- Email: [email protected]
Self-similar solutions to the hyperbolic mean curvature flow
Acta Mathematica Scientia 2017,37B(3):657–667 http://actams.wipm.ac.cn
SELF-SIMILAR SOLUTIONS TO THE HYPERBOLIC MEAN CURVATURE FLOW∗
ÛSZ)
Chunlei HE (
Å)
Shoujun HUANG (
0¡¯)
Xiaomin XING (
School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China E-mail : [email protected]; [email protected]; [email protected] Abstract This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity. Key words
Hyperbolic mean curvature flow; self-similar solutions; curvature
2010 MR Subject Classification
1
35L10; 35C06
Introduction
In 2009, Kong, Liu, and Wang [6, 7] proposed the hyperbolic mean curvature flow (HMCF) for plane curves. More precisely, they considered the following initial value problem: 2 ∂ F 2 (z, t) = k(z, t)N (z, t) + ρ(z, t)T (z, t), ∀ (z, t) ∈ R × [0, T ), ∂t (1.1) F (z, 0) = F0 (z), ∂F (z, 0) = h(z)N0 (z), ∂t where F (z, t) denotes the unknown vector-valued function standing for the curve at time t, k(z, t) the mean curvature of F (z, t), N (z, t) the unit normal vector of F (z, t), T (z, t) the unit tangent vector of F (z, t), and F0 (z) the initial curve; while h(z) and N0 (z) are the magnitude of initial velocity and unit normal vector of initial curve F0 (z), respectively; the function ρ(z, t) is defined by 2 ∂ F ∂F ρ(z, t) = − , , (1.2) ∂s∂t ∂t in which s is the arclength parameter. The flow described by (1.1) is always normal, that is, the velocity filed is perpendicular to the curve during the evolution. Kong, Liu, and Wang [6] investigated closed plane curves under the HMCF. They found that system (1.1) can be reduced into a hyperbolic Monge-Amp`ere equation for the support function and showed that ∗ Received
July 22, 2015; revised October 24, 2016. This work was supported in part by a grant from China Scholarship Council, the National Natural Science Foundation of China (11301006), and the Anhui Provincial Natural Science Foundation (1408085MA01).
658
ACTA MATHEMATICA SCIENTIA
Vol.37 Ser.B
either the curve converges to a point, or shocks and other singularities are generated in finite time. Moreover, Kong and Wang [7] considered the motion of periodic plane curves under the HMCF. It was proved that if the total variation of initial curve is small enough in one period and some certain condition is satisfied, then singularities must develop in finite time and the lifespan was also given. Recently, Wang [9] extended the above discussions on HMCF to the case in Minkowski space, wherein the motion of immersed spacelike closed curves and the corresponding singularities were investigated. In this article, we study the self-similar solutions to the HMCF for plane curves. More precisely, we consider the following immersed plane curve: F (z, t) = g(t)eif (t) F0 (z) + H(t) ∀ (z, t) ∈ R × I,
(1.3)
which is a complex valued function. Here, F0 (z) : R → C is the immersed initial curve and is assumed to be nonzero; f, g : I → R and H : I → C are differentiable with f (0) = 0, g(0) = 1, and H(0) = 0. Obviously, the function f determines the rotation, g relates to the scaling, and H is the translation term. We shall show that all self-similar solutions evolving under the HMCF are either straight lines or circles. Moreover, one immersed circle always shrinks into a point, whenever the initial velocity vanishes or not. In addition, the time when a circle converges to a point is derived explicitly. These results presumably are reasonable because the HMCF is a normal one, however, the proof seems nontrivial. The geometric flow has been investigated for many years and can be dated back at least to Gage and Hamilton [2], wherein a curve shortening flow was studied. They showed that under the curve shortening flow, a strictly convex plane curve must shrink to a unit circle in a certain sense. Recently, Halldorsson [3] concerned all self-similar solutions to the curve shortening flow and gave a complete description and classification. It was shown that apart from straight lines and circles, there are many other immersed curves in the plane under this flow. Huisken and Ilmanen [5] developed a theory of the inverse mean curvature flow and succeeded in proving the Riemann Penrose inequality in general relativity. For hyperbolic version of geometric flows, He, Kong, and Liu [4] introduced the following hyperbolic mean curvature flow ∂2X (u, t) = H(u, t)N (u, t), ∀ u ∈ M, ∀ t > 0, ∂t2 where M is a Riemannian manifold, X(·, t) : M → Rn+1 , and H is the mean curvature and N is the unit inner normal vector. He, Kong, and Liu [4] derived a system of hyperbolic equations and obtained a uniquely short-time smooth solution. Some nonlinear wave equations for curvatures are also provided. Recently, Lefloch and Smoczyk [8] proposed an interesting hyperbolic geometric evolution equations, which describe the motion of a hypersurface along the direction of its mean curvature vector. The governing equations read ⊤ ∂2 ∂X ~ − ∇e, X = eH(u, t) N = 0, ∂t2 ∂t |t=0 ~ denotes its where the scalar H is the mean curvature of the hypersurface and the vector N 1 d 2 unit normal, e = 2 (| dt X| + n) is the local energy density. The second equation in the above equations means that the tangential part of initial velocity vanishes. They showed that this
No.3
C.L. He et al: SELF-SIMILAR SOLUTIONS TO HMCF
659
kind of curvature flow must blow up in finite time. Moreover, they pointed out that for a graph, the classical solution can exist uniquely in a larger class of entropy solutions. For onedimensional graphs of this flow, a global-in-time existence result is discussed. Moreover, an existence theorem for weak solutions has been established under the assumption that the BV norm of initial data is small. Most recently, Chou and Wo [1] proposed a new hyperbolic curvature flow for convex hypersurfaces. This flow is most suited when the Gauss curvature is involved. For different driving forces, they derived many new nonlinear hyperbolic equations. Local solvability to the hyperbolic Gauss curvature flow has been established and some results on finite time blow-up and asymptotic behavior are also given. This article is organized as follows. In Section 2, some preliminaries are provided. In Section 3, we consider the self-similar solutions to the HMCF when the translation term vanishes and derive a nonlinear ordinary differential equation of second order, from which we can obtain the evolution of plane curves under the HMCF. Section 4 is devoted to studying the self-similar curves under the HMCF by including the translation terms.
