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The bounded area class of negatively curved surfaces

  • Roberto Frigerio
  • Ervin Hadziosmanovic
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Abstract

Let S be an oriented surface, possibly of infinite type, endowed with a complete Riemannian metric with pinched negative curvature. We prove that the area form defines a non-trivial class in the second bounded cohomology group of S, unless S is diffeomorphic to the disc or the cylinder. This is in sharp contrast with the n-dimensional case, n>2, where the non-triviality of the volume form in bounded cohomology is related to the Cheeger constant of the manifold. We also discuss how the bounded area class depends on the metric: we prove that, for compact surfaces, it recognizes constant curvature metrics among pinched negatively curved ones, while, even in the case of surfaces of infinite type, it does not distinguish non-isometric hyperbolic structures. More precisely, when S is compact we show that, among the negatively curved structures of fixed area, the ones with constant curvature provide the unique minimizers for the norm of the bounded area class.

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Cite

BibTeXRIS
BibTeX
@misc{frigerio2026bounded,
  title = {{The bounded area class of negatively curved surfaces}},
  author = {Roberto Frigerio and Ervin Hadziosmanovic},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01830},
  eprint = {2609.01830},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01830},
}
RIS
TY  - UNPB
TI  - The bounded area class of negatively curved surfaces
AU  - Roberto Frigerio
AU  - Ervin Hadziosmanovic
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01830
UR  - https://arxiv.org/abs/2609.01830
AB  - Let S be an oriented surface, possibly of infinite type, endowed with a complete Riemannian metric with pinched negative curvature. We prove that the area form defines a non-trivial class in the second bounded cohomology group of S, unless S is diffeomorphic to the disc or the cylinder. This is in sharp contrast with the n-dimensional case, n>2, where the non-triviality of the volume form in bounded cohomology is related to the Cheeger constant of the manifold. We also discuss how the bounded area class depends on the metric: we prove that, for compact surfaces, it recognizes constant curvature metrics among pinched negatively curved ones, while, even in the case of surfaces of infinite type, it does not distinguish non-isometric hyperbolic structures. More precisely, when S is compact we show that, among the negatively curved structures of fixed area, the ones with constant curvature provide the unique minimizers for the norm of the bounded area class.
ER  -