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Hamiltonian stationary self-similar solutions for Lagrangian mean curvature flow in complex Euclidean plane

We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.


Introduction
The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. By the first variation formula, the mean curvature vector points to the direction where the volume decreases most rapidly. Classically it has been studied by several approaches (partial differential equations, geometric measure theory, level sets or numerical methods) and its possible applications in symplectic topology and mirror symmetry are quite important.
There are very interesting results on regularity, global existence and convergence of the mean curvature flow in several ambient spaces. The flow for hypersurfaces in arbitrary Riemannian manifolds is well understood whereas not so much is known when the codimension is greater than one. In higher codimension the mean curvature H is a vector whose direction we do not know how to control, as a contrast with the codimension one case, where H is essentially a scalar function whose sign is preserved along the flow. In the last few years, mean curvature flow of higher codimension submanifolds has attracted special attention, mainly when the initial submanifold is Lagrangian in complex Euclidean space C n ; see [6], [7], [15], [16], [21] and [22] for example. One reason of this growing interest is that the Lagrangian condition is preserved by mean curvature flow (see [18]). In addition, as the gradient flow of the volume functional, the mean curvature flow seems to be a potential approach to the construction of special Lagrangians, which are volume minimizers that play a critical role in the T-duality formulation of mirror symmetry [20].
When the ambient space is Euclidean, the mean curvature flow is the solution to a system of parabolic equations that can be considered as the heat equation for submanifolds. In general, the mean curvature flow fails to exist after a finite time. The singularities are completely determined by the blow up of the second fundamental form. In geometric flows, like mean curvature flow or Ricci flow, the singularities often model on soliton solutions. In this article we are interested in a class of special solutions of the mean curvature flow that preserve the shape of the evolving submanifolds: the self-similar solutions, for which the evolution is a homothety. Eliminating the time variable, these self-similar solutions reduce the parabolic equation to the non-linear elliptic equation H = aφ ⊥ , where φ ⊥ stands for the projection of the position vector φ onto the normal space. When a is a negative constant, the submanifolds shrinks in finite time to a single point under the action of the mean curvature flow remaining its shape unchanged. If a is positive, the submanifold will expand with the same shape and in this case is necessarily non-compact. If a vanishes, the submanifold is minimal and stationary under the action of the flow.
The study of this type of solutions is hoped to give a better understanding of the flow at a singularity since by Huisken's monotonicity formula [10], any central blow-up of a finite-time singularity of the mean curvature flow is a self-similar solution.
Examples of self-similar solutions for mean curvature flow in C n were first constructed by Anciaux [1] in 2006. In order to produce eternal solutions of the Brakke flow (a weak formulation of the mean curvature flow), Lee and Wang [12]- [13] constructed in 2007 interesting examples of Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to Hamiltonian stationary cones generalizing Schoen-Wolfson ones [19]. All of them have been generalized very recently by Joyce, Lee and Tsui in [11] providing examples with different topologies.
It is expectable that the understanding of the singularities of the mean curvature flow will rely on the classification of self-similar solutions. But it is a hard and open problem classify all the self-similar solutions for the mean curvature flow. Using strongly techniques of complex analysis, our contribution in this paper is the classification of those ones in complex Euclidean plane that are, in addition, Hamiltonian stationary Lagrangians, i.e. critical points of the area functional among all Hamiltonian deformations. The main result (Theorem 1) characterizes in this way three one-parameter families of examples which include Lee and Wang examples in dimension two (see Remarks 2, 3 and 5). We provide (see Propositions 2 and 3) not only Hamiltonian stationary Lagrangian conformal immersions of cylinders, Moebius strips, tori and Klein bottles but also embedded Lagrangian nontrivial planes all of them self-similar solutions for mean curvature flow. Our construction is based essentially in a method given in [3] for obtaining Lagrangian surfaces in C 2 starting from spherical and hyperbolic curves; in this setting our examples are constructed using simple particular geodesics and parallels (see Remarks 4 and 6).
Acknowledgments: The authors would like to thank Francisco Urbano for his valuable suggestions and constant availability and Henri Anciaux for helpful discussions.