2
Some Preliminaries
In this section, we derive an equivalent form for the initial value problem (1.1) and then provide some preliminaries for convenience of discussion. First, we have the following lemma. Lemma 2.1 Finding the solutions to the initial value problem (1.1) for the HMCF is equivalent to considering the solutions to the following equations: 2 ∂ F ∂F , N = k (t > 0), , T = 0 (t ≥ 0). (2.1) ∂t2 ∂t Proof We first prove the necessity. By (1.1), it is easy to see that the first equation in (2.1) is valid. At initial time t = 0, the second equation in (2.1) holds because of the initial data in (1.1). Now, we need to show that it is still true for t > 0. By direct computation, we have 2 ∂ ∂F ∂F ∂ F ∂F ∂F ∂ 2 F , = , + , ∂t ∂t ∂z ∂t2 ∂z ∂t ∂z∂t ∂F ∂ 2 F ∂F = ρT, + , ∂z ∂t ∂z∂t ∂F ∂s ∂F ∂ 2 F ∂s = ρT, · + , · ∂s ∂z ∂t ∂s∂t ∂z ∂s ∂F ∂ 2 F = ρ+ , = 0, ∂z ∂t ∂s∂t where we use (1.1) and (1.2), and s denotes the arclength parameter. Then, ∂F ∂F ∂F ∂F , (z, t) = , (z, 0) = 0. ∂t ∂z ∂t ∂z Noting
∂F ∂F , ∂t ∂z
=0⇔
∂F ,T ∂t
= 0,
660
ACTA MATHEMATICA SCIENTIA
Vol.37 Ser.B
∂s ∂z
6 0, we complete the proof of necessity. = Now, we prove the sufficiency. By the second equation in (2.1), the initial data in (1.1) are valid. Because of the first equation in (2.1), we can assume
if
∂ 2F = kN + ρT. (2.2) ∂t2 Next, we only need to show that (1.2) is true. By the second equation in (2.1), we have ∂ ∂F ∂F , = 0, ∂t ∂t ∂s that is,
∂ 2 F ∂F , ∂t2 ∂s
+
∂F ∂ 2 F , ∂t ∂t∂s
= 0.
So, with the aid of (2.2), we have ρ+
∂F ∂ 2 F , ∂t ∂t∂s
= 0,
which is nothing but (1.2). The proof is completed.
By Lemma 2.1, we only need to consider the self-similar solutions to (2.1). By (1.3), direct calculation gives
and
∂F = [g ′ (t) + ig(t)f ′ (t)] eif (t) F0 (z) + H ′ (t), ∂t ∂2F = g ′′ (t) + 2ig ′ (t)f ′ (t) + ig(t)f ′′ (t) − g(t)(f ′ (t))2 eif (t) F0 (z) + H ′′ (t), ∂t2 g(t) if (t) ∂F/∂z = e T0 (z), T (z, t) = |∂F/∂z| |g(t)| k(z, t) =
′′ 1 |Fz′ × Fzz | = k0 (z), ′ 3 |Fz | |g(t)|
(2.3) (2.4) (2.5)
(2.6)
where T0 (z), k0 (z) are the unit tangent vector and curvature of the initial curve F0 (z), respectively. Remark 2.2 From (2.3), we see that for the self-similar solution (1.3), the initial velocity has the special form: Ft (z, 0) = (A + Bi)F0 (z) + C, (2.7) where A = g ′ (0), B = f ′ (0), and C = H ′ (0).
and
Therefore, by (2.3)–(2.6), Equation (2.1) can be reduced into D E g ′ (t) hF0 (z), T0 (z)i − g(t)f ′ (t) hF0 (z), N0 (z)i + e−if (t) H ′ (t), T0 (z) = 0, t ≥ 0, g(t)g ′′ (t) − (g(t)f ′ (t))2 hF0 (z), N0 (z)i + g(t) [2f ′ (t)g ′ (t) + g(t)f ′′ (t)] D E hF0 (z), T0 (z)i + g(t) e−if (t) H ′′ (t), N0 (z) = k0 (z), t > 0,
where we use the following facts:
N (z, t) = iT (z, t), Summarizing the above discussions, we have
N0 (z) = iT0 (z).
(2.8)
(2.9)
No.3
C.L. He et al: SELF-SIMILAR SOLUTIONS TO HMCF
661
Lemma 2.3 For the self-similar solution given by (1.3), Equation (2.1) are equivalent to the equations (2.8) and (2.9). From now on, we parameterize the curves by arc length, denoted by s. For every initial curve F0 (s), by the Frenet formulas, we easily have d d hF0 , T0 i = 1 + k0 hF0 , N0 i , hF0 , N0 i = −k0 hF0 , T0 i . ds ds By the same method in [3], we define the functions as x = A hF0 , T0 i − B hF0 , N0 i ,
(2.10)
y = B hF0 , T0 i + A hF0 , N0 i ,
which results in x + iy = (A + iB) (hF0 , T0 i + i hF0 , N0 i) .
(2.11)
Then, by (2.10) we find that x, y satisfy that x′ = A + k0 y,
y ′ = B − k0 x.
(2.12)
Moreover, if A2 + B 2 6= 0, it always holds that F0 = hF0 , T0 i T0 + hF0 , N0 i N0 = T0 (hF0 , T0 i + i hF0 , N0 i) x + iy = eiθ , A + iB Rs where we use (2.11) and θ(s) = 0 k0 (s′ )ds′ + θ0 , T0 (0) = eiθ0 .
3
(2.13)
Self-Similar Solutions With Vanishing Translation Terms
This section considers the self-similar solutions for the case that the translation term vanishes, that is, H(t) ≡ 0. The other case that H(t) 6= 0 will be discussed in the next section. As H(t) ≡ 0, equations (2.8)–(2.9) become g ′ (t) hF0 , T0 i − g(t)f ′ (t) hF0 , N0 i = 0, t ≥ 0, and g(t)g ′′ (t) − (g(t)f ′ (t))2 hF0 , N0 i + g(t) [2f ′ (t)g ′ (t) + g(t)f ′′ (t)] hF0 , T0 i = k0 , t > 0.
(3.1)
(3.2)
At time t = 0, Equation (3.1) can be simplified as
A hF0 , T0 i − B hF0 , N0 i = 0.
(3.3)
As we assume F0 (z) = hF0 (z), T0 (z)i T0 + hF0 (z), N0 (z)i N0 6= 0, so, hF0 (z), T0 (z)i and hF0 (z), N0 (z)i can not be zero simultaneously. Thus, it follows from (3.1) and (3.3) that Ag(t)f ′ (t) = Bg ′ (t). (3.4) Under the assumption H(t) ≡ 0, we have C = 0 and the initial velocity Ft (z, 0) = (A + Bi)F0 (z). We shall divide the following discussion into two cases: the initial velocity does not vanish, that is, A2 + B 2 6= 0 and the initial velocity equals zero, that is, A = B = 0. Case I A2 + B 2 6= 0.
