Complete Willmore Legendrian surfaces in $\mathbb{S}^5$ are minimal Legendrian surfaces

In this paper we continue to consider Willmore Legendrian surfaces and csL Willmroe surfaces in $\mathbb{S}^5$, notions introduced by Luo in \cite{Luo}. We will prove that every complete Willmore Legendrian surface in $\mathbb{S}^5$ is minimal and construct nontrivial examples of csL Willmore surfaces in $\mathbb{S}^5$.


Introduction
Let Σ be a Riemann surface, (M n , g) = S n or R n (n ≥ 3) the unit sphere or the Euclidean space with standard metrics and f an immersion from Σ to M. Let For a smooth and compactly supported variation f : Σ × I → M with φ = ∂ t f we have the following first variational formula (cf. [24,25]) where {e α : 3 ≤ α ≤ n} is a local orthonormal frame of the normal bundle of f (Σ) in M and When (M, g) = R 3 , Willmore [27] proved that the Willmore energy of closed surfaces are larger than or equal to 4π and equality holds only for round spheres. When Σ is a torus, Willmore conjectured that the minimum is 2π 2 and it is attained only by the Clifford torus, up to a conformal transformation of R 3 [6,26], which was verified by Marques and Neves in [15]. When (M, g) = R n , Simon [22], combined with the work of Bauer and Kuwert [1], proved the existence of an embedded surface which minimizes the Willmore functional among closed surfaces of prescribed genus. Motivated by these mentioned papers, Minicozzi [16] proved the existence of an embedded torus which minimizes the Willmore functional in a smaller class of Lagrangian tori in R 4 . In the same paper Minicozzi conjectured that the Clifford torus minimizes the Willmore functional in its Hamiltonian isotropic class, which he verified has a close relationship with Oh's conjecture [19,20]. We should also mention that before Minicozzi, Castro and Urbano proved that the Whitney sphere in R 4 is the only minimizer for the Willmore functional among closed Lagrangian sphere. This result was further generalized by Castro and Urbano in [4] where they proved that the Whitney sphere is the only closed Willmore Lagrangian sphere (a Lagrangian sphere which is also a Willmore surface) in R 4 . Examples of Willmore Lagrangian tori (Lagrangian tori which also are Willmroe surfaces) in R 4 were constructed by Pinkall [21] and Castro and Urbano [5]. Motivatied by these works, Luo and Wang [13] considered the variation of the Willmore functional among Lagrangian surfaces in R 4 or variation of a Lagrangian surface of the Willmore functional among its Hamiltonian isotropic class in R 4 , whose critical points are called LW or HW surfaces respectively. We should also mention that Willmroe type functional of Lagrangian surfaces in CP 2 were studied by Montiel and Urbano [18] and Ma, Mironov and Zuo [14].
Inspired by the study of the Willmore functional for Lagrangian surfaces in R 4 , Luo [11] naturally considered the Willmore functional of Legendrian surfaces in S 5 . Luo [11] proved that Willmore Legendrian surfaces in S 5 are csL surfaces (see Definition 2.6). In this paper, we continue to study Willmore Legendrian surfaces and csL Willmore surfaces in S 5 . Surprisingly we will prove that every complete Willmore Legendrian surface in S 5 must be a minimal surface (Theorem 3.2). We will also construct nontrivial examples of csL Willmore surfaces from csL surfaces in S 5 for the first time, by exploring relationships between them (Proposition 4.1).
The method here we used to construct nontrivial csL Willmore surfaces in S 5 in Section 4 should also be useful in constructing nontrivial HW surfaces in R 4 introduced by Luo and Wang in [13]. We will consider this problem in a forthcoming paper.

