A Compactness Theorem and Some Gap Results for Free Boundary Minimal Surfaces

Abstract

This thesis consists of several results about minimal surfaces. In the first part we study free boundary minimal surfaces in the Euclidean ball B^n. We prove that if (...) is a kdimensional free boundary minimal surface in Bn satisfying (...), then (...) is diffeomorphic to either (...) orto (...). Further geometric information is given in the codimension one case. Moreover, in case (...) is a 2-dimensional free boundary minimalsurface, then either (...) and (...) is an equatorial disk (...) B n or (...) at a point (...) and (...) is isometric to a critical catenoid. We also prove the existence of a gap for the area of free boundary minimal surfaces in the ball. Namely, there exists (...) so that whenever (...) is a free boundary minimal surface in B n satisfying (...), then (...) is an equatorial disk (...). To prove this gap result we compare the excess of free boundary minimal surfaces with the excess of the associated cones over the boundaries. As a corollary, we show that (...) is the only free boundary minimal surface in B n whose boundary is minimal in (...). In the second part we prove two results about closed minimal surfaces in 3-manifolds. The main result is a compactness theorem for the space of minimal surfaces with area bounded from above and injective radius bounded from below. Finally, we prove a weak result for positively curved 3-manifolds withsymmetries containing stable minimal surfaces..