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PreprintCC BY 4.0arXiv

Minimal tubes of the sphere as the dimension grows: Volume and explosion of the mean exit time

  • Vicent Gimeno i Garcia
  • Martín Torres-Valverde
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Abstract

The mean exit time of a Brownian motion from a tube around a closed, minimal embedded n-dimen\-sional submanifold of the (n+m)-dimen\-sional sphere is studied, establishing two-sided bounds via radial transplant. In accordance with known results when n = 0 or m = 1, the mean exit time goes to infinity as n → ∞ and to 0 as m → ∞. Simultaneously, bounds for the volume of such tubes are derived and compared to Weyl's tube formula.

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Cite

BibTeXRIS
BibTeX
@misc{garcia2026minimal,
  title = {{Minimal tubes of the sphere as the dimension grows: Volume and explosion of the mean exit time}},
  author = {Vicent Gimeno i Garcia and Martín Torres-Valverde},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08222},
  eprint = {2610.08222},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08222},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Minimal tubes of the sphere as the dimension grows: Volume and explosion of the mean exit time
AU  - Vicent Gimeno i Garcia
AU  - Martín Torres-Valverde
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08222
UR  - https://arxiv.org/abs/2610.08222
AB  - The mean exit time of a Brownian motion from a tube around a closed, minimal embedded n-dimen\-sional submanifold of the (n+m)-dimen\-sional sphere is studied, establishing two-sided bounds via radial transplant. In accordance with known results when n = 0 or m = 1, the mean exit time goes to infinity as n → ∞ and to 0 as m → ∞. Simultaneously, bounds for the volume of such tubes are derived and compared to Weyl's tube formula.
ER  -