662
ACTA MATHEMATICA SCIENTIA
Vol.37 Ser.B
When A = 0, by (3.4) and B 6= 0, we have g(t) ≡ 1. In addition, it follows from (3.3) that hF0 , N0 i ≡ 0. Substituting g(t) ≡ 1 and hF0 , N0 i ≡ 0 into (3.2) yields f ′′ (t)hF0 , T0 i = k0 , which implies that f (t) =
k0 t2 + Bt, 2hF0 , T0 i
0 where hF0k,T is constant. Moreover, by (2.10) we have k0 = 0, hF0 , T0 i = s+s0 , s0 is a constant. 0i So, we obtain the initial curve from (2.13): F0 (s) = (s + s0 )eiθ0 . Thus, we have the following result.
Theorem 3.1 When the translation term H(t) ≡ 0, the HMCF (1.1) admits the following self-similar solution: F (s, t) = (s + s0 )ei(Bt+θ0 ) , (3.5) where B is an arbitrary nonzero constant. Obviously, this solution is nothing but a straight line. When A 6= 0, noting Equation (3.4), it follows from (3.2) that g(A2 f ′ g ′ + ABg ′′ ) hF0 , T0 i + (A2 gg ′′ − B 2 (g ′ )2 ) hF0 , N0 i = A2 k0 , which can be reduced into B(g ′ )2 [A hF0 , T0 i − B hF0 , N0 i] + Agg ′′ [B hF0 , T0 i + A hF0 , N0 i] = A2 k0 .
(3.6)
Here, we use (3.4) again. Thus, noting (3.3), we obtain from (3.6) gg ′′ [B hF0 , T0 i + A hF0 , N0 i] = Ak0 .
(3.7)
Remark 3.2 Here, we point out that B hF0 , T0 i + A hF0 , N0 i 6= 0. Otherwise, a combination of B hF0 , T0 i + A hF0 , N0 i = 0 and (3.3) gives hF0 , T0 i = hF0 , N0 i = 0 and then F0 (s) = hF0 , T0 i T0 + hF0 , N0 i N0 = 0, which contradicts to our previous assumption that F0 (s) does not vanish. Then, by the definitions of x, y at the end of Section 2, Equation (3.3), and Equation (3.7), we have x(s) ≡ 0, g(t)g ′′ (t)y(s) = Ak0 (s). (3.8) Moreover, the nonlinear ordinary differential equations (2.12) are reduced to A + k0 (s)y(s) = 0,
y ′ (s) = B.
(3.9)
A combination of (3.8)2 and (3.9)1 gives g(t)g ′′ (t) = −k02 (s),
(3.10)
which implies that the curvature k0 (s) is a constant. So, it is obtained from (3.9) that y(s) = − kA0 is a nonzero constant and B = 0, where we assume k0 6= 0, because if k0 = 0, we have
No.3
C.L. He et al: SELF-SIMILAR SOLUTIONS TO HMCF
663
A = 0 and this case is reduced into the previous one. Thus, from (2.13) we have the following initial curve 1 F0 (s) = − iei(k0 s+θ0 ) . k0 To completely determine the self-similar solution (1.3), it suffices to solve g(t), which satisfies the following initial value problem for an ODE of second order: g(t)g ′′ (t) = −k02 , (3.11) t = 0 : g = 1, g ′ = A. Multiplying g ′ on both sides of (3.11)1 and integrating, we have (g ′ )2 = −2k02 log g + A2 , and then
q dg = ± −2k02 log g + A2 . (3.12) dt When the initial velocity A > 0, we can see from (3.12) that g(t) will increase to a maximum, A2 2
where −2k02 log g + A2 =0, that is, g = e 2k0 and then decrease to zero; see Figure 1.
Figure 1
The solution curve for (3.11) when A > 0,
A2
where t1 =
R e 2k02 1
A2
√ 2dv −2k0 log v+A2
and T1 = t1 +
R e 2k02 0
√
dv 2 log v+A2 −2k0
Before the function g reaches the maximum, we obtain from (3.12) q dg = −2k02 log g + A2 dt, and then t=
Z
1
g
dv p . 2 −2k0 log v + A2
So, the time when g takes the maximum is given by A2 2
t1 =
Z
1
e 2k0
dv p . −2k02 log v + A2
Furthermore, after the time t1 , g begins to decrease. So, it follows from (3.12) that q dg = − −2k02 log g + A2 dt,
664
ACTA MATHEMATICA SCIENTIA
and then
Vol.37 Ser.B
A2 2
t − t1 =
Z
e 2k0 g
Thus, the time when g = 0 can be given as A2 2
T1 =
Z
dv p . −2k02 log v + A2
A2 2
e 2k0
+
1
e 2k0
Z
0
dv p < ∞. 2 −2k0 log v + A2
(3.13)
When the initial velocity A < 0, we can observe that g(t) will decrease directly to zero; see Figure 2.
Figure 2
The solution curve for (3.11) when A < 0, where T2 =
R1 0
√
dv 2 log v+A2 −2k0
So, by (3.12) we have dg = −
q −2k02 log g + A2 dt,
and then, t=
Z
1
1
dv p < ∞. 2 −2k0 log v + A2
g
The time when g = 0 is given by T2 =
Z
0
dv p . 2 −2k0 log v + A2 (3.14)
As a consequence, once the dilation fucntion g(t) is obtained, one can determine the rotation fucniton f (t) from (3.4) and get f (t) ≡ 0. From the above argument, we have Theorem 3.3 When the translation term H(t) ≡ 0, the HMCF (1.1) admits the following self-similar solution: g(t) i(k0 s+θ0 ) F (s, t) = − ie , (3.15) k0 where k0 is the nonzero constant curvature of initial curve and g(t) is determined by (3.11). Obviously, this solution is a circle, which expands to a bigger one and then shrinks to a point as t → T1 when A > 0, or just shrinks to a point as t → T2 when A < 0, where the constant A relates to the initial velocity because Ft (s, 0) = AF0 (s). Moreover, by (2.6), we see that in both cases, the curvature for the limit circle approaches to infinity as t → T1 or t → T2 .
No.3
C.L. He et al: SELF-SIMILAR SOLUTIONS TO HMCF
665
These results are consistent with that in Theorem 1.2 by Kong, Liu, and Wang [6]. However, here we can describe the evolution of a circle under the HMCF much more clear. Case II A = B = 0. For this case, Equation (3.3) is satisfied automatically and we only need to investigate Equations (3.1) and (3.2) for t > 0. When g ′ (t) ≡ 0, that is, g(t) ≡ 1, it follows from (3.1) that f ′ (t)hF0 , N0 i = 0. If f ′ (t) = 0, then f (t) ≡ 0 and by (3.2), we have k0 = 0. It means that the initial curve F0 (s) is a straight line. So, the corresponding self-similar solution to the HMCF (1.1) is given by F (s, t) = F0 (s).