Basic material and formulas
In this section we record some basic material of contact geometry. We refer the reader to consult [7] and [2] for more materials.
2.1. Contact Manifolds. Definition 2.1. A contact manifold M is an odd dimensional manifold with a one form α such that α ∧ (dα) n 0, where dim M = 2n + 1.
Assume now that (M, α) is a given contact manifold of dimension 2n + 1. Then α defines a 2n−dimensional vector bundle over M, where the fibre at each point p ∈ M is given by Sine α ∧ (dα) n defines a volume form on M, we see that ω := dα is a closed nondegenerate 2-form on ξ ⊕ ξ and hence it defines a symplectic product on ξ, say ω, such that (ξ, ω| ξ⊕ξ ) becomes a symplectic vector bundle. A consequence of this fact is that there exists an almost complex bundle structureJ : ξ → ξ compatible with dα, i.e. a bundle endomorphism satisfying: Since M is an odd dimensional manifold, ω must be degenerate on T M, and so we obtain a line bundle η over M with fibres Definition 2.2. The Reeb vector field R is the section of η such that α(R) = 1.
Thus α defines a splitting of T M into a line bundle η with the canonical section R and a symplectic vector bundle (ξ, ω|ξ ⊕ ξ). We denote the projection along η by π, i.e.
Using this projection we extend the almost complex structureJ to a section J ∈ Γ(T * M ⊗ T M) by setting J(V) =J(π(V)), We have special interest in a special class of submanifolds in contact manifolds.
For algebraic reasons the dimension of an isotropic submanifold of a 2n + 1 dimensional contact manifold can not bigger than n.
2.2. Sasakian manifolds. Let (M, α) be a contact manifold. A Riemannian metric g α defined on M is said to be associated, if it satisfies the following three conditions: We should mention here that on any contact manifold there exists an associated metric on it, because we can construct one in the following way. We introduce a bilinear form b by is Kähler with respect to the following canonical almost complex structure J on TCM = R ⊕ R ⊕ ξ : Furthermore if g α is Einstein, M is called a Sasakian Einstein manifold.
We record more several lemmas which are well known in Sasakian geometry. These lemmas will be used in the subsequent sections.
for X, Y ∈ T M, where∇ is the Levi-Civita connection on (M, g α ). For a proof of this lemma we refer to [10,Proposition A.2], and [23, lemma 2.8]. In fact they proved this result under the weaker assumption that (M, α, g α , J) is a weakly Sasakian Einstein manifold, where weakly Einstein means that g α is Einstein only when restricted to the contact hyperplane. Proof. For any X, Y ∈ T Σ, where in the third equality we used (2.1).
In particular this lemma implies that the mean curvature H of Σ is orthogonal to the Reeb field R. This fact is important in our following argument. A canonical example of Sasakian Einstein manifolds is the standard odd dimensional sphere S 2n+1 .
Example 2.1 (The standard sphere S 2n+1 ). Let C n = R 2n+2 be the Euclidean space with coordinates (x 1 , . . . , x n+1 , y 1 , . . . , y n+1 ) and S 2n+1 be the standard unit sphere in R 2n+2 . Define x j dy j − y j dx j , then α := α 0 | S 2n+1 defines a contact one form on S 2n+1 . Assume that g 0 is the standard metric on R 2n+2 and J 0 is the standard complex structure of C n . We define g α = g 0 | S 2n+1 , then (S 2n+1 , α, g α ) is a Sasakian Einstein manifold. The contact hyperplane is characterized by 2.3. Legendrian submanifolds in the unit sphere. Assume φ : Σ n → S 2n+1 ⊆ C n+1 is a Legendrian immersion. Let B be the second fundamental form, A ν be the shape operator with respect to the norm vector ν ∈ T ⊥ Σ and H be the mean curvature vector. The shape operator A ν is a symmetric operator on the tangent bundle and satisfies the following Weingarten equations The Gauss equations, Codazzi equations and Ricci equations are given by Let {e 1 , e 2 } be a local orthonormal frame of Σ. Then {Je 1 , Je 2 , Jφ} is a local orthonormal frame of the normal bundle T ⊥ Σ, where J is the complex structure of C n+1 . Set Recall that Definition 2.6. Σ is a csL submanifold if it is a critical point of the volume functional among Legendrian submanifolds.
CsL submanifolds satisfy the following Euler-Lagrangian equation ( [3,8]): It is obvious that Σ is csL when Σ is minimal. The following observation is very important for the study of csL submanifolds.
By using the Bochner formula for harmonic vector fields (cf. [9]), we get From (2.6) it is easy to see that we have

