# Variations of the mean curvature

Good evening everyone,

I am facing a technical problem, maybe one of you can help.

Given a spacelike surface $S$ with mean curvature 0 in a lorentzian $3$-manifold with constant sectionnal curvature -1, I am trying to understand how the mean curvature evolves when one misshapes a bit $S$. Precisly, if one take $f : S \rightarrow \mathbb{R}$ a little smooth function, and we consider $S'$ the surface we get when we follow at $p\in S$ the orthogonal geodesic on lenght $f(p)$. What is the mean curvature of $S'$ ?