Time-like surfaces with zero mean curvature vector in 4-dimensional neutral space forms

Abstract

Let $M$ be a Lorentz surface and $F:M\rightarrow N$ a time-like and conformal immersion of $M$ into a 4-dimensional neutral space form $N$ with zero mean curvature vector. We see that the curvature $K$ of the induced metric on $M$ by $F$ is identically equal to the constant sectional curvature $L_0$ of $N$ if and only if the covariant derivatives of both of the time-like twistor lifts are zero or light-like. If $K\equiv L_0$, then the normal connection $\nabla^{\perp}$ of $F$ is flat, while the converse is not necessarily true. We see that a holomorphic paracomplex quartic differential $Q$ on $M$ defined by $F$ is zero or null if and only if the covariant derivative of at least one of the time-like twistor lifts is zero or light-like. In addition, we see that $K$ is identically equal to $L_0$ if and only if not only $\nabla^{\perp}$ is flat but also $Q$ is zero or null.