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Comparison between renormalized volume and area in the Weil--Petersson topology

  • Dorian Martino
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Abstract

We prove that the renormalized volume between the two Epstein surfaces associated with a Weil--Petersson quasicircle is continuous with respect to the Weil--Petersson topology. As a consequence, we establish that the universal Liouville action is proportional to this renormalized volume for every Weil--Petersson quasicircle. We then establish an inequality relating this renormalized volume to the renormalized area introduced by Alexakis--Mazzeo. To this end, we compare both quantities with the Hilbert--Schmidt norm of the associated Grunsky operator, which can be expressed in terms of a nonlocal Lagrangian introduced by O'Hara--Solanes.

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Cite

BibTeXRIS
BibTeX
@misc{martino2026comparison,
  title = {{Comparison between renormalized volume and area in the Weil--Petersson topology}},
  author = {Dorian Martino},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.03723},
  eprint = {2609.03723},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03723},
}
RIS
TY  - UNPB
TI  - Comparison between renormalized volume and area in the Weil--Petersson topology
AU  - Dorian Martino
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.03723
UR  - https://arxiv.org/abs/2609.03723
AB  - We prove that the renormalized volume between the two Epstein surfaces associated with a Weil--Petersson quasicircle is continuous with respect to the Weil--Petersson topology. As a consequence, we establish that the universal Liouville action is proportional to this renormalized volume for every Weil--Petersson quasicircle. We then establish an inequality relating this renormalized volume to the renormalized area introduced by Alexakis--Mazzeo. To this end, we compare both quantities with the Hilbert--Schmidt norm of the associated Grunsky operator, which can be expressed in terms of a nonlocal Lagrangian introduced by O'Hara--Solanes.
ER  -