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I have been working on minimal surfaces, only recently learnt about maximal surfaces and "maxfaces" in Lorentz spaces.

I can clearly see the mathematical motivations. But I wonder if zero-mean-curvature hypersurfaces in Lorentz spaces ($\mathbb{R}^2_1$ or $\mathbb{R}^3_1$), like minimal surfaces in Euclidean spaces, also find applications in nature sciences or other branches of mathematics.

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Maybe you already encountered such maximal surfaces in the context of General Relativity. Still, the one application of spacelike maximal surfaces that I am aware of is as special kinds of initial data (Cauchy) surfaces for the Einstein equations (a popular broader class are constant mean curvature (CMC) surfaces, of which the maximal condition (vanishing mean curvature) is a special case). The vanishing (or constant) mean curvature condition simplifies the algebraic structure of the Einstein equations when they are split into constraint and evolution subsystems, hence also simplifying their analysis. The existence of maximal Cauchy surfaces in asymptotically flat Lorentzian geometries (spacetimes) was studied for instance in

Bartnik, Robert, Existence of maximal surfaces in asymptotically flat spacetimes, Commun. Math. Phys. 94, 155-175 (1984). ZBL0548.53054. (cf. Thm.5.4)

That work has quite a few citations.

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