Junior Geometry Seminar – Natasha Diederen

Abstract: Minimal surfaces are important objects in geometry, and their study draws on methods from both differential geometry and the analysis of partial differential equations. They are defined as critical points of the area functional, or equivalently as surfaces whose mean curvature vanishes identically. This definition makes sense in any ambient Riemannian manifold, and the case of minimal surfaces in spheres is particularly interesting. Indeed, while there are no closed, compact minimal surfaces in Euclidean 3-space, there are many such examples in the 3-sphere, the simplest being the 2-sphere and the Clifford torus. In the 1970s, Lawson conjectured that the Clifford torus is the only embedded minimal torus in the 3-sphere. Brendle gave an affirmative answer to this in 2012 using an application of the maximum principle to a two-point function. In this talk, I will introduce the basic theory of minimal surfaces in the 3-sphere, including a uniqueness result for embedded minimal surfaces of genus zero, before outlining the key ideas underlying Brendle’s proof.