial\eta_{k}}{\partial
t}(\theta,t)=k\left(\frac{\partial^{2}\eta_{k}}{\partial\theta^{2}}(\theta,t)-\eta_{k}(\theta,t)\right)$
(2.14)

with a smooth initial value
$\eta_{k}(\theta,0)=\frac{\partial\rho_{k}}{\partial\theta}(\theta,0)$.
Solving the linear parabolic equation (2.14) with the initial value, we get a
smooth function $\eta_{k}(\theta,t)$ on the domain
$[0,2m\pi]\times[0,+\infty)$. Since
$\rho_{k}(\theta,t)-\int_{0}^{\theta}\eta_{k}(\widetilde{\theta},t)d\widetilde{\theta}$
is independent of $\theta$, there is a function $\lambda(t)$, to be
determined, so that

$\displaystyle\rho_{k}(\theta,t)=\int_{0}^{\theta}\eta_{k}(\widetilde{\theta},t)d\widetilde{\theta}+\lambda(t).$
(2.15)

By observing the system (2.12), we may compute to obtain
$\frac{\partial}{\partial t}(\rho_{k}-k\rho)\equiv 0$. So
$\rho_{k}(\theta,t)-k\rho(\theta,t)$ is independent of time $t$, i.e, we have

$\displaystyle\rho_{k}(\theta,t)-k\rho(\theta,t)=\rho_{k}(\theta,0)-k\rho(\theta,0).$
(2.16)

Substituting (2.15) into (2.16), we have

$\displaystyle\rho(\theta,t)=\frac{1}{k}\left[\int_{0}^{\theta}\eta_{k}(\widetilde{\theta},t)d\widetilde{\theta}+\lambda(t)-\rho_{k}(\theta,0)+k\rho(\theta,0)\right].$
(2.17)

Using the definition of $f(t)$ (see (1.4)) and the evolution equation of
$\rho$, we get

$\displaystyle\frac{d}{dt}\int_{0}^{2m\pi}\frac{1}{\rho(\theta,t)}=-\int_{0}^{2m\pi}\frac{1}{\rho^{2}(\theta,t)}\frac{\partial\rho}{\partial
t}d\theta\equiv 0.$ (2.18)

Therefore, the function $\lambda(t)$ is uniquely determined by the indentity

$\displaystyle
k\int_{0}^{2m\pi}\frac{d\theta}{\int_{0}^{\theta}\eta_{k}(\widetilde{\theta},t)d\widetilde{\theta}+\lambda(t)-\rho_{k}(\theta,0)+k\rho(\theta,0)}\equiv\int_{0}^{2m\pi}\frac{1}{\rho_{0}(\theta)}d\theta.$
(2.19)

Once we have the function $\lambda(t)$, we get the values of
$\rho_{k}(\theta,t)$, $\rho(\theta,t)$ and $f(t)$. So, integrating the
evolution equations, we obtain $w_{k}(\theta,t)$ and $p(\theta,t)$,
respectively. By the continuity of $\rho_{k}(\theta,t)$, $\rho(\theta,t)$,
$w_{k}(\theta,t)$ and $p(\theta,t)$, the system (2.12) has positive and smooth
solutions on some small time interval. ∎

######  Remark 2.3.

The equation (2.16) says that $\rho$ and $\rho_{k}$ have a concise relation.
The support function $p$ and the width function $w_{k}$ have a similar
relation as shown in the equation (2.16). This fact will be used in the proof
of Theorem 4.5.

Lin and Tsai [12] have considered a relative linear equation (compare to the
system (2.12)) which can be used to answer Yau’s problem of evolving one curve
to another. Recent progress on this problem can be found in the papers [9, 10,
14].

##  3 Long term existence

In this section, we prove that the flow (2.1) exists on the time interval
$[0,+\infty)$. The main idea is to show that the radius of curvature $\rho$
has both uniformly positive lower and upper bounds. If so, then the flow can
be infinitely extended, and in the evolution process the evolving curve
$X(\cdot,t)$ is smooth and locally convex. Let $f(\theta,t)$ be a continuous
function defined on $[0,2m\pi]\times[0,T_{\max})$. We define

$\displaystyle
f_{\max}(t)=\max\\{f(\theta,t)|\theta\in[0,2m\pi]\\},~{}~{}f_{\min}(t)=\min\\{f(\theta,t)|\theta\in[0,2m\pi]\\}.$

######  Lemma 3.1.

Every order derivative of $\rho$ with respect to $\theta$ has uniform bounds
if the flow (2.1) preserves the local convexity of the evolving curve.

