Some global minimizers of a symplectic Dirichlet energy

The variational problem for the functional $F=frac12|phi^*omega|_{L^2}^2$ is considered, where $phi:(M,g)to (N,omega)$ maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of the Faddeev-Hopf energy, and may be regarded as a symplectic analogue of the Dirichlet energy familiar from harmonic map theory. The Hopf fibration $pi:S^3to S^2$ is known to be a locally stable critical point of $F$. It is proved here that $pi$ in fact minimizes $F$ in its homotopy class and this result is extended to the case where $S^3$ is given the metric of the Bergers sphere. It is proved that if $phi^*omega$ is coclosed then $phi$ is a critical point of $F$ and minimizes $F$ in its homotopy class. If $M$ is a compact Riemann surface, it is proved that every critical point of $F$ has $phi^*omega$ coclosed. A family of holomorphic homogeneous projections into Hermitian symmetric spaces is constructed and it is proved that these too minimize $F$ in their homotopy class.

The nature of hypervelocity stars as inferred from their galactic trajectories

We have computed the galactic trajectories of twelve hypervelocity stars (HVSs) under the assumption that they originated in the Galactic Centre. We show that eight of these twelve stars are bound to the Galaxy. We consider the subsequent trajectories of the bound stars to compute their characteristic orbital period, which is 2 Gyr. All eight bound stars are moving away from the centre of the Galaxy, which implies that the stars lifetimes are less than 2 Gyr. We thus infer that the observed HVSs are massive main sequence stars, rather than blue horizontal branch stars. The observations suggest that blue HVSs are ejected from the Galactic Centre roughly every 15 Myr. This is consistent with the observed population of blue stars in extremely tight orbits round the central super-massive black hole (SMBH), the so-called S-stars, if we assume that the HVSs are produced by the breakup of binaries. One of the stars in such a binary is ejected at high velocities to form a HVS; the other remains bound to the SMBH as an S-star. We further show that the one high-velocity system observed to be moving towards the Galactic Centre, SDSS J172226.55+594155.9, could not have originated in the Galactic Centre; rather, we identify it as a halo object.