Index of Lagrangian submanifolds of \(\mathbb{C} \mathbb{P}^ n\) and the Laplacian of 1-forms
type: Item
instance of: scholarly article
title: Index of Lagrangian submanifolds of \(\mathbb{C} \mathbb{P}^ n\) and the Laplacian of 1-forms
zbMATH Open document ID: 0802.53013
DOI: 10.1007/BF01264074
author: Francisco Urbano
published in: Geometriae Dedicata
publication date: 1994-12-11T00:00:00Z
review text: Let \(\mathbb{C}\mathbb{P}^ n\) be complex projective space with the Fubini Study metric. Let \(\phi: M \to \mathbb{C}\mathbb{P}^ n\) be a minimal Lagrangian immersion of a compact manifold \(M\) and let \(\lambda_ i\) be the first two eigenvalues of the Laplacian acting on 1-forms. The author shows that \(\lambda_ 1 \leq (n - 1)/2\) with equality if and only if \(\phi\) is either the totally geodesic embedding of \(\mathbb{R}\mathbb{P}^ n(1/4)\) in \(\mathbb{C}\mathbb{P}^ n\) or is the totally geodesic immersion of \(S^ 2(1/4)\) in \(\mathbb{C}\mathbb{P}^ 2\). The author shows that \[ \{(n - 1)/2 - \lambda_ 1\}\{(n+1)/2 - \lambda_ 2\} + \lambda_ 1 \geq 2\int_ M \overline{\rho}/\text{vol}(M) \] where \(\overline{\rho}\) is the normalized scalar curvature of \(M\) with equality if and only if \(\phi\) is totally geodesic or if \(M\) is the generalized Clifford torus. The author uses these results to show that if \(M\) is an orientable surface, then the index of \(M\) is at least 2 with equality if and only if \(M\) is the Clifford torus. The author also shows the only Hamiltonian stable minimal Lagrangian torus in \(\mathbb{C}\mathbb{P}^ 2\) is the Clifford torus.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 493320
zbMATH Keywords: Fubini Study metric
zbMATH Keywords: Lagrangian immersion
zbMATH Keywords: Laplacian
zbMATH Keywords: totally geodesic embedding
zbMATH Keywords: totally geodesic immersion
zbMATH Keywords: generalized Clifford torus
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01264074
OpenAlex ID: W1967320832
description: scientific article; zbMATH DE number 493320