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PreprintCC BY 4.0arXiv

Rigidity of Euclidean Minimal Hypersurfaces under Nonuniform Diagonal Dilations

  • Jongha Lee
  • Suhwan Lee
  • Jae Won Lee
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Abstract

Let n≥3 and Dt=diag(tg1,…,tgn) be a positive diagonal dilation family. We study connected embedded Euclidean hypersurfaces whose diagonal images are minimal. The level-set minimality operator splits into coefficients indexed by the pair sums gi+gj. Under pair-sum nonresonance, minimality at only binom n2 distinct dilation parameters forces all pair coefficients to vanish. A dimension-reduction argument then shows, without any hypothesis on the coordinate components of the normal, that the second fundamental form vanishes identically. This yields an affine characterization. Repeated-weight helicoidal examples in every dimension and a resonant quadratic cone show that curvature cancellation can survive in genuinely nonuniform families. An application gives a finite-output-level rigidity criterion and an explicit representation for weighted-homogeneous production functions with minimal isoquants.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026rigidity,
  title = {{Rigidity of Euclidean Minimal Hypersurfaces under Nonuniform Diagonal Dilations}},
  author = {Jongha Lee and Suhwan Lee and Jae Won Lee},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00668},
  eprint = {2609.00668},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00668},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Rigidity of Euclidean Minimal Hypersurfaces under Nonuniform Diagonal Dilations
AU  - Jongha Lee
AU  - Suhwan Lee
AU  - Jae Won Lee
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00668
UR  - https://arxiv.org/abs/2609.00668
AB  - Let n≥3 and D_t=diag(t^g_1,…,t^g_n) be a positive diagonal dilation family. We study connected embedded Euclidean hypersurfaces whose diagonal images are minimal. The level-set minimality operator splits into coefficients indexed by the pair sums g_i+g_j. Under pair-sum nonresonance, minimality at only binom n2 distinct dilation parameters forces all pair coefficients to vanish. A dimension-reduction argument then shows, without any hypothesis on the coordinate components of the normal, that the second fundamental form vanishes identically. This yields an affine characterization. Repeated-weight helicoidal examples in every dimension and a resonant quadratic cone show that curvature cancellation can survive in genuinely nonuniform families. An application gives a finite-output-level rigidity criterion and an explicit representation for weighted-homogeneous production functions with minimal isoquants.
ER  -