Skip to content
DraconiticResearch
Search Sources API
190,860 studies31 sourcesUpdated 51 min ago
PreprintarXiv licencearXiv

The Height of Choe's Higher-Dimensional Enneper Surface

  • Yike He
Open at sourcePDFDOICite

Abstract

We prove that Choe's higher-dimensional Enneper hypersurfaces in ℝn+1, 3≤ n≤6, cannot be confined between two parallel hyperplanes, disproving his slab confinement conjecture. We further strengthen this conclusion by determining their precise height growth exponent. If H(R) denotes the maximal absolute height above the asymptotic hyperplane within the horizontal ball of radius R, then H(R)=Rα1+o(1) as R→∞, where α1=(√((n−2)2+4λ1(Ω))−(n−2))/2, Ω=Sn−1∖[0,∞)n, and λ1(Ω) is the first Dirichlet eigenvalue of the spherical Laplacian. In particular, 0<α1<1, so the height is unbounded but grows sublinearly. The proof combines the blow-down behavior of a fundamental piece of Choe's hypersurface with the rigidity of positive harmonic functions on cones. In dimension two, the formula gives 2/3, recovering the classical Enneper height-growth exponent.

Related

  1. Cops and Robber game in higher-dimensional manifolds with spherical and Euclidean metric
    Vesna Iršič, Bojan Mohar, Alexandra Wesolek
    Comptes Rendus Mathématiques. Repères de l'Académie des Sciences du CanadaPeer-reviewedLicence not statedGeneral mathematicsHAL
  2. The Isoperimetric Inequality in Higher-Dimensional Heisenberg Groups
    Wei-Chun Chen, Wei-Ting Kao, Chun-Kai Lien
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  3. The Bernstein problem in higher dimensional Heisenberg groups
    Gianmarco Giovannardi, Simone Verzellesi
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyarXiv
  4. Relative Du Bois complexes over higher dimensional bases
    Dora Kassabova, Haoming Ning
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Rigidity from the strong unstable foliation for higher-dimensional Anosov and partially hyperbolic flows
    Andrey Gogolev
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv

Cite

BibTeXRIS
BibTeX
@misc{he2026height,
  title = {{The Height of Choe's Higher-Dimensional Enneper Surface}},
  author = {Yike He},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14343},
  eprint = {2610.14343},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14343},
}
RIS
TY  - UNPB
TI  - The Height of Choe's Higher-Dimensional Enneper Surface
AU  - Yike He
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.14343
UR  - https://arxiv.org/abs/2610.14343
AB  - We prove that Choe's higher-dimensional Enneper hypersurfaces in ℝ^n+1, 3≤ n≤6, cannot be confined between two parallel hyperplanes, disproving his slab confinement conjecture. We further strengthen this conclusion by determining their precise height growth exponent. If H(R) denotes the maximal absolute height above the asymptotic hyperplane within the horizontal ball of radius R, then H(R)=R^α_1+o(1) as R→∞, where α_1=(√((n−2)^2+4λ_1(Ω))−(n−2))/2, Ω=S^n−1∖[0,∞)^n, and λ_1(Ω) is the first Dirichlet eigenvalue of the spherical Laplacian. In particular, 0<α_1<1, so the height is unbounded but grows sublinearly. The proof combines the blow-down behavior of a fundamental piece of Choe's hypersurface with the rigidity of positive harmonic functions on cones. In dimension two, the formula gives 2/3, recovering the classical Enneper height-growth exponent.
ER  -