Geometry of manifolds of nonnegative scalar curvature

The aim of this proposal is to investigate some important open questions concerning the volume, boundary area, and other basic geometric quantities of manifolds of nonnegative scalar curvature. The expected outcomes include obtaining lower and upper estimates of the volume of manifolds using their boundary geometry, finding sufficient conditions for the existence of closed minimal surfaces, and estimating the area of the outermost minimal surfaces. This proposed research is significant not only because it will produce the above important results, but also because it will introduce new ideas and methods to the current study of manifolds with boundary.