On the K?hler form of complex \(L^{p}\) space and its Lagrangian subspaces

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• 作者：Yang Liu
• 关键词：Complex space ; K?hler form ; Lagrangian subspace ; Differential geometry ; Complex structure ; 70S05 ; 32Q15 ; 53C56 ; 47A15 ; 53C65
• 刊名：Journal of Pseudo-Differential Operators and Applications
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：6
• 期：2
• 页码：265-277
• 全文大小：456 KB
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• 作者单位：Yang Liu (1)
  
  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：None Assigned
• 出版者：Birkh盲user Basel
• ISSN：1662-999X

文摘

In this article, we obtain the K?hler forms for complex \(L^{p}\) spaces, \(1\le p<\infty \), and we find and describe explicitly the set of all Lagrangian subspaces of the complex \(L^{p}\) space. The results in this article show that the Lagrangians of complex \(L^{2}\) space are distinct from those of complex \(L^{p}\) spaces for \(1\le p<\infty \), \(p\ne 2\). As an application, the symplectic structure determined by the K?hler form can be used to determine the symplectic form of the complex Holmes–Thompson volumes restricted on complex lines in integral geometry of complex \(L^{p}\) space.