Wednesday, July 4, 2012

1207.0368 (Weiping Yan)

On the motion of timelike minimal surfaces in the Minkowski space

Weiping Yan
In this paper we are devoted to the study of the motion of the timelike minimal surfaces in the Minkowski space $\textbf{R}^{1+n}$. Those surfaces are known as membranes or relativistic strings, and described by a system with $n$ nonlinear wave equations of Born-Infeld type. We construct a global timelike Sobolev regularity torus in $\textbf{R}^{1+n}$, which time slice are evolved by a rigid motion. A Lyapunov-Schmidt decomposition reduces this problem to an infinite dimensional bifurcation equation and a range equation. To overcome the higher order derivative perturbation in bifurcation equation and the "small divisor" phenomenon in range equation, a suitable Nash-Moser iteration is constructed.
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