Long time existence for the bosonic membrane in the light cone gauge

This paper mainly aims to establish the well-posedness on time interval $[0,\varepsilon^{-\frac{1}{2}}T]$ of the classical initial problem for the bosonic membrane in the light cone gauge. Here $\varepsilon$ is the small parameter measures the nonlinear effects. In geometric, the bosonic membrane are timelike submanifolds with vanishing mean curvature. Since the initial Riemannian metric may be degenerate or non-degenerate, the corresponding equation can be reduce to a quasi-linear degenerate or non-degenerate hyperbolic system of second order with an area preserving constraint via a Hamiltonian reduction.