We describe a new family of representations of 1() in PU(2,1), where  is a hyperbolic Riemann surface with at least one deleted point. This family is obtained by a bending process associated to an ideal triangulation of . We give an explicit description of this family by describing a coordinates system in the spirit of shear coordinates on the Teichm¨uller space. We identify within this family new examples of discrete, faithful, and type-preserving representations of 1(). In turn, we obtain a 1-parameter family of embeddings of the Teichm¨uller space of  in the PU(2,1)-representation variety of 1(). These results generalise to arbitrary  the results obtained in [42] for the 1-punctured torus.

We give a complete diffeomorphism classification of 1-connected closed manifolds M with integral homology H¤(M) ∼= Z ⊕ Z ⊕ Z, provided that dim(M) 6= 4.

We consider an embedded convex compact ancient solution t to the curve shortening flow in R2. We prove that t is either a family of contracting circles, which is a type I ancient solution, or a family of evolving Angenent ovals, which is of type II.

We study analytic properties of harmonic maps from Riemannian polyhedra into CAT(κ) spaces for κ∈{0,1}. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT(κ) spaces. We compute a target variation formula that captures the curvature bound in the target, and use it to prove an Lp Liouville-type theorem for harmonic maps from admissible polyhedra into convex CAT(κ) spaces. Another consequence we derive from the target variation formula is the Eells–Sampson Bochner formula for CAT(1) targets.

We show that it is possible to perturb arbitrary vacuum asymptotically flat spacetimes to new ones having exactly the same energy and linear momentum, but with center of mass and angular momentum equal to any preassigned values measured with respect to a fixed affine frame at infinity. This is in contrast to the axisymmetric situation where a bound on the angular momentum by the mass has been shown to hold for black hole solutions. Our construction involves changing the solution at the linear level in a shell near infinity, and perturbing to impose the vacuum constraint equations. The procedure involves the perturbation correction of an approximate solution which is given explicitly.