UC Berkeley

The subject of this thesis is the initial data sets on R 3 for the vacuum Einstein equation, i.e., pairs (g, k) = (gij , kij ) of symmetric tensor fields on R 3 (where g is positive definite) solving the vacuum constraint equationIndeed, any triple (M3 , g, k) of a 3-manifold, Riemannian metric g and a covariant symmetric 2-tensor k satisfying (0.0.1) is called an initial data set for the vacuum Einstein equationVarious applications of initial data construction include global stability of Minkowski spacetime [CK93], weak cosmic censorship and naked singularity formation [Chr94], disproving the third law of black hole thermodynamics [KU22], and formation of black holes ([LY15] as one example), etc.To understand the nature of the nonlinear PDE system (0.0.1), it is instructive to consider the direct linearization of this system around the simplest solution – i.e., the flat solution (δ, 0) – that takes the form, respectively,This system is comprised of underdetermined PDEs for a symmetric tensor ˙g and a symmetric tensor ˙k. In analogy with vector fields v obeying the divergence-free condition ∂jv j = 0 – the simplest such equation – one should expect quite a bit of flexibility for the set of solutions to (3.1.1).The author has developed with his collaborators explicit integral formulas for solving the linearized system 0.0.3 that enjoy special localization properties so that they can be used to study the nonlinear system 0.0.1.Although being underdetermined, the system 0.0.1 is known to be rigid: one famous example is the celebrated positive mass theorem first proved by [SY79] which states that only Minkowski initial data can have rapid decay. Thus, an interesting question is in what special unbounded regions the initial data can be localized. This will be the main topic of Chapter 2.Another way to construct initial data sets is through gluing. Gluing means, by definition, given two initial data sets (g1, k1) and (g2, k2) defined on disjoint regions A1, A2, respectively, finding a region A ⊃ A1 ∪ A2 and initial data set (g, k) defined on A such that (g, k)|Ai = (gi , ki), i = 1, 2. Because the divergence-type operators have nontrivial cokernel (especially on the flat background), we expect conservation laws that these linear equations should satisfy, called linear obstructions. One can ask under what conditions two initial data sets can be glued together, especially if they violate the linear obstructions. The answer might be surprising: Czimek–Rodnianski [CR22] first proved obstruction-free gluing of characteristic data defined on a null hypersurface, as well as spacelike obstruction-free gluing but under a non-sharp positivity condition. The author and his collaborators have developed a method using the aforementioned integral solution operators and manipulating the nonlinearity to achieve obstruction-free gluing on spacelike hypersurfaces in the asymptotically flat regime under a sharp positivity condition. This will be the topic of Chapter 3.It is then natural to ask if the integral solution operators can also be constructed for curved geometry. The author and his collaborators have developed a general abstract theory for a wide class of underdetermined (and their adjoint) linear differential operators where an integral solution (representation) formula can be constructed. This will be the topic of Chapter 4. Some applications of this theory in geometry and physics are included in Appendix B.