Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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The geometric data on the boundary of convex subsets of hyperbolic manifolds
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by Qiyu Chen and Jean-Marc Schlenker;
Trans. Amer. Math. Soc. 378 (2025), 4225-4301
DOI: https://doi.org/10.1090/tran/9391
Published electronically: March 19, 2025

Abstract:

Let $N$ be a geodesically convex subset in a convex co-compact hyperbolic manifold $M$ with incompressible boundary. We assume that each boundary component of $N$ is either a boundary component of $\partial _\infty M$, or a smooth, locally convex surface in $M$. We show that $N$ is uniquely determined by the boundary data defined by the conformal structure on the boundary components at infinity, and by either the induced metric or the third fundamental form on the boundary components which are locally convex surfaces. We also describe the possible boundary data. This provides an extension of both the hyperbolic Weyl problem and the Ahlfors-Bers Theorem.

Using this statement for quasifuchsian manifolds, we obtain existence results for similar questions for convex domains $\Omega \subset \mathbb {H}^3$ which meets the boundary at infinity $\partial _{\infty }\mathbb {H}^3$ either along a quasicircle or along a quasidisk. The boundary data then includes either the induced metric or the third fundamental form in $\mathbb {H}^3$, but also an additional “gluing” data between different components of the boundary, either in $\mathbb {H}^3$ or in $\partial _\infty \mathbb {H}^3$.

References
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Bibliographic Information
  • Qiyu Chen
  • Affiliation: School of Mathematics, South China University of Technology, 510641 Guangzhou, People’s Republic of China
  • MR Author ID: 1248105
  • ORCID: 0009-0009-7656-4228
  • Email: qiyuchen@scut.edu.cn
  • Jean-Marc Schlenker
  • Affiliation: Department of Mathematics, FSTM, University of Luxembourg, Maison du nombre, 6 avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
  • MR Author ID: 362432
  • ORCID: 0000-0002-0853-4512
  • Email: jean-marc.schlenker@uni.lu
  • Received by editor(s): May 24, 2023
  • Received by editor(s) in revised form: April 22, 2024, and November 21, 2024
  • Published electronically: March 19, 2025
  • Additional Notes: The first author was partially supported by NSFC (No. 12471075, 12101244), Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515012225), and Guangzhou Science and Technology Program (No. 202201010464)
    The second author was partially supported by FNR project O20/14766753
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 4225-4301
  • MSC (2020): Primary 57K32; Secondary 58J05, 58J32
  • DOI: https://doi.org/10.1090/tran/9391