Curvature, Minimal Surfaces, and Bubbles

Einstein’s theory of general relativity says that energy = curvature, roughly speaking. However, this equation is a tautology unless we understand what curvature actually means. In this talk we’ll develop intuition for curvature by focusing on examples we can easily visualize: surfaces in R^3. Our approach will be pleasantly 19th century. We introduce the first and second fundamental forms, and then define the Gaussian and mean curvatures of a surface in several ways. We’ll then consider variations of a surface’s area and relate area to curvature. Next, we’ll discuss minimal surfaces, critical points of the area functional. As a fun application of what we’ve learned, we’ll see how curvature and minimal surfaces can be used to describe soap bubbles!