(3.16)
If f ′ (t) 6= 0, then hF0 , N0 i = 0. Because of (2.10)2 , we have k0 = 0. At this time, (3.2) is simplified to f ′′ (t)hF0 , T0 i = 0 and in turn implies f ′′ (t) = 0. As B = f ′ (0) = 0, we obtain f ′ (t) = 0, a contradiction. When g ′ (t) 6= 0, it follows from (3.1) that g(t)f ′ (t) = µ g ′ (t),
(3.17)
where µ is a constant. Substituting (3.17) into (3.1) gives hF0 , T0 i − µ hF0 , N0 i = 0.
(3.18)
By a similar discussion in Case I and (3.18), Equation (3.2) is reduced to g(t)g ′′ (t) [µ hF0 , T0 i + hF0 , N0 i] = k0 (s).
(3.19)
Introduce x = hF0 , T0 i − µ hF0 , N0 i ,
y = µ hF0 , T0 i + hF0 , N0 i .
Then, by Frenet formulas we have 1 + k0 y = 0,
y ′ = µ.
(3.20)
Note that the first equation in (3.20), Equation (3.19) can be rewritten as g(t)g ′′ (t) = −k02 (s), which indicates that k0 (s) is a constant. Then, it follows from (3.20) that k0 6= 0, y = − k10 , and µ = 0. By the same method at the end of Section 2, we still obtain the initial curve as in (3.15). Now, we need to consider the following initial value problem for g(t): g(t)g ′′ (t) = −k02 , (3.21) t = 0 : g = 1, g ′ = 0. This initial value problem is different from (3.11) because here the initial velocity of g vanishes. It is easy to deduce from (3.21) that (g ′ )2 = −2k02 log g. So, after t = 0, g becomes less than 1. This means that g begins to decrease, that is, we have q g ′ = − −2k02 log g.
666
ACTA MATHEMATICA SCIENTIA
Furthermore, t=
Z
1
1
dv p < ∞. −2k02 log v
g
Thus, the time when g = 0 is given as T3 =
Z
0
Vol.37 Ser.B
dv p . −2k02 log v (3.22)
See Figure 3. So, by (3.17), it turns out that f (t) ≡ 0.
Figure 3
The solution curve for (3.21), where T3 =
R1 0
√
dv 2 log v −2k0
Summarizing the above discussions leads to Theorem 3.4 When the translation term H(t) ≡ 0 and the initial velocity vanishes, the HMCF (1.1) admits the following self-similar solutions: F (s, t) = F0 (s),
or F (s, t) = −
g(t) i(k0 s+θ0 ) ie , k0
(3.23)
where F0 (s) is a straight line, k0 is the nonzero constant curvature of initial curve, and g(t) is determined by (3.21). Obviously, the second one in (3.23) is a circle, which shrinks to a point and the curvature for the limit circle approaches to infinity as t → T3 .
4
Self-Similar Solutions With Nonvanishing Translation Terms
This section is devoted to studying the self-similar solution (1.3) to the HMCF (1.1) by including the translation terms H(t). Without any confusion, we change the notations of initial data in (1.1) into F (s, 0) = F¯ (s), Ft (s, 0) = (A + Bi)F¯0 (s) + C, where A = g ′ (0), B = f ′ (0), and C = H ′ (0). The self-similar solution (1.3) is rewritten as F (s, t) = g(t)eif (t) F¯0 (s) + H(t). Thus, f (t), g(t), H(t), and F¯0 (s) satisfy the equations (2.8) and (2.9). For the case that A2 + B 2 6= 0, we set 1 iB eif (t) − 1 C, A = 0, C F¯0 (z) = F0 (z) − , H(t) = 1 A + iB g(t)eif (t) − 1 C, A = 6 0; A + iB
(4.1)
No.3
C.L. He et al: SELF-SIMILAR SOLUTIONS TO HMCF
667
while, for the case that A = B = 0, let F¯0 (s) = F0 (s) − D,
H(t) = g(t)eif (t) − 1 D,
(4.2)
where D is a constant vector. Then, a straightforward calculation shows that for these two cases, f (t), g(t), H(t) and F¯0 (s) satisfy (2.8)–(2.9) if and only if f (t), g(t), F0 (z) satisfy Equations (3.1), (3.2). For simplicity, the details are omitted here. Thus, by Theorems 3.1, 3.3, and 3.4 in Section 3, we obtain the following results for the case H(t) 6= 0. Theorem 4.1 The HMCF (1) admits the following self-similar solutions: F (s, t) = F0 (s) − D,
C F (s, t) = (s + s0 )ei(Bt+θ0 ) − (B 6= 0), iB g(t) i(k0 s+θ0 ) C F (s, t) = − (A 6= 0), ie − k0 A and F (s, t) = −
g(t) i(k0 s+θ0 ) ie − D, k0
(4.3) (4.4) (4.5)
(4.6)
where F0 (s) is a straight line, D a constant vector. Moreover, the dilation function g(t) in (4.5) and (4.6) is determined by (3.11) and (3.21), respectively. Comparing Theorem 4.1 with Theorems 3.1, 3.3, and 3.4, we find that introducing the translation term does not provide any new solution from the view point of geometry. Acknowledgements This work was completed when C.-L. He and S.-J. Huang were visiting Professor Huai-Dong Cao at Lehigh University. They would like to thank Professor Cao and the Mathematics Department for great hospitality. References [1] Chou K S, Wo W F. On hyperbolic Gauss curvature flows. J Differential Geom, 2011, 89: 455–485 [2] Gage M, Hamilton R. The heat equation shrinking concex plane curves. J Differential Geom, 1986, 23: 417–491 [3] Halldorsson H P. Self-similar solutions to the curve shortening flow. Transactions of the American Mathematical Society, 2012, 364: 5285–5309 [4] He C L. Kong D X, Liu K F. Hyperbolic mean curvature flow. J Differential Equations, 2009, 246: 373–390 [5] Huisken G, Ilmanen T. The inverse mean curvature flow and the Riemann Penrose inequality. J Differential Geom, 2001, 59: 353–437 [6] Kong D X, Liu K F, Wang Z G. Hyperbolic mean curvature flow: evolution of plane curves. Acta Mathematica Scientia, 2009, 29: 493–514 [7] Kong D X, Wang Z G. Formation of singularities in the motion of plane curves under hyperbolic mean curvature flow. J Differential Equations, 2009, 247: 1694–1719 [8] LeFloch P G, Smoczyk K. The hyperbolic mean curvature flow. J Math Pures Appl, 2008, 90: 591–614 [9] Wang Z G. Hyperbolic mean curvature flow in Minkowski space. Nonlinear Analysis, 2014, 94: 259–271 [10] Wang Z G. Blow-up of periodic solutions to reducible quasilinear hyperbolic systems. Nonlinear Analysis, 2010, 73: 704–712
SELF-SIMILAR SOLUTIONS TO THE HYPERBOLIC MEAN CURVATURE FLOW∗
ÛSZ)
Chunlei HE (
Å)
Shoujun HUANG (
0¡¯)
Xiaomin XING (
School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China E-mail : [email protected]; [email protected]; [email protected] Abstract This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity. Key words
Hyperbolic mean curvature flow; self-similar solutions; curvature
2010 MR Subject Classification
1
35L10; 35C06
Introduction
