We apply the mean curvature flow to deform symplectomorphisms of $\mathbb{CP}^n$. In particular, we prove that, for each dimension $n$, there exists a constant , explicitly computable, such that anypinched symplectomorphism of $\mathbb{CP}^n$ is symplectically isotopic to a biholomorphic isometry.

A geometric classification of the compact four-dimensional Einstein-Weyl manifolds with at least four-dimensional symmetry group is given. Our results also sharpen previous results on four-dimensional Einstein metrics and correct Parker's topological classification of cohomogeneity-one four-manifolds.

A well-known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a zero-order pseudodifferential projection on a closed manifold vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in the odd dimensional case an analogous statement holds for the algebra of projective pseudodifferential operators, i.e., the noncommutative residue of a projective pseudodifferential projection vanishes. Our strategy of proof also simplifies the argument in the classical setting. We show that the noncommutative residue factors to a map from the twisted K-theory of the co-sphere bundle and then use arguments from twisted K-theory to reduce this question to a known model case. It can then be verified by local computations that this map vanishes in odd dimensions.

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 discuss our recent work [4] in which gravitational radiation was studied by evaluating the Wang-Yau quasi-local mass of surfaces of fixed size at the infinity of both axial and polar perturbations of the Schwarzschild spacetime, \`{a} la Chandrasekhar [1].