Canonical metrics and geometric evolutions

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Differential geometry is the study of space, its shape, and the interaction between the two. One of the most effective ways to study differential geometry is through the use of so called geometric evolution equations. In my research, I mainly focus on a geometric evolution equation known as the Ricci flow. I will now describe, in broad terms, the main goals of my research in the Ricci flow, together with their techniques and related problems. ***The first goal of my research is to understand what the shape of space tells us about the underlying fabric of the space. Results in this direction are often known as Uniformization Theorems in geometry. Roughly, the Ricci flow is used to deform the shape of the space to become simpler, and in so doing reveal the nature of the underlying fabric of space. One of my main results here states that when a complete Kähler manifold is positively curved in an appropriate sense, and becomes sufficiently flat at its horizon, then this space can be deformed by the Ricci flow equation to assume a flat shape, thereby identifying the nature of the underlying space. Our results have provided one of the strongest links so far in support of the Uniformization Conjecture of S.T. Yau, which states that this result is true, regardless of the behavior of curvature at the horizon. Our study of the above problem is based on an in depth study of the Ricci flow on non-compact Kähler manifolds. ***The second goal of my research involves 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 complete Einstein metrics describe one such class of shapes, and a fundamental question is whether or not a given a complete Kähler manifold admits an Einstein metric. An elegant way to answer this question would be to show the given metric converges to a limit when deformed along the Ricci flow as any limit to the flow is necessarily Einstein. A closely related question is the stability of the Kähler Ricci flow at Einstein metrics. Here we consider convergence of Kähler Ricci flow starting from a metric which is a priori close to being an Einstein metric. Such results are fundamental to understanding convergence under more general hypothesis and the key is to allow for as weak a notion of the above ``closeness" as possible. ***A parallel goal is to study how Ricci flow deforms shapes which are poorly behaved in the sense of unbounded curvature. This extends the classical theory of Ricci flow, which demands bounded curvature, and such a study is key to addressing the above geometric problems on non-compact manifolds in full generality.*** **