Abstract
In this paper, we study Lagrangian surfaces satisfying nabla ^*T=0 , where T=-2nabla ^*(check{A}lrcorner omega ) and check{A} is the Lagrangian trace-free second fundamental form. We obtain a gap lemma for such a Lagrangian surface.
Highlights
Gap phenomena form an interesting topic in differential geometry, with many related results to be found
The following result by Kuwert-Schätzle for the Willmore surfaces immersed in Euclidean space is one of our motivations: Theorem (Gap lemma for Willmore surfaces, [3, Th. 1.1] or [7, Th. 2.7]) Let f : be a properly immersed Willmore surface, and let f −1(B (0))
Our personal interest is on Lagrangian submanifolds which often play an important role in symplectic geometry where objects often have a natural presentation as Lagrangian manifolds
Summary
Gap phenomena form an interesting topic in differential geometry, with many related results to be found. Followed by Kuwert-Schätzle’s idea, Luo-Wang proved a similar result under Lagrangian settings: Theorem (Gap lemma for HW surfaces, [9, Th. 4.3]) Let f : → C2 be a properly immersed HW surface, there exists 0(n) > 0 such that if the norm of the second fundamental form A L2 < 0(n), it must be a Lagrangian plane. Combining the previous result with a classification theorem from Lagrangian geometry (see [4] or [5]), we obtain a gap theorem which answers the above question: Theorem 1.2 Assume f : → C2 is a properly immersed Lagrangian surface (compact or non-compact) that satisfies ∇∗T = 0, and let (0) = f −1(B (0)), there exists 0 > 0 such that if.
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