Willmore Legendrian surfaces in S 5
In this section we prove that every complete Willmore Legendrian surface in S 5 must be a minimal surface. Firstly, we rewrite the Willmore operator acting on Legendrian surfaces, i.e., we prove the following Proposition 3.1. Assume that Σ is a Legendrian surface in S 5 , then its Willmore operator can be written as

In particular, the Euler-Lagrangian equation of Willmore Legendrian surfaces in
Proof. Let {ν 1 , ν 2 , R} be a local orthonormal frame of the normal bundle of Σ, then the Willmore equation (1.1) can be rewritten as . Choose a local orthonormal frame field around p with ∇e i | p = 0, then where in the last equality we also used (2.1). Therefore we obtain which implies that Σ satisfies the following equation In addition, by Lemma 2.2, the dual one form of JH is harmonic. By the Ricci identity we have The Proposition is then a consequence of the following Claim together with above two identities. Claim.
Proof. The first equation is obvious by the Gauss equation (2.3). The second equation can be proved by the Gauss equation (2.3) and the tri-symmetry of the tensor σ (see (2.5)). To be precise, for every tangent vector field Z ∈ T Σ we have This completes the proof of the second equation.
Now we are in position to prove the following Theorem 3.2. Every complete Willmore Legendrian surface in S 5 is a minimal surface.
Proof. We prove by a contradiction argument. Assume that Σ is a complete Willmore Legendrian surface in S 5 which is not a minimal surface. If H 0, then let e 1 = JH |H| , e 2 be a local orthonormal frame field of Σ. From (3.1) we have which also implies that Then by the Gauss equation (2.3) we have Since Σ is a Willmore Legendrian surface, from (3.1) we see that div(JH) = 0. By Lemma 2.7 the minimal points of Σ are discrete and so the Gauss curvature of Σ equals 1 everywhere on Σ, therefore Σ is compact by Bonnet-Myers theorem. Apply Lemma 2.6 to obtain that on Σ Then the maximum principle implies that H ≡ 0 which is a contradiction. Therefore Σ is a minimal surface.

Examples of csL Willmore surfaces in S 5
From the definition we see that complete Willmore Legendrian surfaces, which are minimal surface by Theorem 3.2 in the last section, are trivial examples of csL Willmore surfaces in S 5 . Thus it is very natural and important to construct nonminimal csL Willmore surfaces in S 5 . This will be done in this section by analyzing a very close relationship between csL Willmore surfaces and csL surfaces in S 5 .
Assume that Σ is a csL Willmore surface in S 5 , then since the variation vector field on Σ under Legendrian deformations can be written as J∇u + 1 2 uR for smooth function u on Σ (cf. where in the last euqality we used − → W(Σ), R = −2 div(JH), by Proposition 3.1. Therefore Σ satisfies the following Euler-Lagrangian equation: Remark 4.1. Note that the coefficient of the Euler-Lagrangian equation (4.1) for csL Willmore surfaces in S 5 is slightly different with (1.7) in [11]. That is because here we use the notation H = trace B, whereas in [11] we defined H = 1 2 trace B. Then by (3.1), Σ satisfies the following equation.
Therefore Σ satisfies the following equation  Example 4.1 (Calabi tori). For every four nonzero real numbers r 1 , r 2 , r 3 , r 4 with r 2 1 + r 2 2 = r 2 3 + r 2 4 = 1, the Calabi torus Σ is a csL surface in S 5 defined as follows. Denote then F(t, s) = (r 1 r 3 φ 1 , r 1 r 4 φ 2 , r 2 φ 3 ). Since the induced metric in Σ is given by 1F} is a local orthonormal frame of S 5 such that {E 1 , E 2 } is a local orthonormal tangent frame and R is the Reeb field. A direct calculation yields

Hence,
Moreover E 1 and E 2 are two parallel tangent vector field. It is obvious that Σ is a csL Willmore surface.
Secondly, we give some examples that JH is not parallel. Mironov [17] constructed some new csL surfaces in S 5 . We can verify that Mironov's examples are in fact csL Willmore surfaces. [17]). Let F : Σ 2 → S 5 be an immersion. Then F is a Legendrian immersion iff

