Minimal tubes of the sphere as the dimension grows: Volume and explosion of the mean exit time
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 \(n \to \infty\) and to \(0\) as \(m \to \infty.\) Simultaneously, bounds for the volume of such tubes are derived and compared to Weyl's tube formula.
math.DG↗