The PI proposes to study various problems in the field of geometric partial differential equations. For example, he would like to study the qualitative behavior of the Ricci flow on manifolds with positive isotropic curvature. In particular, the PI would like to analyze what kinds of singularities can occur along the flow. In another project, the PI intends to study the Yamabe problem on manifolds with boundary. The goal here is to construct conformal metrics that have constant scalar curvature in the interior and zero mean curvature along the boundary. The PI also intends to study ancient solutions to the Yamabe flow on the n-dimensional sphere. This question is motivated by a classification result, due to Hamilton, Daskalopoulos, and Sesum, for ancient solutions to the Ricci flow on the two-sphere. This project is concerned with questions at the crossroads of analysis and differential geometry. The use of analytical techniques in geometry has been extremely successful in recent years. In particular, Hamilton's Ricci flow plays a key role in Perelman's solution of the Poincare conjecture, as well as in the proof of the Differentiable Sphere Theorem by Richard Schoen and myself. The goal of the project is to gain a better understanding of these geometric evolution equations, and their qualitative properties.
The PI has worked on various problems at the crossroads of analysis and geometry. The following is a list of some specific topics that were studied, and their outcomes: 1. Minimal surfaces in the 3-sphere. Minimal surfaces are surfaces that locally have smallest area. They arise as models for soap films in physics, but also play an important role in geometry. Usually, these surfaces lie in some Euclidean space, but it is also interesting to classify minimal surfaces that lie in a curved ambient manifold, such as a 3-dimensional sphere. The PI gave a complete description of all such surfaces which are topologically a torus and in addition are free of self-intersections. This answered a question posed by H. Blaine Lawson, Jr., in 1970. 2. Constant mean curvature surfaces in Riemannian manifolds. Constant mean curvature surfaces are more general than minimal surfaces. They are stationary for area among all surfaces that enclose a given amount of volume. In Euclidean space, Alexandrov proved that round spheres are the only surfaces of constant mean curvature that are free of self-intersections. A natural question is what happens if we consider surfaces in a curved ambient manifold? To study this question, we studied the special case when the ambient manifold is rotationally symmetric. Under certain natural structure conditions, we were able to show that the only constant mean curvature surfaces that are free of self-intersections are the obvious ones, namely spheres of symmetry. Moreover, our conditions are essentially optimal. 3. Geometric inequalities. In joint work with Pei-Ken Hung and Mu-Tao Wang, we established a sharp inequality for surfaces in the Anti-deSitter Schwarzschild manifold. This inequality generalizes a classical inequality due to Minkowski, and is closely related to the Penrose inequality for collapsing null shells of dust. In another direction, we considered minimal surfaces in the unit ball which meet the boundary at a 90 degree angle. In 2011, we showed that any such surface has at least as much area as a flat disk, thereby answering a question posed by Schoen. 4. Analysis of singularities of mean curvature flow. Mean curvature flow is a process that deforms a surface with a speed given by its mean curvature. One can think of this as the flow which reduces area as quickly as possible. For convex surfaces, the qualitative behavior of this flow has been completely understand through the work of Huisken in the 1980s. However, once we drop the convexity assumption, the flow will form singularities, which pose challenging problems. In the case of embedded, mean convex surfaces, we proved a sharp estimate for the inscribe radius along this deformation, which improves earlier work by Andrews. In a joint work with Gerhard Huisken, we used this estimate to extend the flow past singularities through a process called 'mean curvature flow with surgery'. A similar process was used by Hamilton and Perelman in their works on the Ricci flow. 5. Uniqueness questions for self-similar solutions to the Ricci flow. The analysis of singularities is one of the most important problems in the study of geometric flows. Of particular importance are self-similar solutions (also called solitons), as they often serve as local models for a singularity. In particular, they were studied extensively by Hamilton and Perelman. An important example of a steady gradient Ricci soliton is the rotationally symmetric soliton discovered by Bryant. It is an interesting question (which goes back to Perelman's work) whether the Bryant soliton is the only steady soliton in dimension 3 which is noncollapsed in a certain sense. In 2012, we gave an affirmative answer to this question. Our work also works for steady solitons in dimension 4 which are noncollapsed and have uniformly positive isotropic curvature. 6. Scalar curvature rigidity problems. The scalar curvature plays an important role in differential geometry, in part due to its connections to general relativity. In particular, it is an interesting question whether one can deform the geometry on a manifold in such a way that the scalar curvature increases. For example, Gromov and Lawson showed that an n-dimensional torus cannot admit a metric with positive scalar curvature. In a similar spirit, Min-Oo conjectured that one cannot deform the metric on a hemisphere in such a way that the scalar curvature increases while the boundary geometry is unchanged. In joint work with Marques and Neves in 2010, we disproved Min-Oo's conjecture in all dimensions greater than 2.
