Minimal tubes of finite integral curvature

The author defines a tube to be an immersed submanifold u:Mp→Rn+1 and the interval of existence τ(Mp) to be the interval of those t for which the intersection Σt of u(Mp) with the hyperplane xn+1=t in Rn+1 is nonempty and compact. The length of τ(Mp) is called the time of existence of the tube. The tube is minimal if u is a minimal immersion. Denote by vT an orthogonal projection of v into the tangent space of M, ν=eTn+1/∥eTn+1∥, and introduce a vector J, called a vector-flow with coordinates, Jk=∫Σt((ek)T,ν),1≤k≤n+1. The angle between J and en+1 is denoted by α. The main result of the article under review is the following estimate: |τ(M)|≤G(M)∥J∥cos(α)16α2, where G(M) denotes the absolute integral Gauss curvature of M