Lagrangian submanifolds in complex space forms satisfying equality in the optimal inequality involving $$\delta (2,\ldot
- PDF / 345,804 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 2 Downloads / 191 Views
Lagrangian submanifolds in complex space forms satisfying equality in the optimal inequality involving ı(2, . . . , 2) Bang-Yen Chen1 · Luc Vrancken2,3 · Xianfeng Wang4 Received: 5 August 2020 / Accepted: 5 October 2020 © The Managing Editors 2020
Abstract It was proved in Chen and Dillen (J Math Anal Appl 379(1), 229–239, 2011) and Chen et al. (Differ Geom Appl 31(6), 808–819, 2013) that every Lagrangian submanifold M of a complex space form M˜ n (4c) with constant holomorphic sectional curvature 4c satisfies the following optimal inequality: δ(2, . . . , 2) ≤
n 2 (2n − k − 2) 2 n 2 − n − 2k H + c, 2(2n − k + 4) 2
(A)
where H 2 is the squared mean curvature and δ(2, . . . , 2) is a δ-invariant on M introduced by the first author, and k is the multiplicity of 2 in δ(2, . . . , 2), where n ≥ 2k +1. This optimal inequality improves an earlier inequality obtained by the first author in Chen (Jpn J Math 26(1), 105–127, 2000). The main purpose of this paper is to study Lagrangian submanifolds of M˜ n (4c) satisfying the equality case of the optimal inequality (A). Keywords Lagrangian submanifold · Optimal inequality · δ-invariants · Ideal submanifolds · H -umbilical Lagrangian submanifold Mathematics Subject Classification Primary 53B25; Secondary 53D12
1 Introduction Let M˜ n be a Kähler n-manifold with the complex structure J , a Kähler metric g and the Kähler 2-form ω. An isometric immersion ψ : M → M˜ n of a Riemannian n-manifold M into M˜ n is called Lagrangian if ψ ∗ ω = 0.
X. Wang was supported in part by NSFC Grant No. 11971244, Natural Science Foundation of Tianjin, China (Grant No. 19JCQNJC14300) and the Fundamental Research Funds for the Central Universities.
B
Xianfeng Wang [email protected]
Extended author information available on the last page of the article
123
Beitr Algebra Geom
A Kähler n-manifold with constant holomorphic sectional curvature 4c, is called a complex space form and we denote it by M˜ n (4c). A complete simply-connected complex space form M˜ n (4c) is holomorphically isometric to the complex Euclidean n-plane Cn , the complex projective n-space CPn (4c), or a complex hyperbolic n-space CHn (4c) according to c = 0, c > 0 or c < 0, respectively. The first author introduced in the 1990s a sequence of new Riemannian invariants δ(n 1 , . . . , n k ) in order to study submanifolds of real space forms and in particular in order to obtain intrinsic restrictions whether or not a given manifold can be minimally and isometrically immersed into a given real space form. For any n-dimensional submanifold M in a real space form R m (c) with constant curvature c, he proved the following sharp general inequality (see Chen (2000b, 2011) for details): k n 2 n + k − 1 − i=1 ni H2 δ(n 1 , . . . , n k ) ≤ k 2 n + k − i=1 n i k 1 + n i (n i − 1) c. n(n − 1) − 2
(1.1)
i=1
As the Gauss equation of a Lagrangian submanifold of a complex space form has the same expression as the one for a submanifold of a real space form, it also immediately follows that Theorem A Let M be an
Data Loading...
