Canonical metrics and geometric flows on non-compact manifolds

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Differential geometry is the study of space, its shape, and the interaction between the two. My research focuses in two categories of this study. The first category involves studying the underlying fabric of space itself and how this is influenced by its shape. Conversely, we also study the problem of determining when an underlying space can assume certain ideal shapes corresponding to geometric objects with beautiful mathematical descriptions, but for which concrete examples are very hard to construct. The second category involves studying ways in which subspaces move within a larger space. The main analytic tools I use to study problems in these two categories are the Ricci flow equation and the mean curvature flow equation respectively. The Ricci flow equation essentially prescribes a way to deform the shape of any given space into a geometrically nicer one. The remarkable thing is that in many cases, the flow actually produces