| Résume | A Jordan curve on the Riemann sphere can be encoded by its conformal welding homeomorphism, which is a circle homeomorphism. The graph of the welding homeomorphism can be naturally viewed as a positive curve onbthe boundary of AdS3 space. For instance, Thurston’s earthquake map associated with a circle homeomorphism has a geometric interpretation in AdS3.
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 Mobius. 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. We show that these two problems exhibit mysterious symmetry. |
