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

TL;DR

This paper proves the long-time existence and well-posedness of the classical bosonic membrane equations in the light cone gauge, accounting for nonlinear effects and degeneracy in the initial metric.

Contribution

It establishes the well-posedness of the bosonic membrane initial value problem over extended time intervals, including degenerate cases, using Hamiltonian reduction and hyperbolic PDE analysis.

Findings

Long-time existence of solutions for the bosonic membrane equations.

Well-posedness of the initial value problem in the light cone gauge.

Extension of results to degenerate and non-degenerate initial metrics.

Abstract

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.