Karpukhin, Mikhail;
Stern, Daniel;
(2024)
Existence of harmonic maps and eigenvalue optimization in
higher dimensions.
Inventiones Mathematicae
, 236
pp. 713-778.
10.1007/s00222-024-01247-3.
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Abstract
We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold Mⁿ, g) of dimension n ⪪ 2 to any closed, non-aspherical manifold N containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres N = Sᵏ ⪫3, we obtain a distinguished family of nonconstant harmonic maps M Sᵏ of index at most k+1, with singular set of codimension at least 7 for k sufficiently large. Furthermore, if 3 ⪪ n ⪪ 5, we show that these smooth harmonic maps stabilize as k becomes large, and correspond to the solutions of an eigenvalue optimization problem on
| Type: | Article |
|---|---|
| Title: | Existence of harmonic maps and eigenvalue optimization in higher dimensions |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00222-024-01247-3 |
| Publisher version: | https://doi.org/10.1007/s00222-024-01247-3 |
| Language: | English |
| Additional information: | © The Author(s), 2024. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. https://creativecommons.org/licenses/by/4.0/ |
| 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/10194583 |
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