Example 4.2 (Mironov's examples
Here {x, y} is a local coordinates of Σ and , stands for the hermitian inner product in C 3 . Set where g is a real positive matrix which is the induce metric of Σ. There is a hermitian matrix Θ such that We compute Similarly, The Lagrangian angle is then given by θ = trℜΘ. The above discussion implies that Let a, b, c are three positive constants and consider the following immersion One can check that F is a Legendrian immersion. Denote Σ := F S 1 × S 1 . Notice that The induced metric g is given by A strait forward calculation yields that We get We get and In particular Therefore, Σ is a csL Willmore surface in S 5 .

Introduction
Let Σ be a Riemann surface, (M n , g) = S n or R n (n For a smooth and compactly supported variation f : Σ×I → M with φ = ∂ t f we have the following first variational formula (cf. [22,23]) where h α i j is the component of B and H α is the trace of h α i j .
A smooth immersion f : Σ → M is called a Willmore immersion, if it is a critical point of the Willmore functional W. In other words, f is a Willmore immersion if and only if it satisfies When (M, g) = R 3 , Willmore [25] proved that the Willmore energy of closed surfaces are larger than or equal to 4π and equality holds only for round spheres. When Σ is a torus, Willmore conjectured that the minimum is 2π 2 and it is attained only by the Clifford torus, up to a conformal transformation of R 3 [24,6], which was verified by Marques and Neves in [13]. When (M, g) = R n , Simon [20], combined with the work of Bauer and Kuwert [1], proved the existence of an embedded surface which minimizes the Willmore functional among closed surfaces of prescribed genus. Motivated by these mentioned papers, Minicozzi [14] proved the existence of an embedded torus which minimizes the Willmore functional in a smaller class of Lagrangian tori in R 4 . In the same paper Minicozzi conjectured that the Clifford torus minimizes the Willmore functional in its Hamiltonian isotropic class, which he verified has a close relationship with Oh's conjecture [17,18]. We should also mention that before Minicozzi, Castro and Urbano proved that the Whitney sphere in R 4 is the only minimizer for the Willmore functional among closed Lagrangian sphere. This result was further generalized by Castro and Urbano in [4] where they proved that the Whitney sphere is the only closed Willmore Lagrangian sphere (a Lagrangian sphere which is also a Willmore surface) in R 4 . Examples of Willmore Lagrangian tori (Lagrangian tori which also are Willmroe surfaces) in R 4 were constructed by Pinkall [19] and Castro and Urbano [5]. Motivatied by these works, Luo and Wang [11] considered the variation of the Willmore functional among Lagrangian surfaces in R 4 or variation of a Lagrangian surface of the Willmore functional among its Hamiltonian isotropic class in R 4 , whose critical points are called LW or HW surfaces respectively. We should also mention that Willmroe type functional of Lagrangian surfaces in CP 2 were studied by Montiel and Urbano [16] and Ma, Mironov and Zuo [12].
Inspired by the study of the Willmore functional for Lagrangian surfaces in R 4 , Luo [9] naturally considered the Willmore functional of Legendrian surfaces in S 5 . Luo [9] proved that Willmore Legendrian surfaces in S 5 are csL surfaces (see Definition 2.1). In this paper, we continue to study Willmore Legendrian surfaces and csL Willmore surfaces in S 5 . Surprisingly we will prove that every complete Willmore Legendrian surface in S 5 must be a minimal surface (Theorem 2.5). We also find nontrivial examples of csL Willmore surfaces from csL surfaces in S 5 for the first time, by exploring relationships between them (Proposition 3.1).
The method here we used to find nontrivial csL Willmore surfaces in S 5 in Section 4 should also be useful in discovering nontrivial HW surfaces in R 4 introduced by Luo and Wang in [11]. We will consider this problem in the future.