###### Proof.

If the evolving curve $X(\cdot,t)$ is locally convex under the flow (2.1),
then we have the evolution equation of $\rho_{k}$ as (2.12). Differentiating
this equation with respect to $\theta$ gives

$\displaystyle\frac{\partial^{2}\rho_{k}}{\partial
t\partial\theta}=k\frac{\partial^{3}\rho_{k}}{\partial\theta^{3}}-k\frac{\partial\rho_{k}}{\partial\theta}.$

Let
$u(\theta,t)=\frac{1}{2}\mid\frac{\partial\rho_{k}}{\partial\theta}\mid^{2}$.
Then this function satisfies

$\displaystyle\frac{\partial u}{\partial
t}=k\left(\frac{\partial^{2}u}{\partial\theta^{2}}-\left(\frac{\partial^{2}\rho_{k}}{\partial\theta^{2}}\right)^{2}\right)-2ku.$

Set $v(\theta,t)=e^{2kt}u(\theta,t)$. Then $v(\theta,0)=u(\theta,0)$ and

$\displaystyle\frac{\partial v}{\partial t}\leq
k\frac{\partial^{2}v}{\partial\theta^{2}}.$

Applying the maximum principle, one obtains $v_{\max}(t)\leq v_{\max}(0)$,
which implies

$\displaystyle u(\theta,t)\leq u_{\max}(0)e^{-2kt}.$

Moreover,

$\displaystyle\left|\frac{\partial\rho_{k}}{\partial\theta}(\theta,t)\right|\leq\max_{\theta}\left|\frac{\partial\rho_{k}}{\partial\theta}(\theta,0)\right|e^{-kt}.$
(3.1)

Denote by $C_{i}$ the constant
$\max\limits_{\theta}\left|\frac{\partial^{i}\rho_{k}}{\partial\theta^{i}}(\theta,0)\right|$,
$i=2,3,\cdots$. Using the evolution equation of
$\frac{\partial^{i}\rho}{\partial\theta^{i}}$, one may similarly prove that
$\left|\frac{\partial^{i}\rho_{k}}{\partial\theta^{i}}(\theta,t)\right|$ is
bounded by $C_{i}e^{-kt}$.

Differentiating the evolution equation of $\rho$, one obtains

$\displaystyle\frac{\partial}{\partial
t}\left(\frac{\partial^{i}\rho}{\partial\theta^{i}}\right)=\frac{\partial^{i+2}\rho_{k}}{\partial\theta^{i+2}}-\frac{\partial^{i}\rho_{k}}{\partial\theta^{i}}.$

Since $\left|\frac{\partial^{i}\rho_{k}}{\partial\theta^{i}}(\theta,t)\right|$
decays exponentially, there exists a constant $M_{i}$, independent of time,
such that

$\displaystyle\left|\frac{\partial^{i}\rho}{\partial\theta^{i}}(\theta,t)\right|\leq
M_{i},~{}~{}(\theta,t)\in[0,2m\pi]\times[0,T_{\max}),~{}~{}i=1,2,\cdots.$
(3.2)

The proof is finished. ∎

######  Lemma 3.2.

If the flow (2.1) preserves the local convexity of the evolving curve, then
the elastic energy is fixed as time goes.

###### Proof.

Under the flow (2.1), the curvature $\kappa(\theta,t)$ of the curve
$X(\cdot,t)$ evolves according to (2.5), i.e., we have

$\displaystyle\frac{\partial\kappa}{\partial
t}=\kappa^{2}\left(-\frac{\partial^{2}\rho_{k}}{\partial\theta^{2}}+\rho_{k}(\theta,t)+f(t)\right).$
(3.3)

So the elastic energy of the evolving curve satisfies

$\displaystyle\frac{dE}{dt}(t)$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\int_{X(\cdot,t)}\kappa^{2}(s,t)ds$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\int_{mS^{1}}\kappa(\theta.t)d\theta$
$\displaystyle=$
$\displaystyle\int_{mS^{1}}\kappa^{2}\left(-\frac{\partial^{2}\rho_{k}}{\partial\theta^{2}}+\rho_{k}(\theta,t)+f(t)\right)d\theta.$

By the definition of $f(t)$, one has $\frac{dE}{dt}\equiv 0$. ∎

Since the elastic energy $E$ equals
$\int_{0}^{2m\pi}\frac{1}{\rho(\theta,t)}\text{d}\theta$, one gets that under
the flow (2.1)

$\displaystyle\frac{2m\pi}{\rho_{\max}(t)}\leq
E\leq\frac{2m\pi}{\rho_{\min}(t)},$ (3.4)

if this flow preserves the local convexity of the evolving curve. This
observation together with the gradient estimate of $\rho$ lead to its uniform
bounds.

