Quasi-convexity of energy functions along Teichmüller geodesics

Abstract

Hyperbolic length functions are among the most fundamental ones on Teichmüller space, and they are quasi-convex along Teichmüller geodesics. In this paper, we investigate the same question for energy functions of harmonic maps in two natural settings, which may be viewed as nonlinear and higher-dimensional analogs of the length functions. For a fixed domain and a varying hyperbolic target, we prove the energy is quasi-convex along Teichmüller geodesics under a filling hypothesis. Furthermore, we generalize Masur's result on asymptotic growth of the length function along the Teichmüller geodesic determined by a Jenkins-Strebel differential to the energy functions. We also prove the quasi-convexity for covering maps between closed hyperbolic surfaces with fixed target and varying domains. We derive first and second variation formulas of energy functions along Teichmüller geodesics and explain why the natural global statement is quasi-convexity rather than genuine convexity.

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BibTeX RIS

BibTeX

@misc{kim2026quasiconvexity,
  title = {{Quasi-convexity of energy functions along Teichmüller geodesics}},
  author = {Inkang Kim and Xueyuan Wan and Genkai Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30752},
  eprint = {2608.30752},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30752},
}

RIS

TY  - UNPB
TI  - Quasi-convexity of energy functions along Teichmüller geodesics
AU  - Inkang Kim
AU  - Xueyuan Wan
AU  - Genkai Zhang
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30752
UR  - https://arxiv.org/abs/2608.30752
AB  - Hyperbolic length functions are among the most fundamental ones on Teichmüller space, and they are quasi-convex along Teichmüller geodesics. In this paper, we investigate the same question for energy functions of harmonic maps in two natural settings, which may be viewed as nonlinear and higher-dimensional analogs of the length functions. For a fixed domain and a varying hyperbolic target, we prove the energy is quasi-convex along Teichmüller geodesics under a filling hypothesis. Furthermore, we generalize Masur's result on asymptotic growth of the length function along the Teichmüller geodesic determined by a Jenkins-Strebel differential to the energy functions. We also prove the quasi-convexity for covering maps between closed hyperbolic surfaces with fixed target and varying domains. We derive first and second variation formulas of energy functions along Teichmüller geodesics and explain why the natural global statement is quasi-convexity rather than genuine convexity.
ER  -

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