Complete Willmore Legendrian surfaces in S5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}^5$$\end{document} are minimal Legendrian surfaces

In this paper, we continue to consider Willmore Legendrian surfaces and csL Willmore surfaces in S5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}^5$$\end{document}, notions introduced by Luo (Calc Var Partial Differ Equ 56, Art. 86, 19, 2017. https://doi.org/10.1007/s00526-017-1183-z). We will prove that every complete Willmore Legendrian surface in S5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}^5$$\end{document} is minimal and find nontrivial examples of csL Willmore surfaces in S5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}^5$$\end{document}.


Introduction
Let be a Riemann surface, (M n , g) = n or ℝ n (n ≥ 3) the unit sphere or the Euclidean space with standard metrics and f an immersion from to M. Let B be the second fundamental form of f with respect to the induced metric, H the mean curvature vector field of f defined by M the Gauss curvature of df (T ) with respect to the ambient metric g and d f the area element on f ( ) . The Willmore functional of the immersion f is then defined by For a smooth and compactly supported variation f ∶ × I ↦ M with = t f , we have the following first variational formula (cf. [22,23]) When (M, g) = ℝ 3 , Willmore [25] proved that the Willmore energy of closed surfaces is larger than or equal to 4 and equality holds only for round spheres. When is a torus, Willmore conjectured that the minimum is 2 2 and it is attained only by the Clifford torus, up to a conformal transformation of ℝ 3 [6,24], 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 Willmore surfaces) in ℝ 4 were constructed by Pinkall [19] and Castro and Urbano [5]. Motivated by these works, Luo and Wang [11] considered the variation in the Willmore functional among Lagrangian surfaces in ℝ 4 or variation in 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 Willmore-type functional of Lagrangian surfaces in ℂℙ 2 were studied by Montiel and Urbano [16] and Ma et al. [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 5 . 5 is called a Willmore Legendrian surface.

Definition 1.2 A Legendrian surface in 5 is called a contact stationary Legendrian
Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations.
Luo [9] proved that Willmore Legendrian surfaces in 5 are csL surfaces (see Definition 2.1). In this paper, we continue to study Willmore Legendrian surfaces and csL Willmore surfaces in 5 . Surprisingly, we will prove that every complete Willmore Legendrian surface in 5 must be a minimal surface (Theorem 2.5). We also find nontrivial examples of csL Willmore surfaces from csL surfaces in 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 5 in Sect. 3 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 5
In this section, we will prove that every complete Willmore Legendrian surface in 5 is minimal. Firstly, we briefly record several facts about Legendrian surfaces in 5 . We refer the reader to consult [2] for more materials about the contact geometry.
Let 5 , the five-dimensional unit sphere, be the standard Sasakian Einstein manifold with contact one form , almost complex structure J, Reed field and canonical metric g. Let be a closed surface of 5 ⊂ ℂ 3 . We say that is Legendrian if where F ∶ ⟶ 5 is the position vector and T , T are tangent and normal bundles of , respectively. We say that is a minimal Legendrian surface of 5 if is a minimal and Legendrian surface of 5 . Define The Weingarten equation implies that Moreover, by definition, one can check that is a three-order symmetric tensor, i.e.,

The Gauss equation, Codazzi equation and Ricci equation become
where e i is an orthonormal basis of T . The Codazzi equation implies , i.e., ∇ is a fourth-order symmetric tensor.
Recall that Definition 2.1 is a csL surface in 5 if it is a critical point of the volume functional among Legendrian surfaces.
CsL surfaces in 5 satisfy the following Euler-Lagrange equation [3,7]: It is obvious that is csL in 5 when is minimal. The following observation is very important for the study of csL surfaces.

Lemma 2.1 is csL in 5 iff JH is a harmonic vector field.
By using the Bochner formula for harmonic vector fields (cf. [8]), we get

Lemma 2.2 If is csL in 5 , then
From Lemma 2.2, it is easy to see that we have

Lemma 2.3 If ⊂ 5 is csL and non-minimal, then the zero set of H is isolated and
provided H ≠ 0 , where is the Gauss curvature of .
We then prove that every complete Willmore Legendrian surface in 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 is a Legendrian surface in 5 , , then its Willmore operator can be written as

In particular, the Euler-Lagrange equation of Willmore Legendrian surfaces in 5 is
Proof Let { 1 , 2 , } be a local orthonormal frames of the normal bundle of , then the Willmore equation (1.1) can be rewritten as Note that by (2.8) in [9], we have div (JH) = 0.
for X, Y ∈ (T ) , where ∇ denotes the covariant derivative of 5 . Choose a local orthonormal frame field around p with ∇ e i e j | p = 0 , then and where in the last equality we used (2.7) in [9]. Therefore, we obtain which implies that 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. 2) and the tri-symmetry of the tensor (see (2.1)). To be precise, for every tangent vector field Z ∈ T we have Δ(JH) = ∇ div (JH) + JH.

3
This completes the proof of the second equation.
Then, the maximum principle implies that H ≡ 0 , which is a contradiction. Therefore, is a minimal Legendrian surface in 5 . That is because here we use the notation H = tr B , whereas in [9] we defined H = 1 2 tr B.

Proposition 3.1 Assume that is a csL surface in
With the aid of Proposition 3.1, we can find the following examples of csL Willmore surfaces from csL surfaces in 5 . Firstly, according to Proposition 3.1, all closed Legendrian surfaces with parallel tangent vector field JH, which are exactly minimal surfaces or the Calabi tori (cf. [10, Proposition 3.2]), are csL Willmore surfaces. For reader's convenience, we give some detailed computations as follows.
Denote then F(t, s) = r 1 r 3 1 , r 1 r 4 2 , r 2 3 . Since the induced metric in is given by orthonormal frame of 5 such that {E 1 , E 2 } is a local orthonormal tangent frame and is the Reeb field. A direct calculation yields Hence,

Thus,
Moreover, E 1 and E 2 are two parallel tangent vector field. It is obvious that is a csL Willmore surface.
Secondly, we give some examples that JH is not parallel. Mironov [15] constructed the following new csL surfaces in 5 . We will verify that Mironov's examples are in fact csL Willmore surfaces. (Mironov's examples [15]) Let F ∶ 2 ↦ 5 be an immersion. Then, F is a

Legendrian immersion iff
Here, {x, y} is a local coordinates of and ⟨, ⟩ stands for the Hermitian inner product in ℂ 3 . Set where g is a real positive matrix which is the induce metric of . There is a Hermitian matrix such that We compute Hence, which implies Similarly, The Lagrangian angle is then given by = trℜ . The above discussion implies that Let a, b, c are three positive constants and consider the following immersion where where One can check that F is a Legendrian immersion. Denote ∶= F 1 × 1 . Notice that The induced metric g is given by A strait forward calculation yields that We get ,

Thus,
We get and In particular, Hence, is csL. Moreover, Therefore, is a csL Willmore surface in 5 .
Acknowledgements Open access funding provided by Projekt DEAL. This work was partially supported by the NSFC of China (Nos. 11501421, 11801420, 11971358) and the Youth Talent Training Program of Wuhan University.
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