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The lower mean curvature bound in Gromov's mean-of-the-mean-curvature conjecture

  • Christian Baer
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Abstract

Gromov conjectured that for a compact Riemannian manifold X with boundary, the total mean curvature ∫∂ X H is bounded above by a constant depending only on the intrinsic geometry of ∂ X and a lower bound on the scalar curvature of X. Previous results towards this conjecture require, in addition, a lower bound on the mean curvature of the boundary. In the present paper, we investigate whether this extra assumption is necessary. In dimension $2$, we show that no lower bound on the geodesic curvature is needed. We estimate the total geodesic curvature of the boundary in terms of its length and a lower bound for the Gauss curvature of the surface. This confirms Gromov's conjecture in $2$~dimensions without any extra assumptions. In contrast, we give examples showing that a lower bound on the mean curvature is genuinely needed in dimensions n ≥ 3.

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Cite

BibTeXRIS
BibTeX
@misc{baer2026lower,
  title = {{The lower mean curvature bound in Gromov's mean-of-the-mean-curvature conjecture}},
  author = {Christian Baer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01189},
  eprint = {2609.01189},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01189},
}
RIS
TY  - UNPB
TI  - The lower mean curvature bound in Gromov's mean-of-the-mean-curvature conjecture
AU  - Christian Baer
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01189
UR  - https://arxiv.org/abs/2609.01189
AB  - Gromov conjectured that for a compact Riemannian manifold X with boundary, the total mean curvature ∫_∂ X H is bounded above by a constant depending only on the intrinsic geometry of ∂ X and a lower bound on the scalar curvature of X. Previous results towards this conjecture require, in addition, a lower bound on the mean curvature of the boundary. In the present paper, we investigate whether this extra assumption is necessary. In dimension $2$, we show that no lower bound on the geodesic curvature is needed. We estimate the total geodesic curvature of the boundary in terms of its length and a lower bound for the Gauss curvature of the surface. This confirms Gromov's conjecture in $2$~dimensions without any extra assumptions. In contrast, we give examples showing that a lower bound on the mean curvature is genuinely needed in dimensions n ≥ 3.
ER  -