######  Lemma 3.3.

Under the condition of Lemma 3.2, there exist two positive constants $m_{0}$
and $M_{0}$ independent of time such that the curvature radius is bounded as

$\displaystyle m_{0}\leq\rho(\theta,t)\leq M_{0}.$ (3.5)

###### Proof.

Under the flow (2.1), the gradient estimate of $\rho$ tells us that
$\mid\frac{\partial\rho}{\partial\theta}\mid\leq M_{1}$, where $M_{1}$ is a
positive constant, independent of $t$. Fix the time $t$. By continuity of
$\rho$, there exist $\theta_{1}$ and $\theta_{2}$ such that
$\rho_{\min}(t)=\rho(\theta_{1},t)$ and $\rho_{\max}(t)=\rho(\theta_{2},t)$.
So

$\displaystyle\ln\rho_{\max}(t)-\ln\rho_{\min}(t)=\int_{\theta_{1}}^{\theta_{2}}\frac{1}{\rho}\frac{\partial\rho}{\partial\theta}\text{d}\theta\leq\int_{0}^{2m\pi}\frac{1}{\rho}\left|\frac{\partial\rho}{\partial\theta}\right|\text{d}\theta\leq
M_{1}E.$

Therefore,

$\rho_{\max}(t)\leq\rho_{\min}(t)e^{M_{1}E}.$ (3.6)

Setting $m_{0}=\frac{2m\pi}{E}e^{-M_{1}E}$ and
$M_{0}=\frac{2m\pi}{E}e^{M_{1}E}$, and combining (3.4) and (3.6), one has the
estimate (3.5). ∎

Using this lemma, we may show that the flow (2.1) preserves the local
convexity of the evolving curve.

######  Lemma 3.4.

If the initial curve $X_{0}$ is locally convex, then the evolving curve
$X(\cdot,t)$ is also locally convex under the flow (2.1).

###### Proof.

Suppose the flow exists on time interval $[0,T_{\max})$ and there is a
positive $t_{0}<T_{\max}$ such that $X(\cdot,t)$ is locally convex on time
interval $[0,t_{0})$ but the minimum of the curvature $\kappa(\theta,t_{0})$,
with respect to $\theta$, is 0.

By the proof of Lemma 3.3, the curvature has a lower bound
$\kappa(\theta,t)\geq\frac{E}{2m\pi}e^{-M_{1}E}$ under the flow (2.1) for
every $(\theta,t)\in[0,2m\pi]\times[0,t_{0})$. The continuity of curvature
implies that $\kappa(\theta,t_{0})\geq\frac{E}{2m\pi}e^{-M_{1}E}>0$ holds for
all $\theta$. A contradiction. ∎

######  Theorem 3.5.

If the initial curve $X_{0}(\theta)$ is locally convex, then the flow (2.1)
has a unique smooth solution $X(\cdot,t)$ on $[0,2m\pi]\times[0,+\infty)$.

###### Proof.

Suppose the flow (2.1) exists on the maximal time interval $[0,T_{\max})$ and
$T_{\max}$ is a finite positive number. It follows from (3.2) and (3.5) that
$\kappa$ and all its derivatives are uniformly bounded on the time interval
$[0,T_{\max})$. So the nonlocal term $f(t)$ has uniform bound which is
independent of $T_{\max}$.

By the evolution equation of the $k$-order width $w_{k}(\theta,t)$, its
derivative $\frac{\partial^{i}w_{k}}{\partial\theta^{i}}$ satisfies

$\displaystyle\frac{\partial}{\partial
t}\left(\frac{\partial^{i}w_{k}}{\partial\theta^{i}}\right)=k\frac{\partial^{i+2}w_{k}}{\partial\theta^{i+2}}-k\frac{\partial^{i}w_{k}}{\partial\theta^{i}}.$

Applying the same trick as in the proof of Lemma 3.1, one may show that
$|\frac{\partial^{i}w_{k}}{\partial\theta^{i}}|^{2}$ decays exponentially,
then $w_{k}(\theta,t)$ is also uniformly bounded on the time interval
$[0,T_{\max})$.

Hence the velocity of the flow has uniform bound which is independent of $w$.
By the unique existence of the flow, one obtains a smooth and locally convex
curve

$\displaystyle
X_{T_{\max}}(\theta):=X_{0}(\theta)+\int_{0}^{T_{\max}}\frac{\partial
X}{\partial t}(\theta,t)dt.$

Let $X_{T_{\max}}(\theta)$ evolve according to the flow (2.1). Then there
exists a family of smooth, locally convex curves $X(\cdot,t)$ on the time
interval $[T_{\max},T_{\max}+\varepsilon)$, where $\varepsilon$ is a positive
number. By the unique existence of the flow (2.1), this flow is extended on a
larger time interval $[0,T_{\max}+\varepsilon)$. This contradicts the
maximality of $T_{\max}$. ∎

##  4 Convergence

In this section, we explore the asymptotic behavior of the flow (2.1) and
complete the proof of Theorem 1.1.