Hamiltonian stationary Lagrangian surfaces in C 2
In the complex Euclidean plane C 2 we consider the bilinear Hermitian product defined by Then , = Re( , ) is the Euclidean metric on C 2 and ω = −Im( , ) is the Kaehler two-form given by ω( · , · ) = J·, · , where J is the complex structure on C 2 . We also have that 2ω = dλ, where λ is known as the Liouville 1-form of C 2 . Let where T M is the tangent bundle of M . The second fundamental form σ of φ is given by σ(v, w) = JA Jv w, where A is the shape operator, and so the trilinear form C(·, ·, ·) = σ(·, ·), J· is fully symmetric.
Suppose M is orientable and let ω M be the area form of M . If Ω = dz 1 ∧ dz 2 is the closed complex-valued 2-form of C 2 , then φ * Ω = e iβ ω M , where β : M → R/2πZ is called the Lagrangian angle map of φ (see [8]). In general β is a multivalued function; nevertheless dβ is a well defined closed 1-form on M and its cohomology class is called the Maslov class. When β is a single valued function the Lagrangian is called zero-Maslov class and if cos β ≥ for some > 0 then the Lagrangian is said to be almost calibrated.
Note that the Lagrangian condition implies that the Liouville 1-form λ is a It is remarkable that β satisfies (see for example [19]) where H is the mean curvature vector of φ, defined by H = (1/2)trace σ.
If β is constant, say β ≡ β 0 (or, equivalently H = 0), then the Lagrangian immersion φ is calibrated by Re(e −iβ 0 Ω) and hence area-minimizing. They are referred as being Special Lagrangian.
A Lagrangian submanifold is called Hamiltonian stationary if the Lagrangian angle β is harmonic, i.e. ∆β = 0, where ∆ is the Laplace operator on M . Hamiltonian stationary Lagrangian (in short HSL) surfaces are critical points of the area functional with respect to a special class of infinitesimal variations preserving the Lagrangian constraint; namely, the class of compactly supported Hamiltonian vector fields (see [17]). Examples of HSL surfaces in C 2 can be found in [2], [5] and [9]. 3. Self-similar solutions for the mean curvature flow Let φ : M → R 4 be an immersion of a smooth surface M in Euclidean 4-space. It is not hard to check that where H is the mean curvature vector and ∆ is the Laplace operator of the induced metric on M .
and [0, T sing ) is the maximal time interval such that (3) holds. Looking at equation (2), this can be considered as the heat equation for submanifolds. The equation (3) is a quasi-linear parabolic system and short time existence is guaranteed when the initial submanifold M is compact.
When φ : M → R 4 ≡ C 2 is Lagrangian it is well known that 0 < T sing < ∞ and being Lagrangian is preserved along the mean curvature flow. The monotone condition defined in Section 2 is also preserved under the flow (see [7]).
In geometric flows such as the Ricci flow or the MCF, singularities are often locally modelled on soliton solutions. In the case of MCF, one type of soliton solutions of great interest are those moved by scaling in the Euclidean space. We recall that solitons moved by scaling must be of the following form: for some nonzero constant a, where φ ⊥ denotes the normal projection of the position vector φ and H is the mean curvature vector of φ. If a < 0, it is called a self-shrinker and if a > 0 it is called a self-expander.
Remark 1. In the Lagrangian setting, using (1) the condition (4) to be a self-similar for MCF can be reformulated in terms of the Lagrangian angle map and the restriction to the surface of the Liouville 1-form by means of dβ = −4aφ * λ, a = 0.
In particular, any Lagrangian self-similar solution for MCF is monotone.
One could normalize the value of a up to dilations, but we prefer not to do it because we will take different particular values of a along the paper to recover some known examples.
It is an exercise to check that, given any a < 0, the right circular cylinder (4). Thus they are examples of self-shrinkers. Both of them are also Lagrangian surfaces with parallel mean curvature vector and hence Hamiltonian stationary Lagrangians.
