Sphere theorems for Lagrangian and Legendrian submanifolds
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Calculus of Variations
Sphere theorems for Lagrangian and Legendrian submanifolds Jun Sun1,2 · Linlin Sun1,2 Received: 6 March 2020 / Accepted: 31 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove some differentiable sphere theorems and topological sphere theorems for Lagrangian submanifolds in Kähler manifold and Legendrian submanifolds in Sasaki space form. Mathematics Subject Classification 53C20 · 53C40
1 Introduction The study of Lagrangian submanifolds in a Kähler manifold, especially in a Calabi-Yau manifold, has attracted much attention in the past few decades [15], partially because of its importance in classical mechanics and mathematical physics. For instance, Strominger, Yau and Zaslow [23] found that mirror symmetry is related closely to special Lagrangian submanifolds in Calabi-Yau manifold. Let (N 2n , g, ¯ ω, ¯ J ) be a Kähler manifold. A submanifold M n in N 2n is called a Lagrangian submanifold, if the restriction of the Kähler form ω¯ to M vanishes. Or equivalently, for any
Communicated by J. Jost. The first author was supported by the National Natural Science Foundation of China (Grant No. 11401440). Part of the work was finished when the first author was a visiting scholar at MIT supported by China Scholarship Council (CSC) and the Youth Talent Training Program of Wuhan University. The author would like to express his gratitude to Professor Tobias Colding for his invitation and to MIT for their hospitality. The second author was supported by the National Natural Science Foundation of China (Grant Nos. 11801420, 11971358) and the Youth Talent Training Program of Wuhan University. The author thanks the Max Planck Institute for Mathematics in the Sciences for good working conditions when this work was carried out. They also thank Dr. Yong Luo for helpful discussions.
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Linlin Sun [email protected] Jun Sun [email protected]
1
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
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Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China 0123456789().: V,-vol
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J. Sun, L. Sun
x ∈ M, J maps Tx M onto N x M, where J is the complex structure on N , and Tx M and N x M are the tangent space and normal space of M at x in N , respectively. Since the tangent bundle and the normal bundle of a Lagrangian submanifold are isomorphic via the complex structure J of the ambient manifold, Lagrangian submanifold has its own special properties in topology and geometry, particularly in its second fundamental form. A result of Gromov [12] implies that every compact embedded Lagrangian submanifold of Cn is not simply-connected. Of course, there exist immersed compact Lagrangian submanifolds in Cn . One standard example is the well-known Whitney sphere, which is given by F : Sn −→ Cn (x0 , · · · , xn ) −→
1 (x1 , · · · , xn , x0 x1 , · · · , x0 xn ). 1 + x02
Gromov [11] also showed that a compact n-manifold M admits a Lagrangian immersion into Cn if and only if the complexification of the tangent
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