Let $X_{0}(\theta)$ be a locally convex plane curve with rotation number $m$
and tangent angle $\theta$, where $\theta\in[0,2m\pi]$. Expand the support
function as

$\displaystyle
p(\theta)=\frac{a_{0}}{2}+\sum\limits_{n=1}^{\infty}\left(a_{n}\cos\frac{n\theta}{m}+b_{n}\sin\frac{n\theta}{m}\right),$

where the coefficients are expressed as

$\displaystyle
a_{0}=\frac{L_{0}}{m\pi},~{}~{}~{}a_{n}=\frac{1}{m\pi}\int_{-m\pi}^{m\pi}p(\theta)\cos\frac{n\theta}{m}d\theta,~{}~{}~{}b_{n}=\frac{1}{m\pi}\int_{-m\pi}^{m\pi}p(\theta)\sin\frac{n\theta}{m}d\theta.$

So the $k$-order width of the curve is

$\displaystyle
w_{k}(\theta)=\frac{ka_{0}}{2}+\sum\limits_{n=1}^{\infty}a_{n}\sum\limits_{l=0}^{k-1}\cos\left(\frac{n\theta}{m}+\frac{2nl\pi}{k}\right)+\sum\limits_{n=1}^{\infty}b_{n}\sum\limits_{l=0}^{k-1}\sin\left(\frac{n\theta}{m}+\frac{2nl\pi}{k}\right).$
(4.1)

For a positive integer $l$, one has the identities

$\displaystyle\sin\left(\frac{2n\pi}{k}\right)+\sin\left(\frac{4n\pi}{k}\right)+\cdots+\sin\left(\frac{2(k-1)n\pi}{k}\right)=0$

and

$\displaystyle\cos\left(\frac{2n\pi}{k}\right)+\cos\left(\frac{4n\pi}{k}\right)+\cdots+\cos\left(\frac{2(k-1)n\pi}{k}\right)=\left\\{\begin{aligned}
-1,\quad n\neq kl,&\\\ k-1,\quad n=kl.\end{aligned}\right.$

So one may compute

$\displaystyle w_{k}(\theta)$ $\displaystyle=$
$\displaystyle\frac{ka_{0}}{2}+\sum\limits_{n=1}^{\infty}a_{n}\cos\frac{n\theta}{m}\sum\limits_{l=0}^{k-1}\cos\frac{2nl\pi}{k}-\sum\limits_{n=1}^{\infty}a_{n}\sin\frac{n\theta}{m}\sum\limits_{l=0}^{k-1}\sin\frac{2nl\pi}{k}$
(4.2)
$\displaystyle+\sum\limits_{n=1}^{\infty}b_{n}\sin\frac{n\theta}{m}\sum\limits_{l=0}^{k-1}\cos\frac{2nl\pi}{k}+\sum\limits_{n=1}^{\infty}b_{n}\cos\frac{n\theta}{m}\sum\limits_{l=0}^{k-1}\sin\frac{2nl\pi}{k}$
$\displaystyle=$
$\displaystyle\frac{ka_{0}}{2}+\sum\limits_{n=1}^{\infty}k\left(a_{nk}\cos\left(\frac{nk\theta}{m}\right)+b_{nk}\sin\left(\frac{nk\theta}{m}\right)\right).$

Hence, one has the following proposition.

######  Proposition 4.1.

Let $X_{0}(\theta)$ be a locally convex plane curve with the rotation number
$m$. If it is of constant $k$-order width, then

$\displaystyle p(\theta)=\frac{a_{0}}{2}+\sum\limits_{n\neq
kl}^{\infty}\left(a_{n}\cos\frac{n\theta}{m}+b_{n}\sin\frac{n\theta}{m}\right).$

######  Definition 4.2.

Let $X_{0}$ be a plane closed curve with the rotation number $m$. If it is
invariant under the rotation of the angle $\frac{2m\pi}{k}$, then it is called
$k$-_symmetric_.

Moreover, we can prove the following proposition.

######  Proposition 4.3.

Let $X_{0}(\theta)$ be a locally convex curve with the rotation number $m$. If
$X_{0}(\theta)$ is $k$-symmetric, then

$\displaystyle
p(\theta)=\frac{a_{0}}{2}+\sum\limits_{l=1}^{\infty}\left(a_{kl}\cos\frac{kl\theta}{m}+b_{kl}\sin\frac{kl\theta}{m}\right).$

###### Proof.

By the Fourier expansion of the support function $p$, one obtains

$\displaystyle p(\theta+\frac{2m\pi}{k})=$
$\displaystyle\frac{a_{0}}{2}+\sum\limits_{n=1}^{\infty}\left(a_{n}\cos\left(\frac{n\theta}{m}+\frac{2n\pi}{k}\right)+b_{n}\sin\left(\frac{n\theta}{m}+\frac{2n\pi}{k}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{a_{0}}{2}+\sum\limits_{n=1}^{\infty}\cos\frac{n\theta}{m}\left(a_{n}\cos\frac{2n\pi}{k}+b_{n}\sin\frac{2n\pi}{k}\right)$
$\displaystyle+$
$\displaystyle\sum\limits_{n=1}^{\infty}\sin\frac{n\theta}{m}\left(b_{n}\cos\frac{2n\pi}{k}-a_{n}\sin\frac{2n\pi}{k}\right).$

