Second variation of compact minimal Legendrian submanifolds of the sphere
type: Item
instance of: scholarly article
title: Second variation of compact minimal Legendrian submanifolds of the sphere
zbMATH Open document ID: 1043.53052
published in: Michigan Mathematical Journal
publication date: 2003-10-27T00:00:00Z
review text: The author computes the Jacobi operator \(L\) of a compact minimal Legendrian submanifold \(M\) of \({\mathbb S}^{2n+1}\) to prove that it is an intrinsic operator on the submanifold and that it can be written in terms of the exterior differential operator \(d,\) its codifferential operator \(\delta\) and the Laplacian \(\triangle\) as \(L(\alpha, f)=(\triangle \alpha+2(n-1)\alpha+2df, \triangle f-2\delta\alpha)\) where \(\alpha\) is an 1-form and \(f\) is a smooth function. He also obtains a formula for the index \(\text{Ind}(M)\) and the nullity \(\text{Null}(M)\) of \(L\) and in particular, he proves the following results for an orientable, compact, non-totally geodesic minimal Legendrian surface in \({\mathbb S}^5\): (1) \(\text{Ind}(M)\geq8\) and the equality holds if and only if \(M\) is an equilateral torus. (2) \(\text{Null}(M)\geq13\) and the equality holds if and only if \(M\) is an equilateral torus.
reviewed by: Sung-Eun Koh
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1997322
zbMATH Keywords: minimal Legendrian submanifolds
zbMATH Keywords: Jabobi operator
zbMATH Keywords: index
zbMATH Keywords: nullity
author: Francisco Urbano
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1060013206
OpenAlex ID: W2019473590
cites work: Totally Real Minimal Immersions of n-Dimensional Real Space Forms into n-Dimensional Complex Space Forms
cites work: The index of minimal immersions of \(S^ 2\) into \(S^{2n}\)
cites work: Riemannian manifolds admitting isometric immersions by their first eigenfunctions
cites work: Spectra and eigenforms of the Laplacian on \(S^n\) and \(P^n(C)\)
cites work: A characterization of minimal Legendrian submanifolds in \({\mathbb{S}}^{2n+1}\)
cites work: Second variation of superminimal surfaces into self-dual Einstein four-manifolds
cites work: Certain conditions for a Riemannian manifold to be isometric with a sphere
cites work: Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds
cites work: Spectral geometry of CR-minimal submanifolds in the complex projective space
cites work: Minimal varieties in Riemannian manifolds
cites work: Minimal Surfaces with Low Index in the Three-Dimensional Sphere
cites work: Index of Lagrangian submanifolds of \(\mathbb{C} \mathbb{P}^ n\) and the Laplacian of 1-forms
cites work: Submanifolds with Constant Mean Curvature
DOI: 10.1307/MMJ/1060013206
description: scientific article; zbMATH DE number 1997322