We finish this section studying a close relationship between the area and the Willmore functional of compact Lagrangian self-shrinkers, that it is even true for monotone compact HSL surfaces (see [14]).
where φ denotes the tangent projection of the position vector φ. In particular, if M is compact then a < 0 and Area(M ) = − 1 a M |H| 2 . Proof. Derivating in the direction of any tangent vector v in the equality φ = φ +H/a and taking tangent components, we get that

The examples
In this section we introduce three one-parameter families of HSL surfaces in C 2 which are self-similar solutions for mean curvature flow satisfying (4) and describe their main geometric properties. In the next result, we provide examples with the topology of a plane, a cylinder or a Moebius strip. Proposition 2. Given any a > 0, let define: with s δ = sinh δ, c δ = cosh δ and t δ = tanh δ. Then Φ δ is a Hamiltonian stationary Lagrangian conformal immersion and a self-similar solution for mean curvature flow satisfying (4). If cosh 2 δ / ∈ Q, Φ δ is -in addition-an embedded self-expander plane asymptotic to the Hamiltonian stationary Lagrangian cone If cosh 2 δ = p/q ∈ Q, (p, q) = 1, p > q, then Φ δ is given by and satisfies Φ p,q (s + 2π √ pq, t) = Φ p,q (s, t), ∀(s, t) ∈ R 2 , inducing a Hamiltonian stationary Lagrangian immersion of a self-expander cylinder. Moreover, if p is odd and q is even, then Φ p,q (s + π √ pq, −t) = Φ p,q (s, t), ∀(s, t) ∈ R 2 , inducing a Hamiltonian stationary Lagrangian immersion of a self-expander Moebius strip.

Remark 3.
By taking where 0 ≤ θ < 2π. In this way we arrive at the examples given in Proposition 2.1 of [13] for n = 2. Restricting θ ∈ [0, π), in Proposition 2.2 of [13] it is proved that these examples are oriented if and only if p − q is even and are embedded if and only q = 1. We finally remark that Φ δ and Υ γ generalize them although they can be included in the observation made in Remark 2.1 of [13].
This shows that Φ is a Lagrangian conformal immersion whose induced metric is given by , = e 2u(t) ds 2 + dt 2 , with e 2u(t) = 1 2a t 2 δ cosh 2 t + s 2 δ sinh 2 t . Moreover we compute e iβ Φ = e −2u det C (Φ s , Φ t ) = e i s 2 δ s/c δ . So we conclude that Φ is HSL since β Φ depends only on s. Using (1) it is not hard to obtain that the mean curvature vector of Φ satisfies that H Φ = s 2 δ e −2u 2c δ JΦ s = aΦ ⊥ . To prove that Φ is an embedding, we start from Φ(s 1 , t 1 ) = Φ(s 2 , t 2 ) and then it is clear that when (s 1 , t 1 ) = (s 2 , t 2 ) necessarily c 2 δ is a rational number.
Finally, Φ is asymptotic to the given HSL cone taking into account that if t → +∞ then cosh t, sinh t e t /2 and if t → −∞ then cosh t e −t /2 and sinh t −e −t /2.
The reasoning for Υ γ is completely similar so that we omit it.
The next result describes examples with the topology of a cylinder, a torus or a Klein bottle. Proposition 3. Given any a < 0, let define: with s ν = sinh ν, c ν = cosh ν and t ν = coth ν. Then Ψ ν is a Hamiltonian stationary Lagrangian conformal immersion and a self-similar solution for mean curvature flow satisfying (4). If sinh 2 ν / ∈ Q, Ψ ν is -in addition-an embedded self-shrinker cylinder.
(3) The Clifford torus T 1,1 is the only one embedded in the above family. On the other hand, it is clear that the HSL tori T m,n admit a oneparameter group of isometries. Using the notation of [5], it is not complicated to check that their universal coverings would correspond to the immersions F π/2 0,β , sin β ∈ Q.
Proof. Given any a < 0, let Ψ = Ψ ν . It is an exercise to check that This shows that Ψ is a Lagrangian conformal immersion whose induced metric is given by , = e 2v(s) ds 2 + dt 2 , with e 2v(s) = 1 −2a t 2 ν cos 2 s + c 2 ν sin 2 s . Moreover we get that e iβ Ψ (t) = −i e i c 2 ν t/sν . Thus we conclude that Ψ is HSL. Using (1) it is easy to obtain that the mean curvature vector of Ψ satisfies that H Ψ = c 2 ν e −2v 2sν JΨ t = aΨ ⊥ . It is straightforward to get that Ψ is an embedding since Ψ(s 1 , t 1 ) = Ψ(s 2 , t 2 ) and (s 1 , t 1 ) = (s 2 , t 2 ) implies that s 2 ν is a rational number. The geometric properties of Ψ m,n are deduced making use of the above data and Remark 4.

