Minimal Submanifolds and Waists of Locally Symmetric Spaces

Abstract

We show that compact locally symmetric manifolds M with universal cover the symmetric space X for SL(n,ℝ) form a topological higher d-expander family for d≤ n/8. We prove the same statement for SL(n,ℝ) replaced by a split simple non-compact real Lie group G and for d linear in the rank of G. We accomplish this by showing that minimal submanifolds of low codimension in such M must have volume comparable to the volume of M. Our proof is based on a new monotonicity formula for minimal submanifolds of X, together with bounds on the decay of matrix coefficients for unitary representations of higher rank Lie groups. We also give the first locally symmetric example of power-law systolic freedom. This paper partially supersedes \cite{fl24}.

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BibTeX RIS

BibTeX

@misc{fraczyk2026minimal,
  title = {{Minimal Submanifolds and Waists of Locally Symmetric Spaces}},
  author = {Mikolaj Fraczyk and Ben Lowe},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.40200},
  eprint = {2609.40200},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40200},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Minimal Submanifolds and Waists of Locally Symmetric Spaces
AU  - Mikolaj Fraczyk
AU  - Ben Lowe
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.40200
UR  - https://arxiv.org/abs/2609.40200
AB  - We show that compact locally symmetric manifolds M with universal cover the symmetric space X for SL(n,ℝ) form a topological higher d-expander family for d≤ n/8. We prove the same statement for SL(n,ℝ) replaced by a split simple non-compact real Lie group G and for d linear in the rank of G. We accomplish this by showing that minimal submanifolds of low codimension in such M must have volume comparable to the volume of M. Our proof is based on a new monotonicity formula for minimal submanifolds of X, together with bounds on the decay of matrix coefficients for unitary representations of higher rank Lie groups. We also give the first locally symmetric example of power-law systolic freedom. This paper partially supersedes \cite{fl24}.
ER  -

Abstract

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