On Maximal surfaces in asymptotically flat spacetimes

Albachiara Cogo (University of Tübingen)

Maximal surfaces are spacelike hypersurfaces of a Lorentzian manifold which are critical points of the area functional. They are very important tools in General Relativity since they somehow reflect some properties of
the ambient spacetime; for example, foliations by maximal slices played a crucial role in the first proofs of the positive mass theorem and in the analysis of the global stability of Minkowski.
The Euler-Lagrange equation of the variational problem of maximization of the area functional is a quasi-linear elliptic PDE that geometrically describes the vanishing of the mean curvature.
We will thereby discuss the powerful role played by the theory of nonlinear partial differential equations in the proof of the existence of maximal surfaces, recalling the main ideas provided by R. Bartnik.
I will additionally present some recent developments of my Ph.D. project under the supervision of Professor G. Huisken, trying to highlight the physical meaning of the objects involved.