In this paper, we construct several kinds of new time-periodic solutions of the vacuum Einsteins field equations whose Riemann curvature tensors vanish, keep finite or take the infinity at some points in these space-times, respectively. The singularities of these new time-periodic solutions are investigated and some new physical phenomena are discovered.

We consider the discrete Laplacian $\Delta$ on the cubic lattice $\mathbb{Z}^d$, and deduce estimates on the group $e^{i t \Delta}$ and the resolvent $(\Delta-z)^{-1}$, weighted by $\ell^q (\mathbb{Z}^d)$-weights for suitable $q \geqslant 2$. We apply the obtained results to discrete Schrödinger operators in dimension $d \geqslant 3$ with potentials from $\ell^p (\mathbb{Z}^d)$ with suitable $p \geqslant1$.

We construct families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed, in particular establishing the existence of waves of large amplitude. A Riemann–Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudodifferential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler–Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation.

We consider Bose gases consisting of N particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of order N−1 (Gross–Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as N→∞. Our results confirm Bogoliubov’s predictions.

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations (“linearised gravity”) around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior, and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components). These latter statements are in fact deduced in the course of the proof by exploiting associated quantities shown to satisfy the Regge–Wheeler equation, for which appropriate decay can be obtained easily by adapting previous work on the linear scalar wave equation. The bounds on the rate of decay to linearised Kerr are inverse polynomial, suggesting that dispersion is sufficient to control the non-linearities of the Einstein equations in a potential future proof of non-linear stability. This paper is self-contained and includes a physical-space derivation of the equations of linearised gravity around Schwarzschild from the full non-linear Einstein vacuum equations expressed in a double null gauge.