In 2009, Kong, Liu, and Wang [6, 7] proposed the hyperbolic mean curvature flow (HMCF) for plane curves. More precisely, they considered the following initial value problem: 2 ∂ F 2 (z, t) = k(z, t)N (z, t) + ρ(z, t)T (z, t), ∀ (z, t) ∈ R × [0, T ), ∂t (1.1) F (z, 0) = F0 (z), ∂F (z, 0) = h(z)N0 (z), ∂t where F (z, t) denotes the unknown vector-valued function standing for the curve at time t, k(z, t) the mean curvature of F (z, t), N (z, t) the unit normal vector of F (z, t), T (z, t) the unit tangent vector of F (z, t), and F0 (z) the initial curve; while h(z) and N0 (z) are the magnitude of initial velocity and unit normal vector of initial curve F0 (z), respectively; the function ρ(z, t) is defined by 2 ∂ F ∂F ρ(z, t) = − , , (1.2) ∂s∂t ∂t in which s is the arclength parameter. The flow described by (1.1) is always normal, that is, the velocity filed is perpendicular to the curve during the evolution. Kong, Liu, and Wang [6] investigated closed plane curves under the HMCF. They found that system (1.1) can be reduced into a hyperbolic Monge-Amp`ere equation for the support function and showed that ∗ Received
July 22, 2015; revised October 24, 2016. This work was supported in part by a grant from China Scholarship Council, the National Natural Science Foundation of China (11301006), and the Anhui Provincial Natural Science Foundation (1408085MA01).
658
ACTA MATHEMATICA SCIENTIA
Vol.37 Ser.B
either the curve converges to a point, or shocks and other singularities are generated in finite time. Moreover, Kong and Wang [7] considered the motion of periodic plane curves under the HMCF. It was proved that if the total variation of initial curve is small enough in one period and some certain condition is satisfied, then singularities must develop in finite time and the lifespan was also given. Recently, Wang [9] extended the above discussions on HMCF to the case in Minkowski space, wherein the motion of immersed spacelike closed curves and the corresponding singularities were investigated. In this article, we study the self-similar solutions to the HMCF for plane curves. More precisely, we consider the following immersed plane curve: F (z, t) = g(t)eif (t) F0 (z) + H(t) ∀ (z, t) ∈ R × I,
(1.3)
which is a complex valued function. Here, F0 (z) : R → C is the immersed initial curve and is assumed to be nonzero; f, g : I → R and H : I → C are differentiable with f (0) = 0, g(0) = 1, and H(0) = 0. Obviously, the function f determines the rotation, g relates to the scaling, and H is the translation term. We shall show that all self-similar solutions evolving under the HMCF are either straight lines or circles. Moreover, one immersed circle always shrinks into a point, whenever the initial velocity vanishes or not. In addition, the time when a circle converges to a point is derived explicitly. These results presumably are reasonable because the HMCF is a normal one, however, the proof seems nontrivial. The geometric flow has been investigated for many years and can be dated back at least to Gage and Hamilton [2], wherein a curve shortening flow was studied. They showed that under the curve shortening flow, a strictly convex plane curve must shrink to a unit circle in a certain sense. Recently, Halldorsson [3] concerned all self-similar solutions to the curve shortening flow and gave a complete description and classification. It was shown that apart from straight lines and circles, there are many other immersed curves in the plane under this flow. Huisken and Ilmanen [5] developed a theory of the inverse mean curvature flow and succeeded in proving the Riemann Penrose inequality in general relativity. For hyperbolic version of geometric flows, He, Kong, and Liu [4] introduced the following hyperbolic mean curvature flow ∂2X (u, t) = H(u, t)N (u, t), ∀ u ∈ M, ∀ t > 0, ∂t2 where M is a Riemannian manifold, X(·, t) : M → Rn+1 , and H is the mean curvature and N is the unit inner normal vector. He, Kong, and Liu [4] derived a system of hyperbolic equations and obtained a uniquely short-time smooth solution. Some nonlinear wave equations for curvatures are also provided. Recently, Lefloch and Smoczyk [8] proposed an interesting hyperbolic geometric evolution equations, which describe the motion of a hypersurface along the direction of its mean curvature vector. The governing equations read ⊤ ∂2 ∂X ~ − ∇e, X = eH(u, t) N = 0, ∂t2 ∂t |t=0 ~ denotes its where the scalar H is the mean curvature of the hypersurface and the vector N 1 d 2 unit normal, e = 2 (| dt X| + n) is the local energy density. The second equation in the above equations means that the tangential part of initial velocity vanishes. They showed that this
No.3
C.L. He et al: SELF-SIMILAR SOLUTIONS TO HMCF
659
kind of curvature flow must blow up in finite time. Moreover, they pointed out that for a graph, the classical solution can exist uniquely in a larger class of entropy solutions. For onedimensional graphs of this flow, a global-in-time existence result is discussed. Moreover, an existence theorem for weak solutions has been established under the assumption that the BV norm of initial data is small. Most recently, Chou and Wo [1] proposed a new hyperbolic curvature flow for convex hypersurfaces. This flow is most suited when the Gauss curvature is involved. For different driving forces, they derived many new nonlinear hyperbolic equations. Local solvability to the hyperbolic Gauss curvature flow has been established and some results on finite time blow-up and asymptotic behavior are also given. This article is organized as follows. In Section 2, some preliminaries are provided. In Section 3, we consider the self-similar solutions to the HMCF when the translation term vanishes and derive a nonlinear ordinary differential equation of second order, from which we can obtain the evolution of plane curves under the HMCF. Section 4 is devoted to studying the self-similar curves under the HMCF by including the translation terms.
