Two optimization problems of the Loewner energy

par Yilin Wang (ETH Zürich)

vendredi 15 mai 2026, 15:30 → 17:00 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Séminaire d'Analyse

A Jordan curve on the Riemann sphere can be encoded by its conformal welding homeomorphism, which is a circle homeomorphism. I will explain that this correspondence should be viewed as a canonical correspondence between a Jordan curve in the boundary of hyperbolic 3-space H³ and a positive curve on the boundary of AdS³ space.

The Loewner energy measures how far a Jordan curve is away from being a circle or, equivalently, how far its welding homeomorphism is away from being Möbius. It arises as the action of random curves SLE, Kähler potential of Weil-Petersson universal Teichmüller space, Fredholm determinant of Grunsky operator, free energy of Coulomb gas on a Jordan curve, and a renormalized volume of H³, etc. All these links refer to either the curve description or the welding description of the Loewner energy.

I will discuss two optimizing problems for the Loewner energy, one under the constraint for the curve to pass through n given points on the Riemann sphere and the other under the constraint for the welding curve to pass through n given points in the boundary of AdS³. These two problems exhibit many symmetries that are poorly understood, but do suggest that the Loewner energy sits right in the middle of two perspectives (curve/welding).

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