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.
(In press).
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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.
| Type: | Article |
|---|---|
| Title: | Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1093/imrn/rnae216 |
| Publisher version: | https://doi.org/10.1093/imrn/rnae216 |
| Language: | English |
| Additional information: | © 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. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10198158 |
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