The goal of this project is to obtain further understanding of the structure of complete noncompact manifolds. The general theme is that understanding the solutions of some partial differential equations in terms of curvature can be used to obtain more information about the geometry and topology of the manifold. In this context, the PI will study bounds on the bottom of spectrum of the Laplacian in terms of the Ricci curvature and use the spectral information to study the structure of the manifold. The PI will also study geometric and analytic properties of complete gradient Ricci solitons, aiming at a better understanding of the singularities of the Ricci flow.
The proposed project will lead to a better understanding of the shape and structure of infinite geometric objects. These objects, called manifolds, appear frequently in science because physical laws can be modeled by certain partial differential equations on manifolds. Understanding the structure of manifolds is important in many fields, such as black holes and worm holes in physics, the structure of molecules in chemistry or the motion of liquid crystal in engineering. Partial differential equations have been a fundamental tool in geometry, for example the Ricci flow has been used to classify the shape of all three dimensional manifolds. Further advances in the study of geometric partial differential equations on manifolds would lead to new discoveries applicable to many areas of mathematics and science.