TY  - JOUR
JF  - Inventiones Mathematicae
PB  - Springer Science and Business Media LLC
A1  - Karpukhin, Mikhail
A1  - Stern, Daniel
SP  - 713
VL  - 236
N1  - © 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/
EP  - 778
UR  - https://doi.org/10.1007/s00222-024-01247-3
ID  - discovery10194583
N2  - We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold  M?, 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 = S? ?3,  we obtain a distinguished family of nonconstant harmonic maps M S?  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 
AV  - public
Y1  - 2024/05//
SN  - 0020-9910
TI  - Existence of harmonic maps and eigenvalue optimization in
higher dimensions
ER  -