eprintid: 10198158
rev_number: 9
eprint_status: archive
userid: 699
dir: disk0/10/19/81/58
datestamp: 2024-10-08 13:15:33
lastmod: 2024-10-08 13:15:33
status_changed: 2024-10-08 13:15:33
type: article
metadata_visibility: show
sword_depositor: 699
creators_name: Karpukhin, Mikhail
creators_name: Métras, Antoine
creators_name: Polterovich, Iosif
title: Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces
ispublished: inpress
divisions: UCL
divisions: B04
divisions: C06
divisions: F59
note: © The Author(s) 2024. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
abstract: Consider a Dirac operator on an oriented compact surface endowed with a Riemannian metric and spin structure. Provided the area and the conformal class are fixed, how small can the $k$-th positive Dirac eigenvalue be? This problem mirrors the maximization problem for the eigenvalues of the Laplacian, which is related to the study of harmonic maps into spheres. We uncover the connection between the critical metrics for Dirac eigenvalues and harmonic maps into complex projective spaces. Using this approach we show that for many conformal classes on a torus the first nonzero Dirac eigenvalue is minimised by the flat metric. We also present a new geometric proof of Bär’s theorem stating that the first nonzero Dirac eigenvalue on the sphere is minimised by the standard round metric.
date: 2024-10-03
date_type: published
publisher: Oxford University Press (OUP)
official_url: https://doi.org/10.1093/imrn/rnae216
oa_status: green
full_text_type: pub
language: eng
primo: open
primo_central: open_green
verified: verified_manual
elements_id: 2324943
doi: 10.1093/imrn/rnae216
lyricists_name: Karpukhin, Mikhail
lyricists_id: MKARP96
actors_name: Karpukhin, Mikhail
actors_id: MKARP96
actors_role: owner
full_text_status: public
publication: International Mathematics Research Notices
article_number: rnae216
issn: 1073-7928
citation:        Karpukhin, Mikhail;    Métras, Antoine;    Polterovich, Iosif;      (2024)    Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces.                   International Mathematics Research Notices      , Article rnae216.  10.1093/imrn/rnae216 <https://doi.org/10.1093/imrn%2Frnae216>.    (In press).    Green open access   
 
document_url: https://discovery.ucl.ac.uk/id/eprint/10198158/7/Karpukhin_rnae216.pdf