Given a nondegenerate minimal hypersurface Σ in a Riemannian manifold, we prove that, for all ε small enough there exists u_ε, a critical point of the Allen-Cahn energy E_ε(u) = ε² ∫ |∇u|² + ∫(1 − u²)², whose nodal set converges to Σ as ε tends to 0. Moreover, if Σ is a volume nondegenerate constant mean curvature hypersurface, then the same conclusion holds with the function u_ε being a critical point of E_ε under some volume constraint.

We define for each $g \geq 2$ and $k \geq 0$ a set $\mathcal{M}_{g,k}$ of orientable hyperbolic 3-manifolds with $k$ toric cusps and a connected totally geodesic boundary of genus $g$. Manifolds in $\mathcal{M}_{g,k}$ have Matveev complexity $g + k$ and Heegaard genus $g+1$, and their homology, volume, and Turaev-Viro invariants depend only on $g$ and $k$. In addition, they do not contain closed essential surfaces. The cardinality of $\mathcal{M}_{g,k}$ for a fixed $k$ has growth type $g^g$.

We completely describe the non-hyperbolic Dehn fillings of each $M$ in $\mathcal{M}_{g,k}$, showing that, on any cusp of any hyperbolic manifold obtained by partially filling $M$, there are precisely 6 non-hyperbolic Dehn fillings: three contain essential discs, and the other three contain essential annuli. This gives an infinite class of large hyperbolic manifolds (in the sense of Wu) with $\partial$-reducible and annular Dehn fillings having distance 2, and allows us to prove that the corresponding upper bound found by Wu is sharp. If $M$ has one cusp only, the three $\partial$-reducible fillings are handlebodies.

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation. Optimal Liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established. The results are then applied to prove several structure theorems on complete noncompact Kähler manifolds with nonnegative bisectional or sectional curvature.