Vol. 293, No. 1, 2018

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Contact stationary Legendrian surfaces in $\mathbb{S}^5$

Yong Luo

Vol. 293 (2018), No. 1, 101–120
Abstract

Let (M5,α,gα,J) be a 5-dimensional Sasakian Einstein manifold with contact 1-form α, associated metric gα and almost complex structure J, and let L be a contact stationary Legendrian surface in M5. We will prove that L satisfies the equation

ΔνH + (K 1)H = 0,

where Δν is the normal Laplacian with respect to the metric g on L induced from gα and K is the Gauss curvature of (L,g).

Using this equation and a new Simons’ type inequality for Legendrian surfaces in the standard unit sphere S5, we prove an integral inequality for contact stationary Legendrian surfaces in S5. In particular, we prove that if L is a contact stationary Legendrian surface in S5 and B is the second fundamental form of L, with S = |B|2, ρ2 = S 2H2 and

0 S 2,

then we have either ρ2 = 0 and L is totally umbilic or ρ20, S = 2,H = 0 and L is a flat minimal Legendrian torus.

Keywords
volume functional, Simons inequality, gap theorem
Mathematical Subject Classification 2010
Primary: 53B25, 53C42
Milestones
Received: 17 November 2016
Accepted: 11 September 2017
Published: 3 November 2017
Authors
Yong Luo
School of Mathematics and Statistics
Wuhan University
Wuhan
China