The capillary problem considers an interface separating two fluids that lie adjacent to each other and do not mix. This imposes certain geometric conditions on the interface surface and its boundary. The existence, uniqueness and regularity of such surfaces has been widely studied in Rn. Here we study the existence of a graph in MxR with prescribed mean curvature and prescribed contact angle, where M is a Riemannian manifold. We follow the work of Korevaar to estimate the gradient of solutions, using the maximum principle. This is joint work with Leili Shariyari.

Roughly speaking, it is expected that the only two types of singular laminations that can occur as limits of closed embedded minimal surfaces in a 3-manifold with positive scalar curvature are accumulations of catenoids and non-properhelicoid-like limits. T. Colding and C. De Lellis constructed an example of the first type. I will present a construction of the second type: we show that there exists a metric with positive scalar curvature on S2xS1 and a sequence of embedded minimal tori that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two helicoid-like singularities on the 2-sphere. This is a joint work with Darren Lee.