Applications of minimal surface techniques in geometry

It is easy to see that the shortest path connecting two given points in the plane is a straight line. If you try to find a surface spanned by a given closed curve in the three-dimensional Euclidean space with the least possible amount of area, things already get more complicated. Yet, this problem can be solved. The corresponding surface of least area is called a minimal surface. If this is too abstract, you can fill in a closed wire using soap film to see a minimal surface in real life. Minimal surfaces are among the most fundamental objects in mathematics. They provide valuable information on the space that surrounds them. This information then gives new insights on the geometry of complicated objects, the properties of the spacetime we live in, and even on other minimal surfaces. The goal of this project is to strengthen our understanding of minimal surfaces and related, more complicated objects such as constant mean curvature surfaces and surfaces of prescribed mean curvature. Can we develop more precise methods to quantify the area of such a surface? Can we better understand the shape of a minimal surface close to its boundary if we know the angle between the minimal surface and another surface that contains its boundary? The answers to these questions would be very valuable. For instance, they would help us quantify the size of a black hole and understand the structure of a minimal surface with no boundary.