Consider a metric space (S,d) with an upper curvature bound in the sense of Alexandrov (i.e.~via triangle comparison).  We show that if (S,d) is homeomorphically equivalent to the 2-sphere, then it is conformally equivalent to the 2-sphere.  The method of proof is through harmonic maps, and we show that the conformal equivalence is achieved by…

The study of extremals for Steklov eigenvalues has revitalised the theory of free boundary minimal surfaces. One of the most basic open  questions can be phrased as follows: Can a surface of any given topology  be realised as an embedded free boundary minimal surface in the 3-dimensional Euclidean unit ball? We will answer this question…