Conformally critical metrics and optimal bounds for Dirac eigenvalues on spin surfaces

04.06.2026, 16:15  –  Raum 0.17, Haus 9
Forschungsseminar Differentialgeometrie

Pavel Martynyuk (ENS Paris)

Optimization of eigenvalues of differential operators on manifolds connects spectral geometry with the theory of minimal submanifolds and harmonic maps. For the Laplacian, this approach traces back to Nadirashvili’s resolution of Berger’s isoperimetric problem, concerning the maximization of eigenvalues on a two-dimensional torus.
Recently, Karpukhin, Métras, and Polterovich showed that metrics extremal for Dirac eigenvalues within a conformal class are closely related to harmonic maps into complex projective spaces. This naturally leads to the question: do such optimal metrics always exist?
In this talk, I will present a criterion for the existence of minimizing metrics and explain its applications to obtaining optimal bounds for Dirac eigenvalues on the two-dimensional sphere, generalizing the classical Bär’s inequality.