2
Some Preliminaries
In this section, we derive an equivalent form for the initial value problem (1.1) and then provide some preliminaries for convenience of discussion. First, we have the following lemma. Lemma 2.1 Finding the solutions to the initial value problem (1.1) for the HMCF is equivalent to considering the solutions to the following equations: 2 ∂ F ∂F , N = k (t > 0), , T = 0 (t ≥ 0). (2.1) ∂t2 ∂t Proof We first prove the necessity. By (1.1), it is easy to see that the first equation in (2.1) is valid. At initial time t = 0, the second equation in (2.1) holds because of the initial data in (1.1). Now, we need to show that it is still true for t > 0. By direct computation, we have 2 ∂ ∂F ∂F ∂ F ∂F ∂F ∂ 2 F , = , + , ∂t ∂t ∂z ∂t2 ∂z ∂t ∂z∂t ∂F ∂ 2 F ∂F = ρT, + , ∂z ∂t ∂z∂t ∂F ∂s ∂F ∂ 2 F ∂s = ρT, · + , · ∂s ∂z ∂t ∂s∂t ∂z ∂s ∂F ∂ 2 F = ρ+ , = 0, ∂z ∂t ∂s∂t where we use (1.1) and (1.2), and s denotes the arclength parameter. Then, ∂F ∂F ∂F ∂F , (z, t) = , (z, 0) = 0. ∂t ∂z ∂t ∂z Noting
∂F ∂F , ∂t ∂z
=0⇔
∂F ,T ∂t
= 0,
660
ACTA MATHEMATICA SCIENTIA
Vol.37 Ser.B
∂s ∂z
6 0, we complete the proof of necessity. = Now, we prove the sufficiency. By the second equation in (2.1), the initial data in (1.1) are valid. Because of the first equation in (2.1), we can assume
if
∂ 2F = kN + ρT. (2.2) ∂t2 Next, we only need to show that (1.2) is true. By the second equation in (2.1), we have ∂ ∂F ∂F , = 0, ∂t ∂t ∂s that is,
∂ 2 F ∂F , ∂t2 ∂s
+
∂F ∂ 2 F , ∂t ∂t∂s
= 0.
So, with the aid of (2.2), we have ρ+
∂F ∂ 2 F , ∂t ∂t∂s
= 0,
which is nothing but (1.2). The proof is completed.
By Lemma 2.1, we only need to consider the self-similar solutions to (2.1). By (1.3), direct calculation gives
and
∂F = [g ′ (t) + ig(t)f ′ (t)] eif (t) F0 (z) + H ′ (t), ∂t ∂2F = g ′′ (t) + 2ig ′ (t)f ′ (t) + ig(t)f ′′ (t) − g(t)(f ′ (t))2 eif (t) F0 (z) + H ′′ (t), ∂t2 g(t) if (t) ∂F/∂z = e T0 (z), T (z, t) = |∂F/∂z| |g(t)| k(z, t) =
′′ 1 |Fz′ × Fzz | = k0 (z), ′ 3 |Fz | |g(t)|
(2.3) (2.4) (2.5)
(2.6)
where T0 (z), k0 (z) are the unit tangent vector and curvature of the initial curve F0 (z), respectively. Remark 2.2 From (2.3), we see that for the self-similar solution (1.3), the initial velocity has the special form: Ft (z, 0) = (A + Bi)F0 (z) + C, (2.7) where A = g ′ (0), B = f ′ (0), and C = H ′ (0).
and
Therefore, by (2.3)–(2.6), Equation (2.1) can be reduced into D E g ′ (t) hF0 (z), T0 (z)i − g(t)f ′ (t) hF0 (z), N0 (z)i + e−if (t) H ′ (t), T0 (z) = 0, t ≥ 0, g(t)g ′′ (t) − (g(t)f ′ (t))2 hF0 (z), N0 (z)i + g(t) [2f ′ (t)g ′ (t) + g(t)f ′′ (t)] D E hF0 (z), T0 (z)i + g(t) e−if (t) H ′′ (t), N0 (z) = k0 (z), t > 0,
where we use the following facts:
N (z, t) = iT (z, t), Summarizing the above discussions, we have
N0 (z) = iT0 (z).
(2.8)
(2.9)
No.3
C.L. He et al: SELF-SIMILAR SOLUTIONS TO HMCF
661
Lemma 2.3 For the self-similar solution given by (1.3), Equation (2.1) are equivalent to the equations (2.8) and (2.9). From now on, we parameterize the curves by arc length, denoted by s. For every initial curve F0 (s), by the Frenet formulas, we easily have d d hF0 , T0 i = 1 + k0 hF0 , N0 i , hF0 , N0 i = −k0 hF0 , T0 i . ds ds By the same method in [3], we define the functions as x = A hF0 , T0 i − B hF0 , N0 i ,
(2.10)
y = B hF0 , T0 i + A hF0 , N0 i ,
which results in x + iy = (A + iB) (hF0 , T0 i + i hF0 , N0 i) .
(2.11)
Then, by (2.10) we find that x, y satisfy that x′ = A + k0 y,
y ′ = B − k0 x.
(2.12)
Moreover, if A2 + B 2 6= 0, it always holds that F0 = hF0 , T0 i T0 + hF0 , N0 i N0 = T0 (hF0 , T0 i + i hF0 , N0 i) x + iy = eiθ , A + iB Rs where we use (2.11) and θ(s) = 0 k0 (s′ )ds′ + θ0 , T0 (0) = eiθ0 .
3
(2.13)
Self-Similar Solutions With Vanishing Translation Terms
This section considers the self-similar solutions for the case that the translation term vanishes, that is, H(t) ≡ 0. The other case that H(t) 6= 0 will be discussed in the next section. As H(t) ≡ 0, equations (2.8)–(2.9) become g ′ (t) hF0 , T0 i − g(t)f ′ (t) hF0 , N0 i = 0, t ≥ 0, and g(t)g ′′ (t) − (g(t)f ′ (t))2 hF0 , N0 i + g(t) [2f ′ (t)g ′ (t) + g(t)f ′′ (t)] hF0 , T0 i = k0 , t > 0.
(3.1)
(3.2)
At time t = 0, Equation (3.1) can be simplified as
A hF0 , T0 i − B hF0 , N0 i = 0.
(3.3)
As we assume F0 (z) = hF0 (z), T0 (z)i T0 + hF0 (z), N0 (z)i N0 6= 0, so, hF0 (z), T0 (z)i and hF0 (z), N0 (z)i can not be zero simultaneously. Thus, it follows from (3.1) and (3.3) that Ag(t)f ′ (t) = Bg ′ (t). (3.4) Under the assumption H(t) ≡ 0, we have C = 0 and the initial velocity Ft (z, 0) = (A + Bi)F0 (z). We shall divide the following discussion into two cases: the initial velocity does not vanish, that is, A2 + B 2 6= 0 and the initial velocity equals zero, that is, A = B = 0. Case I A2 + B 2 6= 0.
