Illinois Journal of Mathematics

Minimal Lagrangian submanifolds in the complex hyperbolic space

Ildefonso Castro, Cristina R. Montealegre, and Francisco Urbano

Full-text: Open access

Abstract

In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomogeneity one. We characterize these submanifolds as the only minimal Lagrangian submanifolds in $\mathbb{C}\mathbb{H}^n$ that are foliated by umbilical hypersurfaces of Lagrangian subspaces $\mathbb{R}\mathbb{H}^n$ of $\mathbb{C}\mathbb{H}^n$. By suitably generalizing this construction, we obtain new families of minimal Lagrangian submanifolds in $\mathbb{C}\mathbb{H}^n$ from curves in $\mathbb{C}\mathbb{H}^1$ and $(n-1)$-dimensional minimal Lagrangian submanifolds of the complex space forms $\mathbb{C}\mathbb{P}^{n-1}$, $\mathbb{C}\mathbb{H}^{n-1}$ and $\mathbb{C}^{n-1}$. We give similar constructions in the complex projective space $\mathbb{C}\mathbb{P}^n$.

Article information

Source
Illinois J. Math., Volume 46, Number 3 (2002), 695-721.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258130980

Digital Object Identifier
doi:10.1215/ijm/1258130980

Mathematical Reviews number (MathSciNet)
MR1951236

Zentralblatt MATH identifier
1032.53052

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53D12: Lagrangian submanifolds; Maslov index

Citation

Castro, Ildefonso; Montealegre, Cristina R.; Urbano, Francisco. Minimal Lagrangian submanifolds in the complex hyperbolic space. Illinois J. Math. 46 (2002), no. 3, 695--721. doi:10.1215/ijm/1258130980. https://projecteuclid.org/euclid.ijm/1258130980


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