Since $X_{0}$ is $k$-symmetric, $p(\theta)=p(\theta+\frac{2m\pi}{k})$ holds
for every $\theta\in[0,2m\pi]$. A comparism of the coefficients in the Fourier
expansion of $p(\theta)$ and $p(\theta+\frac{2m\pi}{k})$ finishes the proof. ∎

Combining the propositions $\eqref{pro:4.1.202302}$ and
$\eqref{pro:4.3.202302}$, one gets

######  Proposition 4.4.

Let $X_{0}$ be a $k$-symmetric, locally convex plane curve with the rotation
number $m$. Then $X_{0}$ is of constant $k$-order width if and only if it is
an $m$-fold circle.

Now we turn to the proof of the remaining part of Theorem 1.1.

######  Theorem 4.5.

The evolving curve of the flow (2.1) converges to a locally convex curve of
constant $k$-order width.

###### Proof.

On one hand, from (4.2) we get the evolving equation of $w_{k}$, namely

$\frac{\partial w_{k}}{\partial
t}(\theta,t)=\frac{k}{2}a_{0}^{\prime}(t)+k\sum\limits_{n=1}^{\infty}\left(a_{nk}^{\prime}(t)\cos\left(\frac{nk}{m}\theta\right)+b_{nk}^{\prime}(t)\sin\left(\frac{nk}{m}\theta\right)\right)$
(4.3)

and

$\frac{\partial
w_{k}}{\partial\theta}(\theta,t)=k\sum\limits_{n=1}^{\infty}\left(-\frac{nk}{m}a_{nk}(t)\sin\left(\frac{nk}{m}\theta\right)+\frac{nk}{m}b_{nk}(t)\cos\left(\frac{nk}{m}\theta\right)\right).$

Moreover,

$\frac{\partial^{2}w_{k}}{\partial\theta^{2}}(\theta,t)=-k\sum\limits_{n=1}^{\infty}\frac{n^{2}k^{2}}{m^{2}}\left(a_{nk}(t)\cos\left(\frac{nk}{m}\theta\right)+b_{nk}(t)\sin\left(\frac{nk}{m}\theta\right)\right).$
(4.4)

Substituting (4.4) into the evolution equation of $w_{k}$ (see (2.10)), one
gets

$\displaystyle\frac{\partial w_{k}}{\partial t}(\theta,t)=-$
$\displaystyle\frac{k^{2}}{2}a_{0}(t)-kf(t)$ (4.5) $\displaystyle-$
$\displaystyle\sum\limits_{n=1}^{\infty}\left(\left(\frac{n^{2}k^{4}}{m^{2}}+k^{2}\right)a_{nk}(t)\cos\left(\frac{nk\theta}{m}\right)+\left(\frac{n^{2}k^{4}}{m^{2}}+k^{2}\right)b_{nk}(t)\sin\left(\frac{nk\theta}{m}\right)\right).$

Comparing the coefficients of the right sides in (4.5) and (4.3), we have

$\left\\{\begin{aligned} a_{0}^{\prime}(t)=&-ka_{0}(t)-2f(t),\\\
a_{nk}^{\prime}(t)=&-\left(\frac{n^{2}k^{3}}{m^{2}}+k\right)a_{nk}(t),\\\
b_{nk}^{\prime}(t)=&-\left(\frac{n^{2}k^{3}}{m^{2}}+k\right)b_{nk}(t).\\\
\end{aligned}\right.$ (4.6)

Integrating the last two equations in (4.6) yields

$\displaystyle a_{nk}(t)=a_{nk}(0)e^{-\frac{n^{2}k^{3}+m^{2}k}{m^{2}}t},\quad
b_{nk}(t)=b_{nk}(0)e^{-\frac{n^{2}k^{3}+m^{2}k}{m^{2}}t}.$

Hence

$w_{k}(\theta,t)=\frac{k}{2}a_{0}(t)+k\sum\limits_{n=1}^{\infty}\left(a_{nk}(0)\cos\left(\frac{nk\theta}{m}\right)+b_{nk}(0)\sin\left(\frac{nk\theta}{m}\right)\right)e^{-\frac{n^{2}k^{3}+m^{2}k}{m^{2}}t}.$
(4.7)

