Complete Willmore Legendrian surfaces in $${\mathbb {S}}^5$$ S 5 are minimal Legendrian surfaces
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Complete Willmore Legendrian surfaces in 𝕊5 are minimal Legendrian surfaces Yong Luo1,2 · Linlin Sun1,2,3 Received: 21 January 2020 / Accepted: 23 May 2020 © The Author(s) 2020
Abstract In this paper, we continue to consider Willmore Legendrian surfaces and csL Willmore surfaces in 𝕊5 , 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 is minimal and find nontrivial examples of csL Willmore surfaces in 𝕊5. Keywords Willmore Legendrian surface · csL surface · csL Willmore surface Mathematics Subject Classification 53C24 · 53C42 · 53C44
1 Introduction Let 𝛴 be a Riemann surface, (M n , g) = 𝕊n or ℝn (n ≥ 3) the unit sphere or the Euclidean space with standard metrics and f an immersion from 𝛴 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
H = tr B, 𝜅M the Gauss curvature of df (T𝛴) with respect to the ambient metric g and d𝜇f the area element on f (𝛴) . The Willmore functional of the immersion f is then defined by ) ( 1 W(f ) = |H|2 + 𝜅M d𝜇f , ∫𝛴 4
* Linlin Sun [email protected] Yong Luo [email protected] 1
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2
Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China
3
Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany
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Vol.:(0123456789)
Annals of Global Analysis and Geometry
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]) ⟨ ⟩ d ���⃗ ), 𝜙 d𝜇f , W(f ) = W(f ∫𝛴 dt ���⃗ ) = ∑n W(f ���⃗ )𝛼 e𝛼 , where {e𝛼 ∶ 3 ≤ 𝛼 ≤ n} is a local orthonormal frame of the with W(f 𝛼=3 normal bundle of f (𝛴) in M and ) ( ∑ 1 𝛽 2 𝛼 𝛽 𝛼 𝛼 𝛼 ���⃗ ) = hij hij H − 2|H| H , 3 ≤ 𝛼 ≤ n, ΔH + W(f 2 i,j,𝛽 ( ) where h𝛼ij is the component of B and H 𝛼 is the trace of h𝛼ij .
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 ∑ h𝛼ij h𝛽ij H 𝛽 − 2|H|2 H 𝛼 = 0, 3 ≤ 𝛼 ≤ n. ΔH 𝛼 + (1.1) i,j,𝛽
When (M, g) = ℝ3 , Willmore [25] proved that the Willmore energy of closed surfaces is 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 ℝ3 [6, 24], which was verified by Marques and Neves in [13]. When (M, g) = ℝn , 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 ℝ4 . In the same p
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