Bachelor's defense: Zaïna Parly

Title: Stability of soap bubbles and constant mean curvature surfaces

Abstract:
This thesis discusses the shape of soap bubbles when they are geometrically stable. Energy considerations are done in the context of surface tension, concluding that energy and surface area are proportional in the physical system of a soap bubble in the absence of gravity. A precise topological definition of a surface is then made in terms of immersions of smooth manifolds. The thesis then introduces the first and second fundamental form and mean curvature of a surface. Using calculus of variations the thesis concludes that the mean curvature of the soap bubble must be constant in order to minimize the energy. Upon that, Poincaré's theorem for indices of singularities of line fields is explained in great detail. Lastly the thesis gives a proof of Hopf's soap bubble theorem, which states that a closed immersed surface of constant mean curvature with genus 0 must be a round sphere, and thus concludes that the stable geometry of the soap bubble is a round sphere.

Thesis advisors:
 Martin Cramer Pedersen (NBI)
 Niels Martin Møller (GeoTop/MATH)