Drawing from ideas in general relativity, the outermost apparent horizon of an asymptotically flat Riemannian manifold is defined as a closed minimal surfaces enclosing all other minimal surfaces. A natural energy condition of general relativity translates to the condition that the Riemannian manifold, thought of as a spacelike slice of spacetime, has non-negative scalar curvature. A classical result by Hawking, completed by Galloway and Schoen, says that under this condition, the outermost apparent horizons must allow metrics of positive scalar curvature. It is conceivable that this is also the only restriction on a bounding manifold to be an outermost apparent horizon. In this talk I want to describe some new examples of outermost apparent horizons with non-trivial topology.  

In this talk, we will study the regularity of sliding almost minimal set. Suppose that $\Omega$ is a closed simply connected subset of $\mathbb{R}^3$ with boundary $\partial \Omega$ a 2-dimensional manifold of class $C^{1,\alpha}$, $E\subset \Omega$ is a sliding almost minimal set with gauge function $h$ satisfying that $h(t)\leq Ct^{\beta}$ and $E\supset \partial \Omega$. It is well known, by Jean Taylor's Theorem, that $E$ is locally $C^{1,\gamma}$ equivalent to a minimal cone away from the boundary $\partial \Omega$. But here we can say that $E$ is also $C^{1,\gamma}$ equivalent to a sliding minimal cone in a half space at the boundary.