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A characterization of submanifolds of ℝm × n satisfying optimal rigidity estimates

  • Malte Borken
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Abstract

Let K ⊂ ℝm × n be a compact C1-submanifold with boundary, p ∈ (1,∞) and Q := (0,1)n. We prove that K satisfies a rigidity estimate of the form |Du − (Du)Q|Lp ≤ C |distK(Du)|Lp, u ∈ W1,p(Q,ℝm), if and only if K satisfies sequential rigidity and for each A ∈ K, the tangent space to K at A satisfies exact rigidity. We further prove that this rigidity estimate is stable under small graphical perturbations of K. The key technical ingredient is proving that outside some small bad set where the maximal function of distKp(Du) is large, the size of the superlevel sets of lvert Du − (Du)Qrvert decays exponentially. This is achieved by an adaptation of the John-Nirenberg inequality for BMO-functions.

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Cite

BibTeXRIS
BibTeX
@misc{borken2026characterization,
  title = {{A characterization of submanifolds of $\mathbbR^m \times n$ satisfying optimal rigidity estimates}},
  author = {Malte Borken},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13218},
  eprint = {2609.13218},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13218},
}
RIS
TY  - UNPB
TI  - A characterization of submanifolds of ℝ^m × n satisfying optimal rigidity estimates
AU  - Malte Borken
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.13218
UR  - https://arxiv.org/abs/2609.13218
AB  - Let K ⊂ ℝ^m × n be a compact C^1-submanifold with boundary, p ∈ (1,∞) and Q := (0,1)^n. We prove that K satisfies a rigidity estimate of the form |Du − (Du)_Q|_L^p ≤ C |dist_K(Du)|_L^p, u ∈ W^1,p(Q,ℝ^m), if and only if K satisfies sequential rigidity and for each A ∈ K, the tangent space to K at A satisfies exact rigidity. We further prove that this rigidity estimate is stable under small graphical perturbations of K. The key technical ingredient is proving that outside some small bad set where the maximal function of dist_K^p(Du) is large, the size of the superlevel sets of lvert Du − (Du)_Qrvert decays exponentially. This is achieved by an adaptation of the John-Nirenberg inequality for BMO-functions.
ER  -