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On minimal Lagrangian submanifolds in complex space forms with semi-parallel second fundamental form

Cece Li

https://orcid.org/0000-0003-1851-6200

School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, P. R. China

E-mail Address: ceceli@haust.edu.cn

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and

Cheng Xing

https://orcid.org/0009-0002-7789-5474

Key Laboratory of Pure Mathematics and Combinatorics (LPMC), School of Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China

E-mail Address: xingcheng@nankai.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0129167X24500563Cited by:1 (Source: Crossref)

Abstract

In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension $n$ in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases $n = 2, 3, 4$ . Second, supposing further that the scalar curvature is constant for $n \geq 5$ , we also give an answer to the problem by applying the classification theorem of [F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012) 79–115]. Finally, for such Lagrangian submanifolds in the above problem with $n \geq 3$ , we establish an inequality in terms of the traceless Ricci tensor, the squared norm of the second fundamental form and the scalar curvature. Moreover, this inequality is optimal in the sense that all the submanifolds attaining the equality are completely determined.

Communicated by Yuguang Shi

Keywords:

AMSC: 53B25, 53C24, 53C42, 53D12

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