By the evolution equation of $p(\theta,t)$ and $w_{k}(\theta,t)$, one gets

$\displaystyle\frac{\partial p}{\partial
t}(\theta,t)=\frac{1}{k}\frac{\partial w_{k}}{\partial t}(\theta,t),$ (4.8)

which implies that

$\displaystyle p(\theta,t)=$ $\displaystyle
p(\theta,0)+\frac{1}{k}\left(w_{k}(\theta,t)-w_{k}(\theta,0)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}a_{0}(0)+\sum\limits_{n=1}^{\infty}\left(a_{n}(0)\cos\frac{n\theta}{m}+b_{n}(0)\sin\frac{n\theta}{m}\right)$
$\displaystyle+\frac{1}{2}a_{0}(t)+\sum\limits_{n=1}^{\infty}\left(a_{nk}(0)\cos\frac{nk\theta}{m}+b_{nk}(0)\sin\frac{nk\theta}{m}\right)e^{-\frac{n^{2}k^{3}+m^{2}k}{m^{2}}t}$
$\displaystyle-\frac{1}{2}a_{0}(0)-\sum\limits_{n=1}^{\infty}\left(a_{nk}(0)\cos\frac{nk\theta}{m}+b_{nk}(0)\sin\frac{nk\theta}{m}\right).$

That is,

$\displaystyle p(\theta,t)=$
$\displaystyle\frac{1}{2}a_{0}(t)+\sum\limits_{n=1}^{\infty}\left(a_{nk}(0)\cos\frac{nk\theta}{m}+b_{nk}(0)\sin\frac{nk\theta}{m}\right)e^{-\frac{n^{2}k^{3}+m^{2}k}{m^{2}}t}$
(4.9) $\displaystyle+\sum\limits_{n\neq
kl}^{\infty}\left(a_{n}(0)\cos\frac{n\theta}{m}+b_{n}(0)\sin\frac{n\theta}{m}\right).$

On the other hand, it follows from (3.2) and (3.5) that $\rho(\cdot,t)$ is
uniformly bounded and equicontinuous. According to the well-known Arzelà-
Ascoli Theorem, the function $\rho(\cdot,t)$ has a convergent subsequence.
Suppose there are two convergent subsequences $\\{\rho(\cdot,t_{i})\\}$ and
$\\{\rho(\cdot,t_{j})\\}$ such that

$\displaystyle\lim_{t_{i}\rightarrow+\infty}\rho(\theta,t_{i})=\widetilde{\rho}(\theta),~{}~{}\lim_{t_{j}\rightarrow+\infty}\rho(\theta,t_{j})=\widetilde{\eta}(\theta),$

where $\widetilde{\rho}$ and $\widetilde{\eta}$ are two positive functions.

By (4.9) and the identity (2.7), the two functions $\widetilde{\rho}$ and
$\widetilde{\eta}$ differ by a constant, i.e.,
$\widetilde{\rho}(\theta)=\widetilde{\eta}(\theta)+c_{0}$ holds for all
$\theta\in[0,2m\pi]$. Since the flow (2.1) preserves the elastic energy
$\int_{0}^{2m\pi}\frac{1}{\rho(\theta,t)}d\theta$, one has

$\displaystyle\int_{0}^{2m\pi}\frac{1}{\widetilde{\rho}(\theta)}d\theta=\int_{0}^{2m\pi}\frac{1}{\widetilde{\eta}(\theta)}d\theta.$
(4.10)

So the constant $c_{0}$ has to be 0. The radius of curvature $\rho(\theta,t)$
converges to a limiting function as $t\rightarrow+\infty$. Since also the
function $\rho_{k}(\theta,t)$ converges, we set

$\displaystyle\lim_{t\rightarrow+\infty}\rho_{k}(\theta,t)=\widetilde{\rho}_{k}(\theta),$
(4.11)

where $\widetilde{\rho}_{k}$ is a positive function. Furthermore, the estimate
(3.1) tells us that this function is a constant function. By the evolution
equation of the $k$-order width $w_{k}(\theta,t)$, its derivative
$\frac{\partial^{i}w_{k}}{\partial\theta^{i}}$ satisfies

$\displaystyle\frac{\partial}{\partial
t}\left(\frac{\partial^{i}w_{k}}{\partial\theta^{i}}\right)=k\frac{\partial^{i+2}w_{k}}{\partial\theta^{i+2}}-k\frac{\partial^{i}w_{k}}{\partial\theta^{i}}.$

Applying the same trick as in the proof of Lemma 3.1, one may show that
$|\frac{\partial^{i}w_{k}}{\partial\theta^{i}}|^{2}$ decays exponentially. So
$w_{k}(\theta,t)$ also converges to a constant as $t\rightarrow+\infty$. Using
the relation (2.8), one has

$\displaystyle\lim_{t\rightarrow+\infty}w_{k}(\theta,t)=\widetilde{\rho}_{k}.$
(4.12)

The equation (4.8) shows that the support function and the width function have
the relation

$\displaystyle
p(\theta,t)=p(\theta,0)+\frac{1}{k}\left(w_{k}(\theta,t)-w_{k}(\theta,0)\right),$
(4.13)

the limit (4.12) implies that $p(\theta,t)$ converges as
$t\rightarrow+\infty$. The equation (2.13) implies that the evolving curve of
the flow (2.1) also converges to a curve $X_{\infty}$ as time goes to
infinity. Finally, since the limit (4.12) says that the $k$-order width
function converges to a constant, the limiting curve $X_{\infty}$ has constant
$k$-order width. ∎

######  Theorem 4.6.

If the initial curve $X_{0}$ is a $k$-symmetric, locally convex closed plane
curve with the rotation number $m$, then the evolving curve $X(\cdot,t)$ under
the flow (2.1) converges to an $m$-fold circle, and vice versa.