662
ACTA MATHEMATICA SCIENTIA
Vol.37 Ser.B
When A = 0, by (3.4) and B 6= 0, we have g(t) ≡ 1. In addition, it follows from (3.3) that hF0 , N0 i ≡ 0. Substituting g(t) ≡ 1 and hF0 , N0 i ≡ 0 into (3.2) yields f ′′ (t)hF0 , T0 i = k0 , which implies that f (t) =
k0 t2 + Bt, 2hF0 , T0 i
0 where hF0k,T is constant. Moreover, by (2.10) we have k0 = 0, hF0 , T0 i = s+s0 , s0 is a constant. 0i So, we obtain the initial curve from (2.13): F0 (s) = (s + s0 )eiθ0 . Thus, we have the following result.
Theorem 3.1 When the translation term H(t) ≡ 0, the HMCF (1.1) admits the following self-similar solution: F (s, t) = (s + s0 )ei(Bt+θ0 ) , (3.5) where B is an arbitrary nonzero constant. Obviously, this solution is nothing but a straight line. When A 6= 0, noting Equation (3.4), it follows from (3.2) that g(A2 f ′ g ′ + ABg ′′ ) hF0 , T0 i + (A2 gg ′′ − B 2 (g ′ )2 ) hF0 , N0 i = A2 k0 , which can be reduced into B(g ′ )2 [A hF0 , T0 i − B hF0 , N0 i] + Agg ′′ [B hF0 , T0 i + A hF0 , N0 i] = A2 k0 .
(3.6)
Here, we use (3.4) again. Thus, noting (3.3), we obtain from (3.6) gg ′′ [B hF0 , T0 i + A hF0 , N0 i] = Ak0 .
(3.7)
Remark 3.2 Here, we point out that B hF0 , T0 i + A hF0 , N0 i 6= 0. Otherwise, a combination of B hF0 , T0 i + A hF0 , N0 i = 0 and (3.3) gives hF0 , T0 i = hF0 , N0 i = 0 and then F0 (s) = hF0 , T0 i T0 + hF0 , N0 i N0 = 0, which contradicts to our previous assumption that F0 (s) does not vanish. Then, by the definitions of x, y at the end of Section 2, Equation (3.3), and Equation (3.7), we have x(s) ≡ 0, g(t)g ′′ (t)y(s) = Ak0 (s). (3.8) Moreover, the nonlinear ordinary differential equations (2.12) are reduced to A + k0 (s)y(s) = 0,
y ′ (s) = B.
(3.9)
A combination of (3.8)2 and (3.9)1 gives g(t)g ′′ (t) = −k02 (s),
(3.10)
which implies that the curvature k0 (s) is a constant. So, it is obtained from (3.9) that y(s) = − kA0 is a nonzero constant and B = 0, where we assume k0 6= 0, because if k0 = 0, we have
No.3
C.L. He et al: SELF-SIMILAR SOLUTIONS TO HMCF
663
A = 0 and this case is reduced into the previous one. Thus, from (2.13) we have the following initial curve 1 F0 (s) = − iei(k0 s+θ0 ) . k0 To completely determine the self-similar solution (1.3), it suffices to solve g(t), which satisfies the following initial value problem for an ODE of second order: g(t)g ′′ (t) = −k02 , (3.11) t = 0 : g = 1, g ′ = A. Multiplying g ′ on both sides of (3.11)1 and integrating, we have (g ′ )2 = −2k02 log g + A2 , and then
q dg = ± −2k02 log g + A2 . (3.12) dt When the initial velocity A > 0, we can see from (3.12) that g(t) will increase to a maximum, A2 2
where −2k02 log g + A2 =0, that is, g = e 2k0 and then decrease to zero; see Figure 1.
Figure 1
The solution curve for (3.11) when A > 0,
A2
where t1 =
R e 2k02 1
A2
√ 2dv −2k0 log v+A2
and T1 = t1 +
R e 2k02 0
√
dv 2 log v+A2 −2k0
Before the function g reaches the maximum, we obtain from (3.12) q dg = −2k02 log g + A2 dt, and then t=
Z
1
g
dv p . 2 −2k0 log v + A2
So, the time when g takes the maximum is given by A2 2
t1 =
Z
1
e 2k0
dv p . −2k02 log v + A2
Furthermore, after the time t1 , g begins to decrease. So, it follows from (3.12) that q dg = − −2k02 log g + A2 dt,
664
ACTA MATHEMATICA SCIENTIA
and then
Vol.37 Ser.B
A2 2
t − t1 =
Z
e 2k0 g
Thus, the time when g = 0 can be given as A2 2
T1 =
Z
dv p . −2k02 log v + A2
A2 2
e 2k0
+
1
e 2k0
Z
0
dv p < ∞. 2 −2k0 log v + A2
(3.13)
When the initial velocity A < 0, we can observe that g(t) will decrease directly to zero; see Figure 2.
Figure 2
The solution curve for (3.11) when A < 0, where T2 =
R1 0
√
dv 2 log v+A2 −2k0
So, by (3.12) we have dg = −
q −2k02 log g + A2 dt,
and then, t=
Z
1
1
dv p < ∞. 2 −2k0 log v + A2
g
The time when g = 0 is given by T2 =
Z
0
dv p . 2 −2k0 log v + A2 (3.14)
As a consequence, once the dilation fucntion g(t) is obtained, one can determine the rotation fucniton f (t) from (3.4) and get f (t) ≡ 0. From the above argument, we have Theorem 3.3 When the translation term H(t) ≡ 0, the HMCF (1.1) admits the following self-similar solution: g(t) i(k0 s+θ0 ) F (s, t) = − ie , (3.15) k0 where k0 is the nonzero constant curvature of initial curve and g(t) is determined by (3.11). Obviously, this solution is a circle, which expands to a bigger one and then shrinks to a point as t → T1 when A > 0, or just shrinks to a point as t → T2 when A < 0, where the constant A relates to the initial velocity because Ft (s, 0) = AF0 (s). Moreover, by (2.6), we see that in both cases, the curvature for the limit circle approaches to infinity as t → T1 or t → T2 .
No.3
C.L. He et al: SELF-SIMILAR SOLUTIONS TO HMCF
665
These results are consistent with that in Theorem 1.2 by Kong, Liu, and Wang [6]. However, here we can describe the evolution of a circle under the HMCF much more clear. Case II A = B = 0. For this case, Equation (3.3) is satisfied automatically and we only need to investigate Equations (3.1) and (3.2) for t > 0. When g ′ (t) ≡ 0, that is, g(t) ≡ 1, it follows from (3.1) that f ′ (t)hF0 , N0 i = 0. If f ′ (t) = 0, then f (t) ≡ 0 and by (3.2), we have k0 = 0. It means that the initial curve F0 (s) is a straight line. So, the corresponding self-similar solution to the HMCF (1.1) is given by F (s, t) = F0 (s).
