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Variational theory for harmonic maps and applications - Lecture 2

By Daniel Stern

Appears in collection : Avancées récentes en analyse géométrique / Recent advances in geometric analysis

I will survey recent progress on the existence and regularity theory for harmonic maps from arbitrary closed manifolds to large classes of positively curved targets, with special emphasis on a natural family of sphere-valued harmonic maps which turns out to be intimately related to isoperimetric problems in spectral geometry, based on joint work with M. Karpukhin. In the case of two-dimensional domains, I will discuss applications of these techniques to the existence, regularity, and stability of metrics maximizing Laplace or Steklov eigenvalues on surfaces, highlighting some of the key ingredients in forthcoming work with Karpukhin, Kusner, and McGrath, in which these methods are employed to produce new families of minimal surfaces in $B^3$ and $S^3$ with prescribed topology.

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Citation data

  • DOI 10.24350/CIRM.V.20110003
  • Cite this video Stern, Daniel (07/11/2023). Variational theory for harmonic maps and applications - Lecture 2. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20110003
  • URL https://dx.doi.org/10.24350/CIRM.V.20110003


  • KARPUKHIN, Mikhail et STERN, Daniel L. Min-max harmonic maps and a new characterization of conformal eigenvalues. arXiv preprint arXiv:2004.04086, 2020. - https://arxiv.org/abs/2004.04086
  • KARPUKHIN, Mikhail et STERN, Daniel. Existence of harmonic maps and eigenvalue optimization in higher dimensions. arXiv preprint arXiv:2207.13635, 2022. - https://arxiv.org/abs/2207.13635

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