Classification
Theorem 1. Let φ : M 2 → C 2 be a Hamiltonian stationary Lagrangian self-similar solution for mean curvature flow.
(a) If φ is a self-expander, i.e. H = aφ ⊥ , a > 0, then φ is locally congruent to some Φ δ : R 2 → C 2 , δ > 0, described in Proposition 2. (b) If φ is a self-shrinker, i.e. H = aφ ⊥ , a < 0, then φ is locally congruent to some of the following: (i) the right circular cylinder Proof. First, following [4] we can associate to any Lagrangian immersion φ : M −→ C 2 a cubic differential form Θ on M defined by and a differential form Λ on M defined by Λ(z) =h(z)dz, with h(z) = 2ω(∂z, H), where z = x+iy is a local isothermal coordinate on M such that the induced metric, also denoted by , , is written as , = e 2u |dz| 2 with |dz| 2 the Euclidean metric, and C and ω are extended C-linearly to the complexified tangent bundles. We remark that our h here ish in [4].
It is straightforward to check that the Frenet equations of φ are given by and it is not difficult (see equation (3.3) in [4]) to get the compatibility equations for (11), obtaining Since φ is a self-similar solution for mean curvature flow, using (4) we have that h = −2a φ z , Jφ , a = 0, and from (11) we deduce that (13) h z = −aRe(h|φ| 2 z ), and, taking into account that φ z , φz = e 2u /2 and φ z , φ z = 0, from (11) we also have that (14) |φ| 2 zz = |h| 2 a + e 2u and (15) As φ is also a Hamiltonian stationary Lagrangian immersion, the second equation of (12) implies thathz = h z = β zz = 0. Hence Λ is a holomorphic differential and we can normalize h ≡ µ, µ > 0, since the zeroes of h and H are the same and a = 0. Thus (13) says that g := |φ| 2 satisfies g x = 0, that is g = g(y). In addition, from (14) g satisfies In particular, we get that u = u(y) too. We can now express f in terms of g and u from (15) by If φ denotes the tangent part of φ, using |h| 2 = e 2u |H| 2 and taking modules in the equality φ = φ + H/a yields to (18) g = e −2u g 2 4 + µ 2 a 2 This implies that g > 0.
Then (20) translates into (21) and (22), (17) implies that that is compatible with (12) and (23). Then we can rewrite the Frenet equations (11) in the following way: After a long computation, using (25), (22) and (23), we get that φ xyy = Eφ x and φ yyy = Eφ y . Up to translations, we can consider (26) φ yy = Eφ and (25) gives In particular, H = µe −2u Jφ x . On the one hand, when E = 0 it is necessarily a < 0 from (20). Using (21), we obtain that e 2u ≡ −2µ 2 /a and (24) gives f ≡ −2µ 3 /a. The integration of the corresponding Frenet equations (25), now simply written as φ xx = 2µJφ x , φ xy = φ yy = 0 leads to the right circular cylinder On the other hand, when E = 0, the discriminant of the second order polynomial P (g) in (20) is 64µ 2 (µ 2 + E)/a 2 . As P (g) = g 2 must be non negative for g > 0 (and observe that P (0) < 0) we distinguish two cases according to the sign of the energy E to reach the following conclusions: • Case (a): If E > 0 then g is bounded from below.
• Case (b): If E < 0, it is necessarily E ≥ −µ 2 , and g is bounded from below and from above. We remark that (21) shows that if E < 0 then it is necessarily a < 0 and we also point out that the case E = −µ 2 corresponds precisely with the constant case g ≡ 1/ − a.
Corollary 1. The tori T m,n (described in Proposition 3) are the only compact orientable Hamiltonian stationary self-similar solutions for mean curvature flow in complex Euclidean plane.