We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness theorem for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $\mathbb{H}^{n+1}$ satisfying $f(k) = \sigma \in (0, 1)$ with a prescribed asymptotic boundary $\Gamma$ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover, if $\Gamma$ is (Euclidean) star-shaped, the solution is unique and also (Euclidean) star-shaped, while if $\Gamma$ is mean convex, the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a “global interior curvature estimate.”

We show that in the fundamental groups of closed manifolds without conjugate points, centralizers of all elements virtually split.

We study the rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature-like quantities for polyhedral surfaces are introduced and are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law. They can be considered as 2-dimensional counterparts of the Schlaefli formula.

We consider the intersections of the complex nodal set $\mathcal{N}_{\lambda_{j}}^{\,\mathbb{C}}$ of the analytic continuation of an eigenfunction of $\Delta$ on a real analytic surface $(M^2, g)$ with the complexification of a geodesic $\gamma$. We prove that if the geodesic flow is ergodic and if $\gamma$ is periodic and satisfies a generic asymmetry condition, then the intersection points $\mathcal{N}_{\lambda_{j}}^{\,\mathbb{C}} \cap \gamma_{x, \xi}^{\mathbb{C}}$ condense along the real geodesic and become uniformly distributed with respect to its arc-length. We prove an analogous result for non-periodic geodesics except that the ‘origin’ $\gamma_{x, \xi}(0)$ is allowed to move with $\lambda_j$.