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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 $S'$ ?

Thank you for your answers

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    $\begingroup$ If all you care about is the infinitesimal change, then you should use the stability operator for minimal surfaces. There should be no change between the lorentzian case and usual riemannian case because you're starting with a spacelike hypersurface. $\endgroup$
    – Otis Chodosh
    Commented Aug 3, 2012 at 23:09
  • $\begingroup$ It would be a start. Thank you for the hint, do you know any references about the stability operator? $\endgroup$
    – Selim G
    Commented Aug 4, 2012 at 13:58
  • $\begingroup$ Leon Simon's book on geometric measure theory is a classic reference but a bit hard to find. This blog article might give some of the rough ideas lamington.wordpress.com/2009/08/25/… .. $\endgroup$
    – Otis Chodosh
    Commented Aug 4, 2012 at 14:30

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