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

“…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.…”

“…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. Luo [9] proved that Willmore Legendrian surfaces in 5 are csL surfaces (see Definition 2.1).…”

“…5 is called a Willmore Legendrian surface. 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 .…”

“…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).…”

“…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].…”