We prove that smooth asymptotically flat solutions to the Einstein vacuum equations which are assumed to be periodic in time, are in fact stationary in a neighborhood of infinity. Our result applies under physically relevant regularity assumptions purely at the level of the initial data. In particular, our work removes the assumption of analyticity up to null infinity in Bičák, Scholtz, and Tod (2010). The proof relies on extending a suitably constructed “candidate” Killing vector field from null infinity, via Carleman-type estimates obtained in Alexakis, Schlue, and Shao (2013).

We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.

Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated theorem of Almgren–Pitts, on the existence of embedded minimal hypersurfaces in compact manifolds. We compare our results with other min–max theories.

The Teichmüller harmonic map flow deforms both a map from an oriented closed surface $$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $$, so as to reduce the energy of the map as quickly as possible [16]. The flow then tries to converge to a branched minimal immersion when it can [16, 18]. The only thing that can stop the flow is a finite-time degeneration of the metric on $$ where one or more collars are pinched. In this paper we show that finite-time degeneration cannot happen in the case that the target has nonpositive sectional curvature, and indeed more generally in the case that the target supports no bubbles. In particular, when combined with [16, 18, 9], this shows that the flow will decompose an arbitrary such map into a collection of branched minimal immersions.