###### Proof.

From (4.13), we have that $p(\theta,t)-\frac{1}{k}w_{k}(\theta,t)$ is constant
independent of $t$, that is,

$\displaystyle
p(\theta,t)-\frac{1}{k}w_{k}(\theta,t)=p(\theta,0)-\frac{1}{k}w_{k}(\theta,0).$

Suppose that the initial curve $X_{0}$ is $k$-symmetric. From Proposition 4.3
and (4.2), one gets

$\displaystyle p(\theta,0)-\frac{1}{k}w_{k}(\theta,0)=0.$

Hence, $p(\theta,t)=\frac{1}{k}w_{k}(\theta,t)$, which together with Theorem
4.5 gives us that $\lim\limits_{t\rightarrow\infty}p(\theta,t)$ is constant,
that is, the limiting curve is an $m$-fold circle.

Conversely, if the flow (2.1) has a global solution on
$[0,2m\pi]\times[0,\infty)$ and the limiting curve is an $m$-fold circle with
center $O$, then $\lim\limits_{t\rightarrow\infty}p(\theta,t)$ is a constant
and (4.9) implies

$\displaystyle a_{n}(0)=b_{n}(0)=0,n\neq kl.$

Therefore, we get

$\displaystyle
p(\theta,t)=\frac{1}{2}a_{0}(t)+\sum\limits_{n=1}^{\infty}\left(a_{nk}(0)\cos\frac{nk\theta}{m}+b_{nk}(0)\sin\frac{nk\theta}{m}\right)e^{-\frac{n^{2}k^{3}+m^{2}k}{m^{2}}t},$

which implies

$\displaystyle
p(\theta,0)=p(\theta+\frac{2m\pi}{k},0)=\cdots=p(\theta+\frac{2m(k-1)\pi}{k},0).$

In this case, $X_{0}$ is $k$-symmetric with respect to the origin. ∎

The combination of Lemma 2.2, Theorem 3.5, Theorem 4.5 and Theorem 4.6 yields
the proof of the main result given in Theorem 1.1.

Acknowledgments Deyan Zhang is supported by University Natural Science
Research Project of Anhui Province (No. 2022AH040067).

## References

  * [1] U. Abresch, J. Langer, _The normalized curve shortening flow and homothetic solutions_. J. Differential Geom. 23 (1986), no. 2, 175-196.
  * [2] H. Anciaux, N. Georgiou, _The Blaschke-Lebesgue problem for constant width bodies of revolution_. 2009\. preprint, arXiv:0903.4284.
  * [3] H. Anciaux, B. Guilfoyle, _On the three-dimensional Blaschke-Lebesgue problem_. Proc. Amer. Math. Soc., 139 (2011), 1831-1839.
  * [4] T. Bayen, T. Lachand-Robert and E. Oudet, _Analytic parametrizations and volume minimization of three-dimensional bodies of constant width_. Arch. Ration. Mech. Anal., 186 (2007), 225-249.
  * [5] K.-S. Chou, X.-P. Zhu, _The Curve Shortening Problem_. CRC Press, Boca Raton, 2001.
  * [6] J.-B. Fang, _Evolving convex curves to constant $k$-order width ones by a perimeter-preserving flow_. Manuscr. Math. 157 (2017) 247–256.
  * [7] L.-Y. Gao, S.-L. Pan, _Evolving convex curves to another constant-width ones by a perimeter-preserving flow_. Pacific J. Math. 272 (2014), 131-145.
  * [8] L.-Y. Gao, Y.-T. Zhang, _Evolving convex surfaces to constant width ones_. Internat. J. Math. 28 (2017), no. 11, 1750082, 18 pp.
  * [9] L.-Y. Gao, Y.-T. Zhang, _On Yau’s problem of evolving one curve to another: convex case_. J. Differential Equations 266 (2019), no. 1, 179-201.
  * [10] L.-Y. Gao, _Whitney-Graustein homotopy of locally convex curves via a curvature flow_. Math. Res. Lett. 30 (2023), no. 4, 1045-1062.
  * [11] H. Halldorsson, _Self-similar solutions to the curve shortening flow_. Trans. Amer. Math. Soc. 364 (2012), no. 10, 5285-5309.
  * [12] Y.-C. Lin, D.-H. Tsai, _Evolving a convex closed curve to another one via a length-preserving linea flow_. J. Differential Equations 9 (2009), 2620-2636.
  * [13] H. Martini, L. Montejano, D. Oliveros, _Bodies of Constant Width. An Introduction to Convex Geometry with Applications_. Birkhäuser/Springer, Cham, 2019. xi+486 pp.
  * [14] J. A. McCoy, P. Schrader, G. Wheeler, _Representation formulae for higher order curvature flows_. J. Differential Equations 344 (2023), 1-43.
  * [15] D. A. Singer, _Lectures on elastic curves and rods_. AIP Conf. Proc. 1002 (2008), 3-32.
  * [16] X.-L. Wang, L.-H. Kong, _Area-preserving evolution of nonsimple symmetric plane curves_. J. Evol. Equ. 14 (2014), no. 2, 387-401.
  * [17] X.-L. Wang, H.-L. Li, X.-Li Chao, _Length-preserving evolution of immersed closed curves and the isoperimetric inequality_. Pacific J. Math. 290 (2017), no. 2, 467-479.
  * [18] X.-L. Wang, W.-F. Wo, M. Yang, _Evolution of non-simple closed curves in the area-preserving curvature flow_. Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), no. 3, 659-668.

