Maximal hypersurfaces and other nonlinear geometric problems

Abstract

In Minkowski spacetime, there is a one-to-one correspondence between inertial observers, Lorentz boosts, and entire spacelike maximal hypersurfaces – namely, those with vanishing mean curvature – established by the renowned Cheng–Yau Bernstein-type theorem. Some years after proving the existence of maximal hypersurfaces in asymptotically flat spacetimes in 1984, Bartnik raised the question of whether a similar correspondence exists in non-flat spacetimes. As a main result of the thesis, we address a first step in this direction. We demonstrate the existence and discuss properties of an entire maximal hypersurface approaching a coordinate-dependent boost in the asymptotically flat ends of the maximally extended Schwarzschild spacetime. Analytically, this consists of finding complete, non-compact solutions with specific prescribed asymptotics to the maximal surface equation, a geometric quasilinear elliptic partial differential equation. We first construct suitable coordinate-dependent boosted hypersurfaces admitting barriers at the asymptotically flat ends, building on the work of Meyers and Bartnik–Chruściel–O'Murchadha. In addition, to apply some of Bartnik’s main results, we introduce a new time function in the maximally extended Schwarzschild spacetime. Notably, this is a Cauchy time function. Moreover, we show that a relaxed version of Bartnik’s uniform interior conditions is satisfied with our choice of time function, employing both the crushing nature of the Schwarzschild singularity and a proof of monotonicity for the relevant geometric quantities involved. As a consequence of the above construction, we can generate a foliation of maximal hypersurfaces in exterior Schwarzschild, corresponding to a geometric choice of time function associated with any boost parameter.

This thesis also comprises three additional articles co-authored with collaborators, reproduced in their original form as published papers or arXiv preprints. The first two concern the renowned black hole and equipotential photon surface uniqueness theorems in asymptotically flat spacetimes: the former tackles the problem in electrovacuum and four dimensions, employing techniques inspired by the works of Agostiniani–Mazzieri; the latter establishes the result in vacuum and generalizes to any dimension the method by Robinson. The final article addresses the classification of ancient solutions to fully nonlinear flows under natural conditions on the speed, showing that every convex, non-collapsing, uniformly 2-convex, ancient, noncompact solution is either a self-similar shrinking cylinder or a rotationally symmetric translating soliton.