Integrable distortion II: the p-Weil-Petersson variational problem
Abstract
We study the hyperbolically weighted distortion energies ℐp(f)=∫D |μf|p/((1−|μf|2)max{2,p}/2) dAη for finite-distortion disk homeomorphisms with prescribed boundary values, together with the associated inverse problem in the monotone Sobolev class. For 1<p≤2, the inverse density is polyconvex precisely when p≥ pc=1+1/√2. In this range, quasisymmetric data with finite-energy monotone barriers yield minimisers with holomorphic Ahlfors--Hopf differentials and strong local Sobolev convergence of minimising subsequences. For pc<p<2, a finite-energy finite-distortion barrier yields a unique absolute monotone minimiser, which is a quasiconformal diffeomorphism and is asymptotically conformal. Together with the boundary-class theorem of Part~I, this characterises the admissible boundary data as Tp. The proof combines a selected barrier approximation, uniform scalar gradient estimates and a weighted Reich--Strebel comparison. For 1<p<pc, finite rank-one comparisons bound distortion on the positive-Jacobian set by a cutoff strictly below the ellipticity cutoff. An attained monotone minimum is either a quasiconformal diffeomorphism or has nonzero rank-one derivatives on a set of positive area. For 0<p≤1, minimisers with finite distortion, or homeomorphic minimisers with a locally Sobolev inverse, are Möbius maps. We also obtain energy-dependent quasiconformal bounds for regular minimisers when p>2. \
Related
- Integrable distortion I: Weil--Petersson circle homeomorphisms and the Schoen conjecture
- Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current
- Closed Response Calculus for SLE Weldings and Weil--Petersson Kähler Geometry
- Scalar curvature, volume, and Weil-Petersson translation length
- The Weil-Petersson diameter of moduli space for large genus
BibTeX
@misc{martin2026integrable,
title = {{Integrable distortion II: the p-Weil-Petersson variational problem}},
author = {Gaven Martin and Cong Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34007},
eprint = {2610.34007},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34007},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Integrable distortion II: the p-Weil-Petersson variational problem
AU - Gaven Martin
AU - Cong Yao
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34007
UR - https://arxiv.org/abs/2610.34007
AB - We study the hyperbolically weighted distortion energies ℐ_p(f)=∫_D |μ_f|^p/((1−|μ_f|^2)^max{2,p}/2) dA_η for finite-distortion disk homeomorphisms with prescribed boundary values, together with the associated inverse problem in the monotone Sobolev class. For 1<p≤2, the inverse density is polyconvex precisely when p≥ p_c=1+1/√2. In this range, quasisymmetric data with finite-energy monotone barriers yield minimisers with holomorphic Ahlfors--Hopf differentials and strong local Sobolev convergence of minimising subsequences. For p_c<p<2, a finite-energy finite-distortion barrier yields a unique absolute monotone minimiser, which is a quasiconformal diffeomorphism and is asymptotically conformal. Together with the boundary-class theorem of Part~I, this characterises the admissible boundary data as T^p. The proof combines a selected barrier approximation, uniform scalar gradient estimates and a weighted Reich--Strebel comparison. For 1<p<p_c, finite rank-one comparisons bound distortion on the positive-Jacobian set by a cutoff strictly below the ellipticity cutoff. An attained monotone minimum is either a quasiconformal diffeomorphism or has nonzero rank-one derivatives on a set of positive area. For 0<p≤1, minimisers with finite distortion, or homeomorphic minimisers with a locally Sobolev inverse, are Möbius maps. We also obtain energy-dependent quasiconformal bounds for regular minimisers when p>2. \
ER -