Maximal tubular hypersurfaces in Minkowski space

V. A. Klyachin ; V. M. Miklyukov

Izvestiya. Mathematics , Tome 38 (1992) no. 1, pp. 203-213

Résumé

Consider $C^2$-solutions of the equations for maximal surfaces in Minkowski space $$ \sum_{i=1}^n \frac\partial{\partial x_i}\left(\frac{fx_i}{\sqrt{1-|\nabla f|^2}}\right)=0. $$ The hypersurface $t=f(x)$ is tubular if for every $\tau$ the level sets $E_\tau=\{x\colon f(x)=\tau\}$ are compact. The girth function of a tubular hypersurface is given by $\rho(\tau)=\max\limits_{x\in E_\tau}|x|$. In this paper it is shown that the girth function of a maximal tubular surface satisfies the differential inequality $\rho(t)\rho ''(t)\geqslant(n-1)(\rho^{'2}(t)-1)$. As a consequence of this assertion it is established that the union of the rays tangent to the hypersurface at an isolated singular point forms the light cone; a bound is obtained, in the neighborhood of an isolated singularity, to the spread of the maximal tube in the direction of the time axis in terms of its deviation from the light cone.