Laiyuan Gao

School of Mathematics and Statistics, Jiangsu Normal University

No.101, Shanghai Road, Xuzhou 221116, Jiangsu, P. R. China

Email<EMAIL_ADDRESS>

Horst Martini

Faculty of Mathematics, University of Technology Chemnitz

09107 Chemnitz, Germany

Email<EMAIL_ADDRESS>

Deyan Zhang

School of Mathematical Sciences, Huaibei Normal University

No.100, Dongshan Road, Huaibei 235000, Anhui, P. R. China

Email<EMAIL_ADDRESS>


* [GK14] Toby Gee and Mark Kisin.  The Breuil-Mézard conjecture for potentially Barsotti-Tate representations.  Forum Math. Pi, 2:56, 2014.  Id/No e1.
  * [Her09] Florian Herzig.  The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations.  Duke Math. J., 149(1):37–116, 2009.
  * [HT15] Yongquan Hu and Fucheng Tan.  The Breuil-Mézard conjecture for non-scalar split residual representations.  Ann. Sci. Éc. Norm. Supér. (4), 48(6):1383–1421, 2015.
  * [Jan03] Jens Carsten Jantzen.  Representations of algebraic groups., volume 107.  Providence, RI: American Mathematical Society (AMS), 2003.
  * [Kap12] Christian Kappen.  Uniformly rigid spaces.  Algebra Number Theory, 6(2):341–388, 2012.
  * [Kis08] Mark Kisin.  Potentially semi-stable deformation rings.  J. Am. Math. Soc., 21(2):513–546, 2008.
  * [Kis09] Mark Kisin.  The Fontaine-Mazur conjecture for $\text{GL}_{2}$.  J. Am. Math. Soc., 22(3):641–690, 2009.
  * [KM18] Karol Kozioł and Stefano Morra.  Serre weight conjectures for $p$-adic unitary groups of rank $2$.  2018\.
  * [LLHLM18] Daniel Le, Bao V. Le Hung, Brandon Levin, and Stefano Morra.  Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows.  Invent. Math., 212(1):1–107, 2018.
  * [LLLM20] Daniel Le, Bao Le Hung, Brandon Levin, and Stefano Morra.  Local models for galois deformation rings and applications.  2020\.
  * [Maz89] B. Mazur.  Deforming Galois representations.  Galois groups over ${\mathbb{Q}}$, Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, 385-437 (1989)., 1989.
  * [Paš15] Vytautas Paškūnas.  On the Breuil-Mézard conjecture.  Duke Math. J., 164(2):297–359, 2015.
  * [Paš16] Vytautas Paškūnas.  On 2-dimensional 2-adic Galois representations of local and global fields.  Algebra Number Theory, 10(6):1301–1358, 2016.
  * [Saa72] Neantro Saavedra Rivano.  Catégories tannakiennes, volume 265.  Springer, Cham, 1972.
  * [Ser77] J.-P. Serre.  Modular forms of weight one and Galois representations.  In Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pages 193–268, 1977.
  * [Til96] Jacques Tilouine.  Deformations of Galois representations and Hecke algebras.  New Delhi: Narosa Publishing House; Publ. for the Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, 1996.
  * [Tun18] Shen-Ning Tung.  On the automorphy of $2$-dimensional potentially semi-stable deformation rings of ${G}_{\mathbb{q}_{p}}$.  2018\.
  * [Tun19] Shen-Ning Tung.  On the modularity of $2$-adic potentially semi-stable deformation rings.  2019\.