(3.16)
If f ′ (t) 6= 0, then hF0 , N0 i = 0. Because of (2.10)2 , we have k0 = 0. At this time, (3.2) is simplified to f ′′ (t)hF0 , T0 i = 0 and in turn implies f ′′ (t) = 0. As B = f ′ (0) = 0, we obtain f ′ (t) = 0, a contradiction. When g ′ (t) 6= 0, it follows from (3.1) that g(t)f ′ (t) = µ g ′ (t),
(3.17)
where µ is a constant. Substituting (3.17) into (3.1) gives hF0 , T0 i − µ hF0 , N0 i = 0.
(3.18)
By a similar discussion in Case I and (3.18), Equation (3.2) is reduced to g(t)g ′′ (t) [µ hF0 , T0 i + hF0 , N0 i] = k0 (s).
(3.19)
Introduce x = hF0 , T0 i − µ hF0 , N0 i ,
y = µ hF0 , T0 i + hF0 , N0 i .
Then, by Frenet formulas we have 1 + k0 y = 0,
y ′ = µ.
(3.20)
Note that the first equation in (3.20), Equation (3.19) can be rewritten as g(t)g ′′ (t) = −k02 (s), which indicates that k0 (s) is a constant. Then, it follows from (3.20) that k0 6= 0, y = − k10 , and µ = 0. By the same method at the end of Section 2, we still obtain the initial curve as in (3.15). Now, we need to consider the following initial value problem for g(t): g(t)g ′′ (t) = −k02 , (3.21) t = 0 : g = 1, g ′ = 0. This initial value problem is different from (3.11) because here the initial velocity of g vanishes. It is easy to deduce from (3.21) that (g ′ )2 = −2k02 log g. So, after t = 0, g becomes less than 1. This means that g begins to decrease, that is, we have q g ′ = − −2k02 log g.
666
ACTA MATHEMATICA SCIENTIA
Furthermore, t=
Z
1
1
dv p < ∞. −2k02 log v
g
Thus, the time when g = 0 is given as T3 =
Z
0
Vol.37 Ser.B
dv p . −2k02 log v (3.22)
See Figure 3. So, by (3.17), it turns out that f (t) ≡ 0.
Figure 3
The solution curve for (3.21), where T3 =
R1 0
√
dv 2 log v −2k0
Summarizing the above discussions leads to Theorem 3.4 When the translation term H(t) ≡ 0 and the initial velocity vanishes, the HMCF (1.1) admits the following self-similar solutions: F (s, t) = F0 (s),
or F (s, t) = −
g(t) i(k0 s+θ0 ) ie , k0
(3.23)
where F0 (s) is a straight line, k0 is the nonzero constant curvature of initial curve, and g(t) is determined by (3.21). Obviously, the second one in (3.23) is a circle, which shrinks to a point and the curvature for the limit circle approaches to infinity as t → T3 .
4
Self-Similar Solutions With Nonvanishing Translation Terms
This section is devoted to studying the self-similar solution (1.3) to the HMCF (1.1) by including the translation terms H(t). Without any confusion, we change the notations of initial data in (1.1) into F (s, 0) = F¯ (s), Ft (s, 0) = (A + Bi)F¯0 (s) + C, where A = g ′ (0), B = f ′ (0), and C = H ′ (0). The self-similar solution (1.3) is rewritten as F (s, t) = g(t)eif (t) F¯0 (s) + H(t). Thus, f (t), g(t), H(t), and F¯0 (s) satisfy the equations (2.8) and (2.9). For the case that A2 + B 2 6= 0, we set 1 iB eif (t) − 1 C, A = 0, C F¯0 (z) = F0 (z) − , H(t) = 1 A + iB g(t)eif (t) − 1 C, A = 6 0; A + iB
(4.1)
No.3
C.L. He et al: SELF-SIMILAR SOLUTIONS TO HMCF
667
while, for the case that A = B = 0, let F¯0 (s) = F0 (s) − D,
H(t) = g(t)eif (t) − 1 D,
(4.2)
where D is a constant vector. Then, a straightforward calculation shows that for these two cases, f (t), g(t), H(t) and F¯0 (s) satisfy (2.8)–(2.9) if and only if f (t), g(t), F0 (z) satisfy Equations (3.1), (3.2). For simplicity, the details are omitted here. Thus, by Theorems 3.1, 3.3, and 3.4 in Section 3, we obtain the following results for the case H(t) 6= 0. Theorem 4.1 The HMCF (1) admits the following self-similar solutions: F (s, t) = F0 (s) − D,
C F (s, t) = (s + s0 )ei(Bt+θ0 ) − (B 6= 0), iB g(t) i(k0 s+θ0 ) C F (s, t) = − (A 6= 0), ie − k0 A and F (s, t) = −
g(t) i(k0 s+θ0 ) ie − D, k0
(4.3) (4.4) (4.5)
(4.6)
where F0 (s) is a straight line, D a constant vector. Moreover, the dilation function g(t) in (4.5) and (4.6) is determined by (3.11) and (3.21), respectively. Comparing Theorem 4.1 with Theorems 3.1, 3.3, and 3.4, we find that introducing the translation term does not provide any new solution from the view point of geometry. Acknowledgements This work was completed when C.-L. He and S.-J. Huang were visiting Professor Huai-Dong Cao at Lehigh University. They would like to thank Professor Cao and the Mathematics Department for great hospitality. References [1] Chou K S, Wo W F. On hyperbolic Gauss curvature flows. J Differential Geom, 2011, 89: 455–485 [2] Gage M, Hamilton R. The heat equation shrinking concex plane curves. J Differential Geom, 1986, 23: 417–491 [3] Halldorsson H P. Self-similar solutions to the curve shortening flow. Transactions of the American Mathematical Society, 2012, 364: 5285–5309 [4] He C L. Kong D X, Liu K F. Hyperbolic mean curvature flow. J Differential Equations, 2009, 246: 373–390 [5] Huisken G, Ilmanen T. The inverse mean curvature flow and the Riemann Penrose inequality. J Differential Geom, 2001, 59: 353–437 [6] Kong D X, Liu K F, Wang Z G. Hyperbolic mean curvature flow: evolution of plane curves. Acta Mathematica Scientia, 2009, 29: 493–514 [7] Kong D X, Wang Z G. Formation of singularities in the motion of plane curves under hyperbolic mean curvature flow. J Differential Equations, 2009, 247: 1694–1719 [8] LeFloch P G, Smoczyk K. The hyperbolic mean curvature flow. J Math Pures Appl, 2008, 90: 591–614 [9] Wang Z G. Hyperbolic mean curvature flow in Minkowski space. Nonlinear Analysis, 2014, 94: 259–271 [10] Wang Z G. Blow-up of periodic solutions to reducible quasilinear hyperbolic systems. Nonlinear Analysis, 2010, 73: 704–712