The Riemannian L^2 topology on the manifold of Riemannian metrics

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L^2 Riemannian metric - so called because it induces an L^2 topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L^1-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically with the completion of the L^2 metric. We also give a user-friendly criterion for convergence (with respect to the L^2 metric) in the manifold of metrics.Comment: 31 pages; v2: minor corrections, published versio

A Tribute to the Ladies

The legendary Cokie Roberts dishes on Martha, Dolley and our country’s Founding Mothers

Influence of viscosity on grinding in ball-mills

Imperial Users onl

Ricci flow and the metric completion of the space of Kahler metrics

We consider the space of Kahler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are equivalent. We also determine the metric completion of the space of Kahler metrics, making contact with recent generalizations of the Calabi-Yau Theorem due to Dinew, Guedj-Zeriahi, and Kolodziej. As an application, we obtain a new analytic stability criterion for the existence of a Kahler-Einstein metric on a Fano manifold in terms of the Ricci flow and the distance function. We also prove that the Kahler-Ricci flow converges as soon as it converges in the metric sense

New K-Indices from South Atlantic Observatories : Port Stanley and Ascension Island

Port Stanley and Ascension Island magnetic observatories have been in continuous operation since the early 1990's. These remote South Atlantic locations provide much needed coverage in the global network of geomagnetic observatories and help to monitor the South Atlantic Anomaly. To enhance the production of longitude-sector planetary magnetic activity indices there is a requirement for local 3-hourly K-index values from Port Stanley (PST) observatory. We describe the process followed to establish an automated routine for the derivation of the indices and we assess the congruence of the indices to those available from other suitably located observatories. A similar procedure has been followed for Ascension Island observatory although this is not shown here