Loewner's equation provides a way to encode a simply connected domain via a real-valued driving function of its boundary and is crucial in the solution to the Bieberbach conjecture, also gives rises to the Schramm Loewner evolution. The Loewner's energy of a simple loop is the Dirichlet energy of its driving function. It depends a priori on the parametrization of the simple loop. However, it was shown in a joint work with Steffen Rohde that there is no such dependence, therefore provides a Moebius invariant quantity for free loops which vanishes only on circles.

In this talk, I will present an intrinsic interpretation of the Loewner energy using the zeta-regularizations of determinants of Laplacians and show that the class of finite energy loops coincides with the Weil-Petersson class in the universal Teichmueller space.

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