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arXiv · 2311.18767

Universal Liouville action as a renormalized volume and its gradient flow

Martin Bridgeman·Kenneth Bromberg·Franco Vargas Pallete·Yilin Wang
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Abstract

The universal Liouville action (also known as the Loewner energy for Jordan curves) is a Kähler potential on the Weil-Petersson universal Teichmüller space, which is identified with the family of Weil-Petersson quasicircles via conformal welding. Our main result shows that, under regularity assumptions, the universal Liouville action equals the renormalized volume of the hyperbolic $3$-manifold bounded by the two Epstein-Poincaré surfaces associated with the quasicircle. We also study the gradient descent flow of the universal Liouville action for the Weil-Petersson metric and show that the flow always converges to the origin (the circle). This provides a bound of the Weil-Petersson distance to the origin by the universal Liouville action.

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Martin Bridgeman, Kenneth Bromberg, Franco Vargas Pallete, Yilin Wang. 2025-01-30. Universal Liouville action as a renormalized volume and its gradient flow. https://arxiv.org/abs/2311.18767

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