Stuart James Hall

Quasi-Einstein metrics on hypersurface families

Abstract We construct quasi-Einstein metrics on some hypersurface families. The hypersurfaces are circle bundles over the product of Fano, Kahler–Einstein manifolds. The quasi-Einstein metrics are related to various gradient Kahler–Ricci solitons constructed by Dancer and Wang and some Hermitian, non-Kahler, Einstein metrics constructed by Wang and Wang on the same manifolds.

Variation of complex structures and the stability of Kähler–Ricci Solitons

We investigate the linear stability of Kahler-Ricci solitons for perturbations induced by varying the complex structure within a fixed Kahler class. We calculate stability for the known examples of Kahler-Ricci solitons.

Perelman’s Entropy for Some Families of Canonical Metrics

We numerically calculate Perelman’s entropy for a variety of canonical metrics on -bundles over products of Fano Kähler–Einstein manifolds. The metrics investigated are Einstein metrics, Kähler–Ricci solitons, and quasi-Einstein metrics. The calculation of the entropy allows a rough picture of how the Ricci flow behaves on each of the manifolds in question.