Willmore Legendrian surfaces in S 5
In this section we will prove that every complete Willmore Legendrian surfaces in S 5 is minimal. Firstly we briefly record several facts about Legendrian surfaces in S 5 . We refer the reader to consult [2] for more materials about the contact geometry.
Let S 5 , the 5-dimensional unit sphere, be the standard Sasakian Einstein manifold with contact one form α, almost complex structure J, Reed field R and canonical metric g. Let Σ be a closed surface of S 5 ⊂ C 3 . We say that Σ is Legendrian if where F : Σ −→ S 5 is the position vector and T Σ, T ν Σ are tangent and normal bundles of Σ respectively. We say that Σ is a minimal Legendrian surface of S 5 if Σ is a minimal and Legendrian surface of S 5 . Define The Weingarten equation implies that Moreover, by definition, one can check that σ is a three order symmetric tensor, i.e., (2.1)

The Gauss equation, Codazzi equation and Ricci equation becomes
i.e., ∇σ is a fourth order symmetric tensor. Recall that It is obvious that Σ is csL in S 5 when Σ is minimal. The following observation is very important for the study of csL surfaces. We then prove that every complete Willmore Legendrian surface in S 5 must be a minimal surface. Firstly, we rewrite the Willmore operator acting on Legendrian surfaces, i.e., we prove the following Proposition 2.4. Assume that Σ is a Legendrian surface in S 5 , then its Willmore operator can be written as In particular, the Euler-Lagrange equation of Willmore Legendrian surfaces in S 5 is Proof. Let {ν 1 , ν 2 , R} be a local orthonormal frames of the normal bundle of Σ, then the Willmore equation (1.1) can be rewritten as Note that by (2.8) in [9] we have , where∇ denotes the covariant derivative of S 5 . Choose a local orthonormal frame field around p with ∇ e i e j | p = 0, then where in the last equality we used (2.7) in [9]. Therefore we obtain which implies that Σ satisfies the following equation In addition, by [9, Lemma 2.9], the dual one form of JH is closed, thus by the Ricci identity we have The Proposition is then a consequence of the following Claim together with above two identities. Claim.
Proof. The first equation is obvious by the Gauss equation ( Since Σ is a Willmore Legendrian surface, from (2.4) we see that div(JH) = 0. By Lemma 2.3 the minimal points of Σ are discrete and so the Gauss curvature of Σ equals one everywhere on Σ, therefore Σ is compact by Bonnet-Myers theorem. Apply Lemma 2.2 to obtain that on Σ Then the maximum principle implies that H ≡ 0, which is a contradiction. Therefore Σ is a minimal Legendrian surface in S 5 .

Examples of csL Willmore surfaces in S 5
From the definition we see that complete Willmore Legendrian surfaces, which are minimal surfaces by Theorem 2.5 in the last section, are trivial examples of csL Willmore surfaces in S 5 . Thus it is very natural and important to find nonminimal csL Willmore surfaces in S 5 . This will be done in this section by analyzing a very close relationship between csL Willmore surfaces and csL surfaces in S 5 .
Assume that Σ is a csL Willmore surface in S 5 , then since the variation vector field on Σ under Legendrian deformations can be written as J∇u + 1 2 uR for smooth function u on Σ (cf. [21, Lemma 3.1]), we have where in the last equality we used  In addition, by the four-symmetric of σ i jk,l (see (2.3)), a direct computation shows div(JB(JH, JH)) = 2 tr B(·, ∇ · (JH)), H + 1 2 ∇ JH |H| 2 .
Here {x, y} is a local coordinates of Σ and , stands for the hermitian inner product in C 3 . Set where g is a real positive matrix which is the induce metric of Σ. There is a hermitian matrix Θ such that We compute Similarly, The Lagrangian angle is then given by θ = trℜΘ. The above discussion implies that J∇θ = H.
Let a, b, c are three positive constants and consider the following immersion where u(x) = c (a + b + (b − a) cos(2x)) 2 .
One can check that F is a Legendrian immersion. Denote Σ := F S 1 × S 1 . Notice that The induced metric g is given by :=e 2p(x) dx 2 + e 2q(x) dy 2 .
A strait forward calculation yields that We get We get and In particular div √ −1H = 0.
Hence Σ is csL. Moreover Therefore, Σ is a csL Willmore surface in S 5 .