Title. Boundary behaviour of sliding almost minimal sets near a smooth, one-dimensional sliding boundary.
Abstract. The sliding almost minimal sets in the title are a version at the boundary of the Almgren minimal sets. We are given a boundary set $L$ (a smooth curve). The sliging almost minimal sets (relative to $L$) are sets $E$, with locally finite Hausdorff measure $H^2$, such that if $F$ is any sliding deformation of $E$ in a ball $B = B(x,r)$, then $H^2(E \cap B(x,r)) \leq H^2(F \cap B(x,r)) + r^2 h(r)$, where $r^2 h(r)$ is a small error term (that accounts for the "almost") and a sliging deformation $F$ is a set of the form $F = \varphi_1(E)$, where $\{ \varphi_t \}$, $0 \leq t \leq 1$ is a continuous one-parameter family of continuous mappings from $E$ to $R^n$, with $\varphi_0(y)=y$, $\varphi_t(y)=y$ on $E \setminus B$, $\varphi_t(E \cap B) \subset B$ and (the sliding condition that describes the sort of Plateau problem we consider here), $\varphi_t(y) \in L$ when $y\in E \cap L$. That is, points on the boundary $L$ are not allowed to leave $L$.
We shall try to describe what we know about these sets near the boundary. We would like to have a nice description like the one given by Jean Taylor away from $L$, that would give a nice classification of the singularities of $E$, but we are not there yet. So we'll only talk about partial results and questions.