We consider Jordan curves passing through a given set of n points with the property such that each edge is the hyperbolic geodesic in the complement of the rest of the curve. Those curves are related to the minimizers of the Loewner energy and give a specific complex projective structure on the n-punctured sphere with real and parabolic holonomy around the punctures. Similar to a result of Takhtajan-Zograf, we also obtain a new type of accessory parameters and show that they can be expressed as differentials of the Loewner energy.

This is based on the joint work with Don Marshall and Steffen Rohde.

The Loewner energy is a conformally invariant quantity that measures the roundness of a Jordan curve, introduced recently, from the study of large deviations of Schramm-Loewner evolutions with vanishing parameter. This perspective naturally connects Loewner energy to determinants of Laplacians and Brownian loop measures.

More surprisingly, we show that a curve has finite Loewner energy if and only if it is a Weil-Petersson quasicircle, a class of Jordan curves studied since the eighties, that has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, and string theory (now we add to the list random conformal geometry for the link to SLE). In these lectures, I will overview the connection of Loewner energy to these probabilistic and analytic concepts.

Main references:
Yilin Wang: Large deviations of Schramm-Loewner evolutions: A survey. ArXiv:2102.07032

Yilin Wang: Equivalent Descriptions of the Loewner Energy.
Invent. Math., Vol. 218. 2, 573-621 (2019)

Leon A. Takhtajan and Lee-Peng Teo: Weil-Petersson metric on the universal
Teichmüller space. Mem. Amer. Math. Soc., 183(861):viii+119 (2006)

The Loewner energy is a conformally invariant quantity that measures the roundness of a Jordan curve, introduced recently, from the study of large deviations of Schramm-Loewner evolutions with vanishing parameter. This perspective naturally connects Loewner energy to determinants of Laplacians and Brownian loop measures.

More surprisingly, we show that a curve has finite Loewner energy if and only if it is a Weil-Petersson quasicircle, a class of Jordan curves studied since the eighties, that has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, and string theory (now we add to the list random conformal geometry for the link to SLE). In these lectures, I will overview the connection of Loewner energy to these probabilistic and analytic concepts.

Main references:
Yilin Wang: Large deviations of Schramm-Loewner evolutions: A survey. ArXiv:2102.07032

Yilin Wang: Equivalent Descriptions of the Loewner Energy.
Invent. Math., Vol. 218. 2, 573-621 (2019)

Leon A. Takhtajan and Lee-Peng Teo: Weil-Petersson metric on the universal
Teichmüller space. Mem. Amer. Math. Soc., 183(861):viii+119 (2006)