We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth $(n + 1)$-dimensional Riemannian manifolds, a theorem proved first by Pitts for $2 \leq n \leq 5$ and extended later by Schoen and Simon to any $n$.

We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three-dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to $\mathbb{R}^3$. We establish a generalization of the Penrose singularity theorem which shows that the presence of an immersed marginally outer trapped surface generically implies the null geodesic incompleteness of any spacetime that satisfies the null energy condition and which admits a non-compact Cauchy surface. Taken together, these results can be viewed as an initial data version of the Gannon-Lee singularity theorem. The first result is a non-time-symmetric version of a theorem of Meeks-Simon-Yau which implies that every asymptotically flat Riemannian 3-manifold that is not diffeomorphic to $\mathbb{R}^3$ contains an embedded stable minimal surface. We also obtain an initial data version of the spacetime principle of topological censorship. Under physically natural assumptions, a 3-dimensional asymptotically flat initial data set with marginally outer trapped boundary and no immersed marginally outer trapped surfaces in its interior is diffeomorphic to $\mathbb{R}^3$ minus a finite number of open balls. An extension to higher dimensions is also discussed.

We show that the first Johnson subgroup of the mapping class group of a surface $\Sigma$ of genus greater than $1$ acts ergodically on the moduli space of representations of $\pi_1(\Sigma)$ in $SU_2$. Our proof relies on a local description of the latter space around the trivial representation and on the Taylor expansion of trace functions.

For two nearby disjoint coassociative submanifolds $C$ and $C'$ in a $G_2$-manifold, we construct thin instantons with boundaries lying on $C$ and $C'$ from regular J-holomorphic curves in $C$. We explain their relationship with the Seiberg-Witten invariants for $C$.

We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudohermitian manifold of dimension $2m + 1 \geq 5$. We prove that the equality holds iff the manifold is equivalent to the CR sphere up to a scaling. For this purpose, we establish an Obata-type theorem in CR geometry that characterizes the CR sphere in terms of a nonzero function satisfying a certain overdetermined system. Similar results are proved in dimension 3 under an additional condition.

We prove the three embeddedness results as follows. (i) Let $\Gamma_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of $\Gamma_{2m+1}\lt 2m\pi$. In particular, the total curvature of $\Gamma_5\lt 4\pi$ and thus any minimal surface $\Sigma \subset \mathbb{R}^n$ bounded by $\Gamma_5$ is embedded. Let $\Gamma_5$ be a piecewise geodesic Jordan curve with $5$ vertices in $\mathbb{H}^n$. Then any minimal surface $\Sigma \subset \mathbb{H}^n$ bounded by $\Gamma_5$ is embedded. If $\Gamma_5$ is in a geodesic ball of radius $\frac{\pi}{4}$ in $\mathbb{S}^n_+$, then $\Sigma \subset \mathbb{S}^n_+$ is also embedded. As a consequence, $\Gamma_5$ is an unknot in $\mathbb{R}^3$, $\mathbb{H}^3$, and $\mathbb{S}^3_+$. (ii) Let $\Sigma$ be an $m$-dimensional proper minimal submanifold in $\mathbb{H}^n$ with the ideal boundary $\partial_{\infty} \Sigma = \Gamma$ in the infinite sphere $\mathbb{S}^{n-1}=\partial_\infty \mathbb{H}^n$. If the Möbius volume of $\Gamma$ $\widetilde{\rm Vol}(\Gamma) \lt 2{\rm Vol}(\mathbb{S}^{m-1})$, then $\Sigma$ is embedded. If $\widetilde{\rm Vol}(\Gamma) = 2{\rm Vol}(\mathbb{S}^{m-1})$, then $\Sigma$ is embedded unless it is a cone. (iii) Let $\Sigma$ be a proper minimal surface in $\mathbb{H}^2 \times \mathbb{R}$. If $\Sigma$ is vertically regular at infinity and has two ends, then $\Sigma$ is embedded.