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Minimal hypersurfaces of finite δ-index in ℝ4 and ℝ5

  • Marcos Ranieri
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Abstract

Let X:Mn→ℝn+1 be a complete, connected, two-sided minimal immersion without boundary, where n=3,4. We prove that finite δ-index, finite Morse index, and finite total curvature are equivalent for δ>((n−1)/n)2, without assumptions on properness, volume growth, or topology. The main estimate shows that δ-stability outside a compact set and finite-dimensional Hc1(M;ℝ) imply intrinsic Euclidean volume growth for δ>(n−2)/n. We also deduce the sharp δ-stable Bernstein theorem in this latter range from results of Hong--Li--Wang and Florit-Simon.

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Cite

BibTeXRIS
BibTeX
@misc{ranieri2026minimal,
  title = {{Minimal hypersurfaces of finite $δ$-index in $\mathbb R^4$ and $\mathbb R^5$}},
  author = {Marcos Ranieri},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14317},
  eprint = {2609.14317},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14317},
}
RIS
TY  - UNPB
TI  - Minimal hypersurfaces of finite δ-index in ℝ^4 and ℝ^5
AU  - Marcos Ranieri
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14317
UR  - https://arxiv.org/abs/2609.14317
AB  - Let X:M^n→ℝ^n+1 be a complete, connected, two-sided minimal immersion without boundary, where n=3,4. We prove that finite δ-index, finite Morse index, and finite total curvature are equivalent for δ>((n−1)/n)^2, without assumptions on properness, volume growth, or topology. The main estimate shows that δ-stability outside a compact set and finite-dimensional H_c^1(M;ℝ) imply intrinsic Euclidean volume growth for δ>(n−2)/n. We also deduce the sharp δ-stable Bernstein theorem in this latter range from results of Hong--Li--Wang and Florit-Simon.
ER  -